Victor Champonnois
email: victor.champonnois@inrae.fr

Olivier Chanel
email: olivier.chanel@univ-amu.fr.

Abul Naga

Dominique Bureau

Katrin Erdlenbruch

Charles Figuières

Emmanuel Flachaire

Bengt Kriström

Justin Leroux

Pierre-Alexandre Mahieu

Michel Lubrano

Stéphane Luchini

Jürgen Meyerhoff

Philippe Quirion

Tina Rambonilaza

Emilie Sartre

Erik Schokkaert

Hubert Stahn

Jean-Christophe Vergnaud

Accounting for subsistence needs in non-market valuation: A simple proposal 0

Keywords: subsistence needs, adjustment factor, non-market valuation, value transfer, population's preferences D60, D70, Q51

Revealed and stated preference techniques are widely used to assess willingness to pay (WTP) for non-market goods as input to public and private decision-making. However, individuals first have to satisfy subsistence needs through market good consumption, which affects their ability to pay. We provide a methodological framework and derive a simple ex post adjustment factor to account for this effect. We quantify its impacts on the WTP for non-market goods and the ranking of projects theoretically, numerically and empirically. This confirms that non-adjusted WTP tends to be plutocratic: the views of the richest -whatever they are -are more likely to impact decision-making, potentially leading to ranking reversal between projects. We also suggest that the subsistence needs-based adjustment factor we propose has a role to play in value transfer procedures. The overall goal is a better representation of the entire population's preferences with regard to non-market goods.

Introduction

Over the past 50 years, non-market goods and services -like ecological services, improved recreation, visual amenities, odours, noise, loss of biodiversity, psychological factors, or the valuation of health -have made their entry into public and private decision-making.

With no marketplace to set economic prices for these goods, their values are often assessed by methods based on revealed or stated preferences (SP). These valuation methods elicit individuals' preferences for a given non-market good, either directly through surveys (stated) or indirectly through exploiting market data (revealed). They then derive shadow prices / willingness to pay (WTP) for the corresponding welfare change, and these fuel collective choices, more comprehensive economic assessments like cost-benefit analyses (CBA), or (international) value transfer from one site to another.

The standard project-assessment criterion in CBA is the Kaldor-Hicks compensation test, an extension of the Pareto test for projects involving losers. It states that if the net present value of a project (measured by the discounted sum of individual benefits and costs) is positive, then social welfare is increased and the gainers can compensate the losers with a monetary transfer [START_REF] Johansson | Cost-Benefit Analysis[END_REF]. The seminal rationale was that CBA targets efficiency (as given by the Lindahl-Samuelson condition for public goods) and not distributional issues, which are handled by the tax-transfer system. However, arguments against this rationale rapidly emerged because there is no guarantee that the transfer between gainers and losers will actually take place once the project is implemented. Equity issues (disadvantaged individuals deserve protection, and the losers should be compensated by the gainers) and distributional issues led to the current commonly accepted view that distributional weights are required, and should correspond to the social marginal utility of income, i.e. society's valuation of the individual marginal utility of income [START_REF] Brent | Applied Cost-Benefit Analysis[END_REF][START_REF] Fleurbaey | The Use of Distributional Weights in Benefit-Cost Analysis: Insights from Welfare Economics[END_REF][START_REF] Johansson | Cost-Benefit Analysis[END_REF].

Following [START_REF] Nyborg | Project Evaluation with Democratic Decision-Making: What Does Cost-Benefit Analysis Really Measure?[END_REF], we consider that while CBA is a tool that can inform decision-makers on the efficiency of projects involving non-market goods, it cannot be used to measure social welfare effects. We hence adopt a positive rather than normative interpretation of CBA by leaving subjective choices, both on how to aggregate gainers' and losers' individual choices and on the implementation of compensation mechanisms, in the hands of the decision-maker [START_REF] Hammitt | Accounting for the Distribution of Benefits and Costs in Benefit-Cost Analysis[END_REF]. S/he will then make choices based on what s/he considers the predominant view of fairness in the population [START_REF] Sager | The Comprehensiveness Dilemma of Cost-Benefit Analysis[END_REF][START_REF] Mouter | The Politics of Cost-Benefit Analysis[END_REF], or possibly by looking at how different fairness views affect the decision [START_REF] Van Wee | How Suitable is CBA for the Ex-Ante Evaluation of Transport Projects and Policies? A Discussion from the Perspective of Ethics[END_REF]. We believe that the major value of CBA lies in providing decision-makers with full and accurate information on individuals' preferences (in particular non-market-related) rather than a definitive final ranking among projects.

Another way in which non-market valuation plays a role in public decision-making is through value transfer (see [START_REF] Johnston | Guidance to Enhance the Validity and Credibility of Environmental Benefit Transfers[END_REF]; for a recent overview). Multiple studies on different populations and different non-market goods are costly and time-consuming, and require the expertise of specialists. Consequently, in Europe and the United States, governmental and non-governmental bodies often transfer values obtained from studies realized at a study site to a policy site of interest, in order to assess the impacts of regulations or inform policy-making. According to OECD (2018), "Value transfers are the bedrock of practical policy analysis" (p. 157). However, this raises the critical issue of how best to account for income differences between the study and the policy sites, especially in international or interregional value transfers.

These two processes -CBA and value transfer -appear very transparent and consistent with mainstream welfare economics, directly feeding the preferences of the whole population into decision-making. Nevertheless, willingness to exchange or receive money for non-market good provision, while acting as an indicator of public preferences, also depends on the ability to pay [START_REF] Brent | Applied Cost-Benefit Analysis[END_REF], with the consequence that the richer are likely to more greatly influence policy decisions [START_REF] Pearce | Cost-Benefit Analysis and the Environment: Recent Developments[END_REF][START_REF] Atherton | Capturing Public Views on Complex and Unfamiliar Goods: Chemical Water Pollution in England and Wales[END_REF]. There have been proposals to take this into account, ex ante by replacing money with a different numéraire (see [START_REF] Brekke | The Numéraire Matters in Cost-Benefit Analysis[END_REF][START_REF] Kassahun | Revisiting Money and Labor for Valuing Environmental Goods and Services in Developing Countries[END_REF]; for a recent review), or ex post, by weighting the WTP (often by a function of income, [START_REF] Donaldson | Valuing the Benefits of Publicly-Provided Health Care: Does 'Ability to Pay' Preclude the Use of 'Willingness to Pay[END_REF][START_REF] Treasury | The Green Book: Central Government Guidance on Appraisal and Evaluation[END_REF] or via a structural model; [START_REF] Lee | Income and Distance-Decay effects on Willingness to Pay Estimated by the Contingent Valuation Method[END_REF], or using a social welfare function (SWF, see [START_REF] Adler | Measuring Social Welfare: An Introduction[END_REF]. However, ex ante approaches call for specific designs, while ex post approaches rely on subjective choices in the weighting process or require additional data and complex procedures rarely used in practice.

We propose an alternative to existing approaches, simple to apply ex post on any existing data and limiting subjective choices. We start by considering that although many income-based corrections try to capture inequalities, they do not allow for the fact that the irreducible costs of meeting subsistence needs (often pre-incurred) limit individuals' ability to pay. The public economic literature contains some work that takes this into account by considering the effects of subsistence requirements when measuring minimum health conditions [START_REF] Russell | Ability to Pay for Health Care: Concepts and Evidence[END_REF] or minimum levels for ecosystem services [START_REF] Baumgärtner | Subsistence, Substitutability and Sustainability in Consumption[END_REF][START_REF] Drupp | Limits to Substitution Between Ecosystem Services and Manufactured Goods and Implications for Social Discounting[END_REF]. However, to our knowledge, the impact of individual subsistence needs on WTP for non-market goods has not been explored either theoretically or empirically.

We propose to fill this gap by investigating how individual subsistence needs affect WTP for non-market goods and then the collective choices that involve them. First, we consider how subsistence needs impact individual WTP, and look for a simple ex post individual adjustment factor. Second, we introduce heterogeneous preferences towards the non-market good, and examine how subsistence needs affect both the mean WTP for a given project and the likelihood of ranking reversal between two projects with different non-market goods. Finally, we provide both numerical and empirical illustrations for various non-market good preferences.

We find that the subsistence needs issue more strongly impacts the WTP of the poorest individuals than that of the richest, and decreases the monetary assessment of a project at the aggregate level. As a consequence, when a population's preferences are heterogeneous, the views of the richest -whatever they are -are more likely to impact decision-making, a standard result that can be considered a plutocratic partiality. We show how the adjustment factor we propose tackles this issue, and can be theoretically obtained with appropriate distributional weights. We find that ranking reversal between two projects can occur, depending on the preferences of the richest and the poorest. Finally, the empirical application estimates adjustment factors on real data for two countries, using both CBA-type and value transfer-type analyses.

Our contribution is to the methodology of policy choice in presence of non-market goods. We propose a framework for assessing the impact of subsistence needs on individual WTP elicited for non-market goods, and a simple and pragmatic way to adjust ex post for this impact. This adjustment factor can be applied to existing data based solely on respondent income and country-and good-specific data. It may precede the application of distributional weights in a CBA or be part of value transfer. This adjustment factor could help decision-makers better account for the preferences of the entire population with regard to non-market goods, which is particularly relevant because poorer households are more exposed to negative environmental externalities (air or water pollution, toxic wastes, noise, natural hazards). Moreover, poor households have disproportionately lower access to positive amenities provided by the natural environment (recreational and aesthetic resources) and it is harder for the poor to combat negative environmental externalities through costly behaviours, like prevention, avoidance or moving away.

The remainder of the paper proceeds as follows. Section 2 briefly sketches the existing approaches to the income-related equity issue relevant to non-market valuation, and introduces subsistence needs. Section 3 describes the relevant models and proposes a simple adjustment factor. Section 4 analyses the impacts of subsistence needs on the monetary assessment of non-market goods: a lower valuation, a plutocratic partiality and potential ranking reversal. Section 5 presents an empirical illustration from Canadian and French income data. The discussion and conclusion are in Section 6.

2. Accounting for the income-related equity issue: a literature review [START_REF] Pearce | Cost-Benefit Analysis and the Environment: Recent Developments[END_REF] remarked that WTP "is heavily influenced by ability to pay (income, wealth). The result is a cost-benefit rule for sanctioning or rejecting projects or policies that is biased in favour of those with higher incomes, raising issues of distributional fairness" (p. 31). Several approaches may help decision-makers tackle the ability to pay issue and better account for the non-market good preferences of the whole population.

Ability to pay

One strand of research attempts to capture inequalities in ability to pay through theoretical or empirical income-based corrections.

Instead of income, the use of discretionary income (i.e. the income above a level of committed expenditure on market goods) has been proposed in non-market valuation modelling. However, this is limited to theoretical models: [START_REF] Mcfadden | Issues in the Contingent Valuation of Environmental Goods: Methodologies for Data Collection and Analysis[END_REF] use discretionary income up to the compensating variation function, subsequently explicitly using income in the remainder of their paper, and [START_REF] Carson | The Discrete Choice Experiment Approach to Environmental Contingent Valuation[END_REF] quickly abandon discretionary income on the grounds that it carries considerable measurement error, which is likely to bias the coefficient estimates. The only noticeable exception we found is an empirical study exploring the effects of a mental accounts-type treatment. Li et al. (2005a,b) use additional reminders about the discretionary income of respondents and its use for environmental causes, and find significant budget constraint effects, although the joint additional reminders make disentangling the income effect difficult.

From an empirical perspective, a few authors tackle the ability-to-pay issue through ex ante or ex post methods. [START_REF] Donaldson | Valuing the Benefits of Publicly-Provided Health Care: Does 'Ability to Pay' Preclude the Use of 'Willingness to Pay[END_REF], for instance, proposes an ex post weighting of individual WTP defined as the ratio of the sample average of incomes to individual income at a given power, the elasticity of the marginal utility of income. This is assumed to help account for inequality aversion. Although the ranking did not change under no or different weightings, he concludes that WTP reflects not only strength of preferences but also ability to pay, which is problematic when preferences are correlated with income.

In SP surveys, [START_REF] Chanel | Décision en Présence d'Incertitude et d' Émotions Face à des Risques de Catastrophes Naturelles (RISKEMOTION)[END_REF] use an ex ante approach with normalised scenarios involving a given (hypothetical) income assumed equal for everyone, and compare them to standard scenarios. They exploit the differences to adjust WTP, which leads to a 20% increase in standard WTP on average. [START_REF] Shono | Willingness to Pay for Public Health Services in Rural Central Java, Indonesia: Methodological Considerations When Using the Contingent Valuation Method[END_REF] explore whether individuals are actually able to pay the amount they declare. They use an ex ante approach to assess two WTPs: "standard" WTP as well as the maximum price respondents can pay without resorting to debt (referred to as "payable"). In their empirical study on three health services, they find a higher demand curve for standard WTP than for payable WTP (from 15.4% to 70.8%). In addition, the gap is negatively affected by income and larger for the poorest for one of these health services. [START_REF] Breffle | Understanding How Income Influences Willingness to Pay for Joint Programs: A More Equitable Value Measure for the Less Wealthy[END_REF] collect WTP ex ante through a specific design and derive an equity adjustment factor, applied ex post. They consider three programs, two independent and one combining the two others, and establish that income effects should make the WTP for the combined program lower than the sum of WTP for the two others. Because this income effect is expected to be larger for the poorest than for the richest, they exploit this difference on the combined program to build a partially equity-adjusted WTP. When applied to the combined program of the poorest, it leads to a WTP increase of about 13%.

Overall, the above approaches tackle the ability-to-pay issue either from a theoretical perspective, using an ex ante specific design with non-monetary contributions or purposely-collected WTP, or through subjective ex post corrections without consensus.

Distributional weights

A second strand of research, frequent in the normative interpretation of CBA, uses ex post distributional weights. It computes the change in social welfare associated with a project (potentially including non-market goods) by applying distributional weights, representing the social marginal utility of income to individual WTP (see [START_REF] Johansson | Cost-Benefit Analysis[END_REF][START_REF] Fleurbaey | Equivalent Income And Fair Evaluation Of Health Care[END_REF]. According to [START_REF] Fleurbaey | Equivalent Income And Fair Evaluation Of Health Care[END_REF], there are two main approaches available to estimate distributional weights: welfarist and non-welfarist.

The welfarist approach estimates the two components of the social marginal utility of income separately: the individual's marginal utility of income, and the weight attributed to every individual in the SWF. Whereas the former only depends on the characteristics of the individual utility function, the latter depends in addition on the choice of SWF, with a parameter that stands for aversion to inequality. Among the many, generally non-independent, sources of inequalities in a population, income and (to a lesser extent) health status are those most frequently taken into account. To avoid choosing an SWF, practitioners sometimes directly specify the social marginal utility of income, generally a function of income alone [START_REF] Donaldson | Valuing the Benefits of Publicly-Provided Health Care: Does 'Ability to Pay' Preclude the Use of 'Willingness to Pay[END_REF][START_REF] Samson | Fairness in Cost-Benefit Analysis: A Methodology for Health Technology Assessment[END_REF][START_REF] Treasury | The Green Book: Central Government Guidance on Appraisal and Evaluation[END_REF]. However, there is no consensus on how to set its parameters [START_REF] Brent | Applied Cost-Benefit Analysis[END_REF][START_REF] Johansson | Cost-Benefit Analysis[END_REF].

The non-welfarist approach directly estimates the social marginal utility of income in monetary terms, which requires tricky interpersonal comparison of utilities [START_REF] Adler | Benefit-Cost Analysis and Distributional Weights: An Overview[END_REF][START_REF] Fleurbaey | The Use of Distributional Weights in Benefit-Cost Analysis: Insights from Welfare Economics[END_REF]. This can be achieved through the equivalent income approach based on the fairness principle: survey respondents are asked to give their preferences on different combinations of resources [START_REF] Fleurbaey | Equivalent Income And Fair Evaluation Of Health Care[END_REF][START_REF] Samson | Fairness in Cost-Benefit Analysis: A Methodology for Health Technology Assessment[END_REF]. The subjective well-being approach can also be used to compare individuals' emotions, feelings or life satisfaction and derive distributional weights (see [START_REF] Fleurbaey | The Use of Distributional Weights in Benefit-Cost Analysis: Insights from Welfare Economics[END_REF][START_REF] Bronsteen | Well-Being Analysis vs. Cost-Benefit Analysis[END_REF]; for a discussion). Although non-welfarist approaches are truly based on individual preferences, decision-makers almost never implement them in practice due to their complexity and the additional data required: they are good-and country-dependent, and require specific surveys.

Overall, the above approaches tackle the income-related equity issue by requiring specific surveys to directly elicit individual preferences, or explicit subjective judgments on the interpersonal comparison of utilities. As pointed out by [START_REF] Hammitt | Accounting for the Distribution of Benefits and Costs in Benefit-Cost Analysis[END_REF], all approaches "depend on normative choices that are fundamentally equivalent judgments about how to evaluate improvements to some people and harms to others" (p. 16). We propose below an approach that offers an ex post alternative to account for the impact of differences in income on non-market preference revelation, simple to implement, and that relies on subsistence needs.

Subsistence needs

The concept of subsistence needs is based on the works of the psychologist [START_REF] Maslow | A Theory of Human Motivation[END_REF] and the economist [START_REF] Georgescu-Roegen | Choice, Expectations and Measurability[END_REF]. It relies on the following hierarchy of needs (or wants), from the bottom upwards: physiological, safety, love and belonging, esteem, and self-actualisation. [START_REF] Strasnick | Neo-Utilitarian Ethics and the Ordinal Representation Assumption[END_REF] noted two interesting points based on Georgescu-Roegen (1954)'s approach. First, because each of the needs must be fulfilled before considering satisfying the next one, this "subordination of wants and satiable wants" leads to the principle of diminishing marginal utility. Second, the "irreducibility of wants" limits substitutability not only between needs, but also between goods for a given type of need [START_REF] Stern | Limits to Substitution and Irreversibility in Production and Consumption: A Neoclassical Interpretation of Ecological Economics[END_REF].

Drawing on this framework, we consider that individuals start by satisfying their basic needs (referred to as subsistence needs) thanks to the minimum expenditure on food, water, energy, housing, clothes, transportation, etc. required to live in a given country.

It is only when these subsistence needs are satisfied that individuals can consider more secondary needs (including non-market goods) by moving up in the hierarchy. This minimum level of subsistence needs is country-dependent, due to differences both in standard of living and in cultural construction [START_REF] Sen | The Standard of Living[END_REF]. It is thus related to the recent concept of societal poverty (World Bank, 2018) accounting for the fact that needs increase with the wealth of a society to allow participation in that society with dignity. It also speaks to the long-standing debate on the impact of universal provision of subsistence (or basic) goods [START_REF] Reinert | No Small Hope: Towards the Universal Provision of Basic Goods[END_REF], for instance through universal basic income [START_REF] Van Parijs | Basic Income: A Radical Proposal, for a Free Society and a Sane Economy[END_REF], currently revitalised by the COVID-19 pandemic.

Consequently, when individuals state their WTP for non-market goods, in revealed or SP methods, they rely on substitutability between market goods and non-market goods depending on their preferences, but consider that there is a minimum level of market goods corresponding to the fulfillment of subsistence needs. This differs from the approaches based on discretionary income or ability to pay, which already embed preferences for market goods leading to various levels of consumption.

As far as we know, no study to date has explored how individual subsistence needs impact WTP for non-market goods. Yet economists have long applied the concept of subsistence requirements, in ecological economics for natural resources [START_REF] Shyamsundar | Tropical Forest Protection: An Empirical Analysis of the Costs Borne by Local People[END_REF][START_REF] Thompson | Costs of Lost Opportunities: Applying Non-Market Valuation Techniques to Potential REDD+ Participants in Cameroon[END_REF][START_REF] Shrestha | Estimating the Local Cost of Protecting Koshi Tappu Wildlife Reserve, Nepal: A Contingent Valuation Approach[END_REF] or ecosystem services [START_REF] Baumgärtner | Subsistence, Substitutability and Sustainability in Consumption[END_REF][START_REF] Drupp | Limits to Substitution Between Ecosystem Services and Manufactured Goods and Implications for Social Discounting[END_REF]. In non-market valuation, subsistence requirements have been applied to lexicographic preferences between market and non-market goods [START_REF] Stevens | Measuring the Existence Value of Wildlife: What Do CVM Estimates Really Show[END_REF][START_REF] Veisten | Lexicographic Preference in Biodiversity Valuation: Tests of Inconsistencies and Willingness to Pay[END_REF][START_REF] Rekola | Lexicographic Preferences in Contingent Valuation: A Theoretical Framework with Illustrations[END_REF] or used to discuss the limits to substitution between them (cf. a thorough albeit old presentation in [START_REF] Stern | Limits to Substitution and Irreversibility in Production and Consumption: A Neoclassical Interpretation of Ecological Economics[END_REF] Let us consider that this preference relation is represented by a utility function u i (x i , q i,j ), where x i represents the quantity of a composite market good and q i,j the quantity of non-market goods, j = 1, ..., J, associated with a project. u i (x i , q i,j ) is assumed to be twice continuously differentiable, strictly increasing and strictly quasi-concave in x i and q i,j , with x i , q i,j ≥ 0. The individual budget constraint is px i = y i , i.e. all his/her income y i is spent on the composite market good at price p,1 and q i,j is considered exogenous to the individual (see [START_REF] Horowitz | Behavioral Foundations of Environmental Economics and Valuation[END_REF][START_REF] Dupoux | Can the Environment be an Inferior Good? A Theory with Context-Dependent Substitutability and Needs[END_REF].

Models

We are interested in the relation between the WTP for the non-market goods and the individual's income. We follow the main trend in the literature on non-market valuation [START_REF] Hanemann | Willingness to Pay and Willingness to Accept: How Much Can They Differ?[END_REF][START_REF] Lankford | Measuring Welfare Changes in Settings with Imposed Quantities[END_REF][START_REF] Ebert | Environmental Goods and the Distribution of Income[END_REF] by defining the marginal WTP as the Lindahl or virtual price for q i,j : W T P i,j (p, q i,j , y i ) = ∂v i (p, q i,j , y i )/∂q i,j ∂v i (p, q i,j , y i )/∂y i (1)

where v i (p, q i,j , y i ) is the indirect utility function obtained from the standard maximization problem. W T P i,j represents the change in income that compensates for the change in the non-market good j. It only depends on the individual's preferences represented by the parameters characterising v i (.), quantities of non-market goods q i,j , p and income y i .

Since we consider that an individual spends all his/her income on market goods, we set

x i ≡ y i /p and p ≡ 1 (without loss of generality in a static framework) in the following.

Thus, composite market good x i can be seen as the numéraire representing the individual's income.

Standard model under a CES utility function

To determine the explicit relationships between the standard and the subsistence needs frameworks, we need to start from a functional form that is as flexible as possible regarding preferences. A relevant, easy to interpret, tractable and well-known specification of the utility function in consumer theory is the CES function first proposed by [START_REF] Arrow | Capital-Labor Substitution and Economic Efficiency[END_REF] and extensively used since then in environmental economics (see for instance [START_REF] Ebert | Environmental Goods and the Distribution of Income[END_REF][START_REF] Sterner | An Even Sterner Review: Introducing Relative Prices into the Discounting Debate[END_REF][START_REF] Baumgärtner | Subsistence, Substitutability and Sustainability in Consumption[END_REF][START_REF] Drupp | Limits to Substitution Between Ecosystem Services and Manufactured Goods and Implications for Social Discounting[END_REF][START_REF] Dupoux | Can the Environment be an Inferior Good? A Theory with Context-Dependent Substitutability and Needs[END_REF][START_REF] Meya | Environmental Inequality and Economic Valuation[END_REF]. For J + 1 goods and a given individual i it is:

u i (x i,k ) = [ k α i,k x θ i i,k ] 1 θ i for θ i ∈]-∞; 1] ; 0 < α i,k < 1 ; k α i,k ≡ 1 ; k = 0, ..., J; ∀i (2)
where x i,k is the quantity of good k, α i,k its share parameter and 1/(1 -θ i ) the elasticity of substitution. Note that the CES function covers a range from perfect complement (θ i → -∞) to perfect substitute (θ i → 1), as well as the Cobb Douglas function (θ i → 0).2

WTP are defined as in Eq.( 1), but based on u i (x i,k ) = u i (x i , q i,j ), where

x i ≡ x i,0
stands for the quantity of composite good / income, and q i,j ≡ x i,j , j = 1, ..., J stands for the quantities of non-market goods.

Model with subsistence needs under a CES function

Let us consider that individuals face minimum subsistence needs defined by level of consumption x s of the composite good. Below the minimum subsistence needs, nobody is willing to trade the composite good for non-market goods, because basic needs must be fulfilled first. This implies that W T P i,j is set at 0. Above the minimum subsistence needs, W T P i,j is defined as in the standard CES utility model but based on the u s i (x i -x s , q i,j ) function that accounts for these subsistence needs.3 3.2. Defining the WTPs under a CES utility function

Case of one non-market good (J=1)

The preferences over x i and q i,1 are expressed in a CES as follows:

u i (x i , q i,1 ) = [α i x θ i i + (1 -α i )q θ i i,1 ] 1 θ i for θ i ∈] -∞; 1] ; 0 < α i < 1; ∀i (3)
where α i represents the preference for market good x i relative to the preference for non-market good q i,1 .

At equilibrium, the WTP for the non-market good in the standard CES model is (from Eq. ( 1) and Eq. ( 3)):

W T P i,1 = ∂u i (.)/∂q i,1 ∂u i (.)/∂x i = (1 -α i )q θ i -1 i,1 α i x θ i -1 i (4)
We extend the CES function to account for minimum subsistence needs as follows:

u s i (x i -x s , q i,1 ) = [α i (x i -x s ) θ i + (1 -α i )q θ i i,1 ] 1 θ i for x i > x s ; θ i ∈] -∞; 1]; 0 < α i < 1; ∀i (5)
The WTP for q i,1 in the CES model with subsistence needs is (from Eq.( 1) and Eq.( 5)):

W T P s i,1 = ∂u s i (.)/∂q i,1 ∂u s i (.)/∂x i = (1 -α i )q θ i -1 i,1 α i (x i -x s ) θ i -1 (6)
It is clear that observed W T P s i,1 , when individuals face minimum subsistence needs, are always lower than W T P i,1 , except when x s = 0, in which case W T P s i,1 ≡ W T P i,1 .

Case of two non-market goods (J=2)

We extend the individual CES function in Eq. ( 3) to the case of two non-market goods, with quantity q i,1 and q i,2 :

u i (x i , q i,1 , q i,2 ) = α i,0 x θ i + α i,1 q θ i i,1 + α i,2 q θ i i,2 1/θ i (7)
with α i,0 the preference for the composite market good, α i,1 and α i,2 respectively the preferences for the first and second non-market goods, and α i,0 + α i,1 + α i,2 ≡ 1 ∀i. The WTPs for the two non-market goods are respectively:

W T P i,1 = ∂u i (.)/∂q i,1 ∂u i (.)/∂x i = α i,1 α i,0 q i,1 x i θ i -1 and W T P i,2 = ∂u i (.)/∂q i,2 ∂u i (.)/∂x i = α i,2 α i,0 q i,2 x i θ i -1 (8)
Extending the CES function similarly to Eq. ( 5) leads to:

W T P s i,1 = α i,1 α i,0 q i,1 x i -x s θ i -1 and W T P s i,2 = α i,2 α i,0 q i,2 x i -x s θ i -1 (9)
As with one non-market good, observed W T P s i,. , when facing minimum subsistence needs, are always lower than W T P i,. , except when x s = 0.

Deriving a subsistence-needs adjustment factor

What we are looking for is the (unobserved) WTP that an individual endowed with the same preferences and the same income would have if her ability to pay equalled her entire income. That is, the practical individual subsistence-needs adjustment factor Z i that makes the observed W T P s i (when facing minimum subsistence needs) equal to W T P i (i.e., Z i W T P s i = W T P i ):

Z i = W T P i W T P s i = (1-α i )q θ i -1 α i x θ i -1 i (1-α i )q θ i -1 α i (x i -xs) θ i -1 = x i -x s x i θ i -1 (10)
Note that Z i does not depend on preferences for the market good α i and is proportionally larger for those who have a smaller income (due to the homothety of the CES). In the case of two non-market goods, it is clear from Eq. ( 8) and ( 9) that Z i is also defined as above for each non-market good.

Z i relies on a country-specific minimum level of subsistence needs and a very flexible utility function, which are indeed normative choices, but is characterized by parameters θ (a non-market-good-specific substitutability) and x s , exogenous to both the individual and the decision-maker. It also relies on a country-specific distribution of income. Consequently, in a value transfer procedure, transferring mean observed WTPs from a study site (indexed by S) to a policy site (indexed by P) may account for differences between the sites, as follows:

E(W T P s,P ) = E(W T P s,S ) E(x P -x P s ) E(x S -x S s ) η ( 11
)
where η is the income elasticity of WTP, which, in a CES, is equal to 1-θ (cf. [START_REF] Meya | Testing Structural Benefit Transfer: The Role of Income Inequality[END_REF].

Overall, we propose a simple and practical way to account for the impact of differences in ability to pay on WTP elicitation. This adjustment factor Z i is only meant to account for the impact of subsistence needs on non-market valuation and is to be separated from the approaches sometimes used under a normative interpretation of CBA to account for health, economic or social inequalities. Consequently, it does not preclude additional equity-based adjustments using distributional weights or an SWF.4

Impacts of subsistence needs on decision-making

We explore how the subsistence needs issue impacts the WTPs of non-market goods (defined in section 3.2) as well as the outcomes of decision-making when income changes.

The closer Z i is to 1, the smaller the adjustment required to account for subsistence needs. We consider first how the average WTP assessed for a given project is affected when preferences are homogeneous over individuals (i.e. same α) and characterise the plutocratic partiality when they are heterogeneous between income groups (i.e. different α). Then we consider how relative project rankings are affected when preferences are heterogeneous.

Homogeneous preferences for one non-market good

When preferences are homogeneous over the population, accounting for minimum subsistence level leads to increased WTP. The size of this increase will depend on the actual distribution of income in the population (see applications to Canada and France in section 5)

Figure 1 illustrates the magnitude of adjustment factor Z i for different combinations of θ and income (represented in terms of x/x s ). It can be seen that the lower the income, the higher Z i for any given value of θ, due to the fact that W T P s i tends towards zero when income tends towards x s . For an income seven times larger than x s , for instance, the adjustment factor is about 1.26 (i.e. a 26% increase in WTP) when θ exhibits complementarity (θ = -0.5), but only about 1.08 for substitutability (θ = 0.5). This means that, at the aggregate level, the monetary assessment of projects including this non-market good will be increased ceteris paribus when adjustment factor Z i is applied to account for subsistence needs.

Heterogeneous preferences for one non-market good

When preferences regarding the non-market good are heterogeneous (i.e. vary over income groups), accounting for subsistence needs also increases the WTP, but the adjustment factor Z i has a greater upward impact on WTP for the poorest fraction of the population than for the richest. This implies that without adjustment, the preferences of the richest will be better represented than those of the poorest whatever they are, a kind of plutocratic partiality.

As an illustration, imagine a population with a bi-modal income distribution (for the sake of simplicity): M individuals have a low income (x L = 1.5 times subsistence needs), and N individuals have a high income (x H = 10 times subsistence needs). 5 Let us consider that the preferences of the two subpopulations regarding the non-market good are respectively measured by α L for the poorest and α H for the richest, varying by stepsize We compute and represent [E(W T P ) -E(W T P s )]/E(W T P s ), the relative change in mean WTP when subsistence needs are accounted for, for three values of substitutability (θ = 0.5, 0, -0.5) in Figure 2, for M = N . The closer to 0, the lower the impact of subsistence needs on the population's non-market monetary assessment.

Whatever the substitutability value θ, we have the following results. For homogeneous preferences (represented by the diagonal black line segment on the three figures), the adjustment factor is clearly constant, with a relative change of about 20%. Provided the preferences of the richest are more non-market oriented than those of the poorest (to the left of the diagonal), the relative change is smaller than in the homogeneous case. The preferences of the richest (for the non-market good) are favoured, because the WTP for the non-market good requires less adjustment to account for subsistence needs. Provided the preferences of the richest are less non-market oriented than those of the poorest (to the right of the diagonal), the relative change is larger than in the homogeneous case. The preferences of the richest (now for the market good) are favoured again, because more adjustment is required. Note that when the number of the poorest differs from the number of the richest, the previous results still hold, the average adjustment factor being higher when M > N and lower when N > M , ceteris paribus.

Overall, whatever the preferences of the richest, they are always better represented in the monetary assessment of a project involving a non-market good than those of the poorest, a kind of plutocratic partiality. In particular, the non-market preferences of the poorest appear never to be properly accounted for unless shared by the richest (i.e. the homogenous case), whereas the non-market preferences of the richest are unfailingly better accounted for, for two reasons. First, their WTPs are higher. Second, their WTPs are also less affected by the subsistence needs: the parts of Figure 2 to the left of the diagonal are always closer to 0 than those to the right. This second fact explains why applying the adjustment factor Z i accounts for the plutocratic partiality.

Heterogeneous preferences for two non-market goods

We will now consider how the issue of subsistence needs may affect the ranking of two projects with different (levels of) non-market goods. If preferences are homogeneous among income groups, there cannot be ranking reversal by definition, because everyone will prefer the same project. However, in the heterogeneous case, there may be ranking reversal between projects depending on whether or not the monetary assessment includes the subsistence-needs adjustment factor. We consider two projects (A and B)6 and introduce heterogeneity in preferences for non-market goods as before, by considering a population composed of N individuals with a high income (with preferences α H,A and α H,B ) and M individuals with a low income (α L,A and α L,B , with

α 0 + α H,A + α H,B ≡ 1 and α 0 + α L,A + α L,B ≡ 1).
Ranking reversal only occurs if the following condition is fulfilled (see proof in Appendix B):

1-α 0 - N M x H -x s x L -x s 1-θ (1-α 0 -2α H,A ) > 2α L,B > 1-α 0 - N M x H x L 1-θ (1-α 0 -2α H,A )
Figure 3 (and C1 and C2 in Appendix C) represents this condition in terms of α H,A and α L,B for three values of θ, and as a function of α 0 . For convenience and ease of comparison with previous results, we set as previously x L at 1.5 times subsistence needs, and x H at 10 areas of the poorest individuals' preferred project grow and cross the 45 • line: the number effect makes it more likely that their preferred project will be chosen, although the subsistence-needs effect still favours the richest. There is still a ranking reversal area, which grows.

Finally, when there are three times more high incomes than low incomes (N = 3M , Figure 3(c)), the number effect reinforces the dominance of the richest individuals' preferences so that there is almost no room for the preferences of the poorest, and ranking reversal areas, albeit limited, are still observed.

The cases where θ = -0.5 (complementarity) and θ = 0 are shown in Appendix C. All previous results hold. A decrease in θ -the substitutability between the composite market good and the non-market goods -leads to decreased likelihood of the poorest individuals' preferences dominating, as well as to smaller ranking reversal areas. The preferences of the richest become increasingly likely to dominate as substitutability decreases and/or their proportion in the population increases.

Overall, we find that the subsistence-needs issue when a population's preferences are heterogeneous creates the conditions for ranking reversal between two projects, always with a bias towards the richest individuals' preferences, whatever they are.

Empirical illustrations

Data

We use real data to examine the extent to which the subsistence needs issue might affect the monetary assessment of non-market goods and value transfer procedures. We rely both on country-specific data for income and the cost of meeting subsistence needs (in Canada and France) expressed in 2016 euros, and on empirical studies eliciting elasticities of substitution for various non-market goods.

Regarding the distribution of income, we compute the distribution of annual (disposable) income by unit of consumption 7 for Canada (Statistics Canada, 2019) and France (Institut National de la Statistique et des Etudes Economiques, 2018), see the first part 7 We use the OECD-modified scale to calculate equivalised disposable income per unit of consumption.

This scale assigns a value of 1 to the household head, of 0.5 to each additional adult member and of 0.3 to each child below 14.

of Table 1. The average income by unit of consumption unit is e27,307 for Canada and e25,030 for France. We then have to set the cost of meeting subsistence needs x s . For Canada, we rely on income supports covering basic expenses for individuals and families in the province of Alberta (AlbertaWorks, 2016), including food, clothing, household and personal needs, transportation and private housing expenses. The monthly core benefits for one single adult expected to work, living in private housing, was $CA627 in 2016 (e439). A French survey estimates the cost of subsistence needs to be about e600 per month for one adult in 2016 (Carrefour des Solidarités, 2011), which is comparable to the active solidarity revenue (RSA) paid by the French government to a single individual with no resources (e545 per month in 2017, in addition to aid for housing, public transport and a free health care scheme). We therefore adopt as a reasonable benchmark an annual cost of meeting subsistence needs per individual of e439x12 = e5,268 for Canada, and e600x12 = e7,200 for France. The distributions of annual income by unit of consumption are given in the first part of Table 1.

Regarding the substitutability of market goods for non-market goods, we merge data from [START_REF] Drupp | Limits to Substitution Between Ecosystem Services and Manufactured Goods and Implications for Social Discounting[END_REF] and [START_REF] Drupp | Relative Prices and Climate Policy: How the Scarcity of Nonmarket Goods Drives Policy Evaluation[END_REF], who gather θ estimates for various environmental non-market goods (air or water quality improvements, forest or marine services, landscape or recreational amenities, biodiversity, etc.) in different countries.

We obtain 28 values in the range [-.42; .90], with a mean empirical estimate of θ = .48 and a standard deviation of .34. This means that, on average, individuals exhibit substitutability between market goods and non-market goods. According to our results in section 4, the subsistence-needs adjustment factor would be lower ceteris paribus than for complementarity, although there would be room for ranking reversal between projects.

Results in Table 1 are presented for three values of θ: mean (0.48), and -/+ 1 standard deviation around the mean, i.e. .14 and .82.

Comparing monetary assessments of a non-market good

We assume that preferences regarding the non-market good are homogeneous over income groups, since we lack relevant data on the population's actual distribution of preferences for the non-market good w.r.t. income. The second part of Table 1 represents the subsistence-needs adjustment factor Z by country and values of θ, and for various levels of income (remember from section 3, that Z does not depend on α in the homogeneous case).

Within a given country, the adjustment factor decreases with income and decreases with the substitutability parameter θ, as expected. Between countries, we find larger adjustment factors for France than for Canada at any given income level, due to the difference In rows (4-5), W T P s are based on Eq. ( 6), mid-values of income classes, q=1 and α=.5. In row (6), results on changes in mean W T P s do not depend on q nor on α. In rows (7-9), the policy site is by convention the other country. In row (7), the true value transfer factors are computed as the ratio of the mean W T P s for the policy site over the mean for the study site. In row (8), the value transfer factors are based on Eq. ( 12), and in row (9), on Eq. ( 11).

in the cost of meeting subsistence needs. We then compute the mean income-weighted WTP over the population without and with subsistence needs adjustment in rows (4) and

(5) as well as the corresponding relative change in row (6). When θ = .48 (respectively .14 and .82), the change in mean WTP is found to be 14.9% (respectively 6.1% and 20.8%) for Canada, and larger for France: 22.3% (respectively 8.3% and 33.5%). This is due to the additional effect of country differences in income distribution.

Comparing value transfer procedures

The subsistence needs-based adjustment factor we propose may have a role to play in value transfer procedures. We start from the commonly used formula for estimating the value transfer factor, which relies on the mean observed WTP from the study site, the mean income levels at both sites and the income elasticity of WTP (see [START_REF] Czajkowski | Choosing a Functional Form for an International Benefit Transfer: Evidence from a Nine-Country Valuation Experiment[END_REF][START_REF] Meya | Testing Structural Benefit Transfer: The Role of Income Inequality[END_REF]:

E(W T P s,P ) = E(W T P s,S ) E(x P ) E(x S ) η ( 12
)
where η is the income elasticity of WTP and equal to 1 -θ.

In Table 1 row (7), we compute the true ratio of study site observed WTPs to those of the other (policy) site (the latter being computed thanks to Eq. ( 6) but unobserved during real value transfers). These values are higher (respectively lower) than 1 when the study site has a lower (respectively higher) mean income level than the policy site, and approach 1 when θ increases. In row (8), we compute the corresponding factor based on Eq (12)

and observed WTP when θ = .48 (respectively .14 and .82). We obtain .956 (respectively .984 and 928) when the transfer is from Canada to France, and 1.046 (respectively 1.016 and 1.078) when the transfer is from France to Canada. We can check that these factors are reasonably close to the figures in row ( 7), but bias towards 1.

Finally, in row (9), we compute the value transfer factors based on Eq. ( 11), i.e.

accounting for the cost of meeting subsistence needs at the study and the policy sites.

These factors differ from the commonly-used factors, but are much closer to the figures in row (8). Note that accounting for the subsistence-needs issue in value transfer can be done ex post, based on the Z distribution in the policy site.

Overall, we find that the value transfer factor accounting for subsistence needs performs better in predicting observed WTP at the policy site than the commonly-used factor, while requiring no additional data other than the cost of meeting subsistence needs in both sites.

Discussion

The monetary assessment of a project containing a non-market good is always driven by the preferences of the richest unless income-related equity is allowed for. This is true not only in absolute terms, but also in relative terms, when the population's preferences differ.

Our findings show that incorporating a subsistence-needs adjustment factor is a simple and practical way of better accounting for the preferences of the poorest, without precluding the additional use of distributional weighting to tackle the equity and distributional issues.

In many economic analyses, differences in the non-market values chosen to assess welfare change may lead to divergent conclusions (see for instance [START_REF] Sterner | An Even Sterner Review: Introducing Relative Prices into the Discounting Debate[END_REF]; on the economic impact of climate change).

Our work is subject to limitations. First, for the sake of interpretation and tractability, we chose to derive our subsistence-needs adjustment factor from one specific utility function. The CES was the best candidate: an absolutely standard and flexible function widely used in consumer theory that fits very different patterns of preferences. Second, what we propose is a way to better account for the non-market preferences of the whole population when comparing several projects under a positive interpretation of CBA.

However, we lack a definitive normative justification for our adjustment factor, like any other adjustment method [START_REF] Hammitt | Accounting for the Distribution of Benefits and Costs in Benefit-Cost Analysis[END_REF]. All we can claim is that ours has the advantage of being exogenously based on country-specific and good-specific parameters, and of being applicable ex post to data already collected. Finally, within CBA, we are aware that the question of whether a given project will indeed pass a cost-benefit test once implemented applies to any adjustment factor. Actually, if the project is to be funded through individual contributions, there is no guarantee that the poorest will be able to contribute as much as they indicate through WTP, regardless of how it is adjusted.

Our findings have implications for empirical research in economics, in addition to raising policy issues.

From an empirical perspective, our findings challenge the use of revealed and SP techniques in CBA involving preferences for a non-market good and measured through WTP. When these preferences are homogeneous, the average preference of the population is properly accounted for. When these preferences depend on income group (or any other socio-demographic characteristics correlated with income, like education, place of residence or age), the preferences of the poorest individuals may be under-represented and those of the richest over-represented. Decision-makers should be aware of the plutocratic partiality issue and the possibility of project ranking reversal. The simple ex post adjustment factor we propose offers an initial guide adaptable to country-specific minimum subsistence needs and to any substitutability value of non-market goods.

Our findings also contribute to the body of research on international value transfer.

Recent evidence suggests that transferring values through simple income adjustment [START_REF] Czajkowski | Choosing a Functional Form for an International Benefit Transfer: Evidence from a Nine-Country Valuation Experiment[END_REF][START_REF] Johnston | Guidance to Enhance the Validity and Credibility of Environmental Benefit Transfers[END_REF][START_REF] Meya | Testing Structural Benefit Transfer: The Role of Income Inequality[END_REF] or income inequality adjustment (Baumgärtner et al., 2017a;[START_REF] Meya | Testing Structural Benefit Transfer: The Role of Income Inequality[END_REF]) may be as efficient as more complex adjustment through benefit functions. The adjustment factor we propose constitutes a utility-based alternative approach that accounts for country differences in both income and levels of subsistence needs, and whose performance w.r.t. existing value / function transfers deserves investigation. So too does its potential to account for differences in the type of non-market good that is the object of the value transfer (through different values for elasticity of substitution / income elasticity of WTP).

In terms of policy implications, our findings contribute to the equity vs. equality debate around whether WTP elicited from the poorest should be adjusted when preferences for a non-market good are known to depend on income groups. Doing so means choosing equity (an as-fair-as-possible representation of preferences) over equality (a common representation of preferences). How would public priorities be affected (see [START_REF] Olsen | Helicopters, Hearts and Hips: Using Willingness to Pay to Set Priorities for Public Sector Health Care Programmes[END_REF]; for a discussion regarding health, where priorities depend on income or age)?

What impact would a change in income inequality have on the assessment of a project? Baumgärtner et al. (2017a) and [START_REF] Meya | Inequality and the Value of Public Natural Capital[END_REF] show that, depending on whether the non-market good is a substitute or a complement for manufactured goods, increased inequality can either decrease or increase mean WTP. Once non-market values are involved in decision-making, we need to be on the lookout for any potential plutocratic partiality or ranking reversal effect that may not be accounted for ex post by distributional weightings.

1 2 1 -α 0 - N M x H -x s x L -x s 1-θ (1 -α 0 -2α H,A ) > α L,B
When ranking is based on adjusted W T P , and by the same reasoning, condition (A2) leads to (from Eq. ( 8)):

1 2 1 -α 0 - N M x H x L 1-θ (1 -α 0 -2α H,A ) < α L,B
Consequently, conditions (A1) and (A2) taken together implies that:

1 2 1 -α 0 - N M x H -x s x L -x s 1-θ (1 -α 0 -2α H,A ) > α L,B > 1 2 1 -α 0 - N M x H x L 1-θ (1 -α 0 -2α H,A )
hence the ranking reversal condition:

1-α 0 - N M x H -x s x L -x s 1-θ (1-α 0 -2α H,A ) > 2α L,B > 1-α 0 - N M x H x L 1-θ (1-α 0 -2α H,A ) AII] Assume that α H,A > α H,B . AII-a] Assume that α L,A ≥ α L,B .
Ranking reversal is meaningless, as both income groups prefer project A.

A-IIb] Assume that α L,A < α L,B

By the same reasoning as in Ib], condition (A1) leads to:

1 2 1 -α 0 - N M x H -x s x L -x s 1-θ (1 -α 0 -2α H,A ) < α L,B
and condition (A2) leads to:

1 2 1 -α 0 - N M x H x L 1-θ (1 -α 0 -2α H,A ) > α L,B
Consequently, we have:

1 2 1 -α 0 - N M x H -x s x L -x s 1-θ (1 -α 0 -2α H,A ) < α L,B < 1 2 1 -α 0 - N M x H x L 1-θ (1 -α 0 -2α H,A )
This condition cannot be met since (1 -α 0 -2α H,A ) < 0 and x H -xs

x L -xs 1-θ > x H x L 1-θ
∀x H , x L , x s , θ in their respective previously-defined domains of definition.

  Below, we detail the standard model and the model incorporating subsistence needs based on a Constant Elasticity of Substitution (CES) utility function, determine the WTPs and propose a subsistence-needs adjustment factor. 3.1. Defining the two utility models under a CES function 3.1.1. Standard model under a general utility function Consider an individual i whose preference relation is continuous, monotonic and convex.

Figure 1 :

 1 Figure 1: Adjustment factor Z i when preferences are homogeneous (by θ and income relative to x s )

Figure 2 :

 2 Figure 2: Average adjustment factor when preferences are heterogeneous (upper panel: θ = .5, middle panel: θ = 0, lower panel: θ = -.5)

Figure 3 :

 3 Figure 3: Pattern of preferences leading to ranking reversal between two projects when θ = .5 (upper panel: N = M , middle panel: M = 3N , lower panel: N = 3M )

Table 1 :

 1 Subsistence-needs adjustment factor Z in the French population, by θ

	Income classes by unit Distribution of income (2)	Subsistence-needs adjustment factor Z (3)
	of consumption (1)	Canada	France	Study site: Canada	Study site: France
	(in e)	(in %)	(in %)	θ=.82 θ=.48 θ=.14 θ=.82 θ=.48	θ=.14
	Less than 5,268	5%	2.9%	-	-	-	-	-	-
	5,268 to 7,200	3.3%	4.2%	1.399 2.637	4.971	-	-	-
	7,200 to 10,000	6.9%	3.4%	1.186 1.637	2.260	1.386	2.570	4.764
	10 to 15,000	13.5%	12.3%	1.104 1.329	1.601	1.167	1.562	2.092
	15 to 20,000	14.2%	25.1%	1.067 1.205	1.361	1.100	1.317	1.578
	20 to 25,000	13.2%	18.3%	1.049 1.149	1.258	1.072	1.222	1.393
	25 to 30,000	11.4%	12.1%	1.039 1.117	1.201	1.056	1.171	1.298
	30 to 35,000	9.0%	5.3%	1.032 1.096	1.164	1.046	1.139	1.240
	35 to 40,000	6.6%	3.6%	1.028 1.082	1.139	1.039	1.117	1.201
	40 to 45,000	4.3%	3.1%	1.024 1.071	1.121	1.034	1.101	1.173
	45 to 50,000	2.6%	1.9%	1.021 1.063	1.106	1.030	1.089	1.152
	50 to 55,000	1.4%	1.4%	1.019 1.057	1.095	1.027	1.080	1.135
	55 to 60,000	0.8%	0.6%	1.017 1.051	1.086	1.024	1.072	1.122
	60 to 65,000	0.8%	0.8%	1.016 1.047	1.079	1.022	1.066	1.111
	65 to 70,000	1.2%	0.8%	1.015 1.043	1.072	1.021	1.060	1.102
	70 to 75,000	1.8%	2.8%	1.014 1.040	1.067	1.019	1.056	1.094
	75 to 80,000	2.3%	0.5%	1.013 1.037	1.062	1.018	1.052	1.087
	80,000 plus	1.8%	1.0%	1.008 1.024	1.039	1.011	1.033	1.055
	(4) Mean W T P s (e): (2)xW T P s		5.5	163	5,213	5.3	146	4,359
	(5) Mean W T P (e): (2)x(3)xW T P s		5.8	187	6,297	5.7	179	5,818
	(6) Change in Mean WTP (%): [(5)-(4)]/(4)	6.1	14.9	20.8	8.3	22.3	33.5
	(7) True value transfer factor on mean W T P s	.955	.897	.836 1.047 1.115	1.196
	(8) Commonly-used value transfer factor for W T P s	.984	.956	.928 1.016 1.046	1.078
	(9) Adjusted value transfer factor for mean W T P s	.963	.896	.833 1.038 1.116	1.200
	Notes:

We restrict to small projects unable to markedly change relative prices of goods or real income, to avoid inconsistencies(Brent,

[START_REF] Brent | Applied Cost-Benefit Analysis[END_REF][START_REF] Boadway | The Welfare Foundations of Cost-Benefit Analysis[END_REF][START_REF] Blackorby | A Review Article: The Case against the Use of the Sum of Compensating Variations in Cost-Benefit Analysis[END_REF].

Indeed, u i (x i,k ) → k x α i,k i,k when θ i → 0.

The rationale can be found in the discretionary income (see section 2.1) or in models that take into account minimum levels of consumption (known as "Stone-Geary" functions, after[START_REF] Geary | A Note on 'A Constant-Utility Index of the Cost of Living[END_REF][START_REF] Stone | Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British Demand[END_REF].

Note that W T P i can also be theoretically obtained with appropriate distribution weights based on a utilitarian SWF (see Appendix A).

The generalisation to more income groups in a discrete or continuous way is straightforward but would make the graphic representation less intuitive. See for instanceBaumgärtner et al. (2017a) or[START_REF] Meya | Environmental Inequality and Economic Valuation[END_REF], who use a log-normally distributed income with a CES function in a continuous framework.

To avoid adding complexity, we assume that the non-market goods of the two projects are different.Otherwise, a function accounting for the proportion of the two goods in each project would be required.

anonymous reviewers for their numerous and valuable comments and suggestions, which helped improve the paper. This work was supported by the French National Research Agency under Grants ANR-17-EURE-0020 ANR-08-RISKNAT-007-01 (Riskemotion) and ANR-16-CE03-0005 (GREEN-Econ), and by the Excellence Initiative of Aix-Marseille University -A*MIDEX.

times subsistence needs. On the X-axis, we measure α H,A , the preference for project A of the high-income individuals: the closer to (1 -α 0 ), the stronger this preference, whereas the closer to 0, the stronger the preference for project B, while (1 -α 0 )/2 indicates indifference between the two projects (i.e. α H,A = α H,B ). On the Y-axis, we measure α L,B , the preference for project B of the low-income individuals: the closer to (1 -α 0 ), the stronger this preference, whereas the closer to 0, the stronger the preference for project A, We observe that when M = N (Figure 3(a)), the area of ranking reversal is below the 45 • line in quadrant 3 (and above it in quadrant 2), which confirms the previous finding that high-income individuals' preferences (for project B in quadrant 3 and project A in quadrant 2) dominate those of low-income individuals (for project A in quadrant 3 and project B in quadrant 2) when WTP is not adjusted for subsistence needs, i.e. there is a Appendix A Deriving W T P i from appropriate ex post distributional weights Let us first consider whether an ex post weight based on the marginal utility of income can properly account for the effect of subsistence needs. For an individual i with preferences (α i , θ i ) and one non-market good q i,1 , the marginal utility of income V s i is equal to:

If we weight the W T P s i in the model with subsistence needs (Eq. ( 6)) by the marginal utility of income, we obtain:

The resulting W T P s w i still depends on the level of subsistence needs x s : accounting for the marginal utility of income does not fully account for the effect of subsistence needs.

Theoretically, we can find a "social weight", denoted SW i , and obtain the same W T P i as that obtained with the subsistence-needs adjustment but after weighting by the marginal utility of income and a utilitarian SWF. This social weight would be equal to:

The implied SWF is defined by the primitive of this social weight with respect to W T P s i . The W T P s w i weighted by the marginal utility of income and by SW i would then be equal to:

In practice, the SW i is almost impossible to elicit without specific surveys asking for individual preferences on different combinations of income and non-market-good levels [START_REF] Fleurbaey | Equivalent Income And Fair Evaluation Of Health Care[END_REF][START_REF] Samson | Fairness in Cost-Benefit Analysis: A Methodology for Health Technology Assessment[END_REF].

Appendix B Condition for ranking reversal

Ranking reversal occurs when the relative ranking of 2 projects (A and B) depends on the way they are evaluated (via observed W T P s or subsistence-needs adjusted WTP). We use the notations previously introduced.

A] Consider that project A is ranked first based on observed W T P s (condition A1) but ranked second based on adjusted WTP (condition A2), i.e.:

(A1) :

Ranking reversal is meaningless, as both income groups prefer project B.

When ranking is based on observed W T P s , condition (A1) leads to (from Eq. ( 9)):

We know that α L,A = 1 -α 0 -α L,B and α H,B = 1 -α 0 -α H,A , hence:

Then, because α H,A < α H,B and α L,A > α L,B , we have 1 -α 0 -2α H A > 0 and 1 -α 0 -2α L,B > 0. Easy manipulations lead to:

Ranking reversal is meaningless: Equation ( 13) cannot be met since We use the symmetry argument to obtain the same valid ranking reversal condition as in A].

Ranking reversal is meaningless, as both income groups prefer project A.


Marko Jankovic Addition Tensor

Goldbach's Conjecture

Marko V Jankovic

Addition Tensor and Goldbach's Conjecture

Introduction

In elementary mathematics, a number line is a picture of the graduated straight line that serves as an abstraction to real numbers. Idea of number line was first introduced by John Napier [START_REF] Napier | A description of the admirable table of logarithms[END_REF], and later, John Wallis [START_REF] Wallis | Treatise of Algebra[END_REF] used this graphical representation to explain operations of addition and subtraction in terms of moving backward and forward under the metaphor of a person walking. However, that type of graphical interpretation is not particularly suitable in other contexts of interest. In order to obtain another useful representation of natural numbers, a multiplication tensor or M N -tensor has been recently introduced [START_REF] Jankovic | Multiplication Tensor and Number of Rational Numbers[END_REF]. Idea came from the fundamental theorem of arithmetic [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF].

In this paper, an addition tensor or A N -tensor is going to be introduced. The addition tensor is going to be created from M N -tensor by using analogy and replacing the operation of multiplication with addition. In order to show usefulness of this presentation of natural numbers it is going to be shown, in an elementary way, that famous (strong) Goldbach's conjecture [START_REF] Guy | Unsolved Problems in Number Theory[END_REF] can not hold.

The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely represented by a product of powers of prime numbers, up to the order of the factors [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF]. Having that in mind, an infinite dimensional tensor M N that contains all natural numbers only once, is going to be constructed [START_REF] Jankovic | Multiplication Tensor and Number of Rational Numbers[END_REF]. In order to do that we are going to mark vector that contains all prime numbers with p.

So, p(1) = 2, p(2) = 3, p(3) = 5,
m i 1 i 2 i 3 ... = p(1) i 1 -1 p (2) i 2 -1 p(3) i 3 -1 ... .
The alternative definition is also possible. Now, the following notation is going to be assumed for some infinite size vectors

2 = [2 0 2 1 2 2 2 3 …], 3 = [3 0 3 1 3 2 3 3 …], 5 = [5 0 5 1 5 2 5 3 …] …
It is simple to be seen that every vector is marked by bold number that is equal to some prime number and that components of the vector are defined as powers of that prime number, including power zero (it can be seen that every vector represents infinite cyclic semi group defined by a primitive that is one of the prime numbers). Now, the M N -tensor can be defined as

M N = 2 ○ 3 ○ 5 ○ 7 ○...,
where ○ stands for outer product.

The tensor M N is of infinite dimension (equal to number of prime numbers) and size, and contains all natural numbers exactly ones. It is easy to understand why it is so, having in mind the fundamental theorem of arithmetic. This type of infinite tensor is called a half infinite tensor [START_REF] Jankovic | Multiplication Tensor and Number of Rational Numbers[END_REF].

The tensor that represents all odd numbers, M NO , contains elements defined as

m i 1 i 2 ... = p( 2) i 1 -1 p(3) i 2 -1 ... , or M NO = 3 ○ 5 ○ 7 ○...,
where ○ stands for outer product. It is simple to be seen that every vector is defined by some prime number and that components of the vector represent all non-negative integer multiples of that prime number. Now, the A N -tensor can be defined as

a i 1 i 2 i 3 ... =(i 1 -1) p(1)+(i 2 -1) p (2)+(i 3 -1) p(3)+...
A N = 2a ○ + 3a ○ + 5a ○ + 7a ○ + ...,
where ○ + stands for outer sum, which is analogous to outer product where operation of interest is addition..

It is interesting to notice that the tensor M N does not contain number 0 that is neutral element for addition, while, on the other hand, the tensor A N does not contain number 1 that is neutral number for multiplication.

Here we will present an additional addition tensor A NO that is created by odd prime numbers, where elements of that tensor are defined as

a i 1 i 2 i 3 ... =(i 1 -1) p( 2)+(i 2 -1) p(3)+(i 3 -1) p(4)+... . or A NO = 3a ○ + 5a ○ + 7a ○ + ...,
where ○ + stands for outer sum.

A proof that (strong) Goldbach's conjecture cannot hold

Goldbach's conjecture (strong version) states that every even natural number bigger than 4 can be expressed as a sum of two odd prime numbers [START_REF] Guy | Unsolved Problems in Number Theory[END_REF]. Here we will show in an elementary way that this conjecture cannot hold. In order to do that we will analyze tensors A NO and M NO contains all odd numbers exactly once, and from that fact it is easy to understand that the number of numbers in the tensor is equal to the number of even numbers. Since the sub-tensor M NO (1:2, 1:2, 1:2, …) together with additional elements that represent squares of odd prime numbers obviously contains smaller number of numbers than the tensor M NO , it can be clearly seen that the number of sums created by two odd prime numbers is smaller than the number of even numbers, which means that (strong) Goldbach's conjecture can not hold. That completes the proof.

Appendix A contains an elementary proof in which it is not necessary to use any reference to tensors.

.

  The edges of that tensor will contain the following vectors 2a = [0 2 4 6 …], 3a = [0 3 6 9 …], 5a = [0 5 10 15 …] …

NO

  1:2, 1:2, …) are going to be compared. It is clear that those two sub-tensors are of the same size by construction. Those sub-tensors contain all sums and all products that are produced by two different odd prime numbers, respectively (they also contain prime numbers themselves and many additional elements that are composed by primes of power one, but this does not affect the final conclusion). The additional numbers that are composed of two prime numbers are on the positions (3 1 1 1 …) (1 3 1 1 1…), (1 1 3 1 1 1…) and so on, and their number is equal in both tensors -in the case of M NO -tensor those numbers represent squares of odd primes and in the case of A NO -tensor they represent doubles of odd primes. It is known that tensor M

Table 1 .

 1 Simple multiplication table for odd prime numbers

	•	p(2)	p(3)	p(4)	p(5)	...
	p(2)	p(2) 2	p(2)•p(3)	p(2)•p(4)	p(2)•p(5)	...
	p(3)	x	p(3) 2	p(3)•p(4)	p(3)•p(5)	...
	p(4)	x	x	p(4) 2	p(4)•p(5)	...
	p(5)	x	x	x	p(5) 2	...
	...	...	...	...	...	...

Table 2 .

 2 Simple addition table for odd prime numbersIt can be seen that lower triangular part of Table1and Table2is filled with x (we are ignoring them), since it would contain the values already contained in upper triangular part of the tables due to the commutative nature of operations of addition and multiplication. From Table1and Table2we can conclude that both tables contain the same number of elements. From fundamental theorem of arithmetic, it is known that all products in Table1have unique values (that is not the case with sums in the Table2, but it has no influence on the final conclusion). From fundamental theorem of arithmetic it is known that all odd numbers can be uniquely expressed in the following form , … are natural numbers. Having this in mind, it can be seen that the unique products in Table1represent a proper subset of odd natural numbers, which means that their number is smaller than the number of odd natural numbers. It is well know fact that the number of odd natural numbers is equal to the number of even natural numbers. Since the number of sums in the Table2is equal to the number of products in the Table1we can conclude that the number of sums in the Table2is smaller than the number of even numbers and then it is easy to conclude that strong version of Goldbach's conjecture cannot hold. It concludes the proof.

	+	p(2)	p(3)	p(4)	p(5)	...
	p(2)	p(2)+p(2)	p(2)+p(3)	p(2)+p(4)	p(2)+p(5)	...
	p(3)	x	p(3)+p(3)	p(3)+p(4)	p(3)+p(5)	...
	p(4)	x	x	p(4)+p(4)	p(4)+p(5)	...
	p(5)	x	x	x	p(5)+p(5)	...
	...	...	...	...	...	...

Appendix A.

Here, a proof that does not reference to tensors is going to be presented.

In order to prove that strong Goldbach's conjecture cannot hold, we are going to analyze a multiplication (Table 1) and an addition table (Table 2). Since we are going to focus on the number of numbers presented in the tables, tables will not contain numbers themselves. Vector p is previously defined as vector that contains all prime numbers with elements p(1) = 2, p(2) = 3, p(3) = 5, and so on.


Almudena Muñoz

Josefina Garrido

Francisco José Melero

Gregorio Cañavate

Carlos González

Leonardo Piccinetti

Florent Bourgeois

Carole Coufort

Petar Kisyov

C Saudejaud

Implementation of environmental technology verification through innovative training in response to urgent envrironmenal challenges

Keywords: Environmental Technology Verification, ETV, environmental challenges, environmental technologies, market, Erasmus Plus, Learning outcomes, Joint Curriculum

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The world faces a range of environmental challenges that will impact on its future prosperity. These include resource depletion, increasing water scarcity, air pollution, climate change and biodiversity loss. Concretely Europe leads on innovation in innovative environmental technologies which can provide solutions while also increasing EU competitiveness. However, breaking into the market with innovations can be a significant problem, because innovations by definition cannot show a successful track-record. Without credible information about innovative technologies, potential buyers are unsure whether or not to trust the claims made about their performance. [START_REF]The European Environment-state and outlook 2020[END_REF] Thus, European Commission with 7 Member States (France, Belgium, Finland, Czech Republic, Denmark, Poland and United Kingdom) intends to launch a solution to address that problem in 2011: Environmental Technology Verification (ETV). This initiative provides a third-party verification of the performance claims made by technology manufacturers in business-to-business relations. By issuing a Statement of Verification, which is the product of a successful ETV process, ETV provides credible information on new technology. Market access for these innovative technologies is significantly enhanced and the technological risk for technology purchasers (whether private or public) is reduced. The information contained in the Statement of Verification summarises the actual performance of the verified technology as well as the results of the tests performed. With proof of performance credibly assured, along with information about the design of the tests, innovations can expect a larger market share. [START_REF] Ratman-Klosinska | A comprehensive guide for proposers to the EU Environmental Technology Verification Pilot Programme[END_REF], [START_REF] Molenda | Quality Assurance in Environmental Technology Verification (ETV): Analysis and impact on the EU ETV Pilot Programme Performance[END_REF] Nowadays, the most common reasons why SMEs don't apply for ETV verification are that they unknown details about this initiative, they assume which it is too complicated and that it is necessary a lot of paperwork to could obtain the verification. In this context, a group of different organisations (SME, university, associations, technological centre and cluster) have created ETV4INNOVATION Consortium. This eco-innovative partnership of 6 European Countries (Ireland, Spain, France, Bulgaria, Poland and Denmark) encompasses in ETV4INNOVATION project (Development of an online resource to train enterprises with the final objective to foster them into Environmental Technology Verification) has the objective of creating a flexible learning pathway in line with the needs of learners and companies in terms of ETV. This e-learning course will fill gaps in relation with ETV for current and future managers and workers in technological sectors. [START_REF]ETV4INNOVATION website[END_REF]

METHODOLOGY

The aim of ETV4INNOVATION project is to design and develop a Joint Curriculum for filling gaps related to ETV for current and future managers and workers in technological sectors. The project methodology followed was focused on the achievement of the following specific objectives: SO1. Identify the lack of knowledge in the organizations to be able to manage and bet on ETV which incorporate green requirements in the development of new eco-technologies. SO2. Design and develop a Joint Curriculum (JCV) on ETV with the participation of stakeholders from the industry, experts in the topic of green technology such as Verification Bodies, Clusters and Universities with expertise in this eco-innovative verification (ETV).

SO3. Deliver the JCV on e-learning OER Platform and protect it under open licenses. SO4. Break borderlines among SMEs and large companies and experts in ETV, creating common procedures and skills. SO5. Equipping enterprises with the right skills and knowledge for the ETV verification of today and tomorrow and fostering a sustainable technology process. It will be performed by promoting better anticipation of ETV skills needs and developing better matching between skills and companies needs and European green requirements too. SO6. Awareness of current environmental problems, the need to carry out a sustainable development and bring more respectful practices with the environment.

In order to reach all the objectives defined in the project, ETV4INNOVATION has comprised the following activities:

-Comparative analysis and evidence-gathering, studies examples of good practices and real-life cases of successful ETV verification already performed, etc. in order to identify the skills needs and competences in the technological sectors required to successfully applied for ETV process: the main objective of this activity was to provide a complete analysis of the current sills and needs to improve the technological sectors towards ETV with high level of competences to boost the competitiveness of the eco-technology sector and the green innovative requirements.

-Development of a Joint Curriculum with the integration of different learning modes and innovative approaches: the main objective of this activity was to define, descript and grouping in units or modules the learning outcomes of the ETV4INNOVATION training course.

-The creation or learning and training materials together with methods, pedagogical approaches and tools: the main objective of this activity was to develop the training methodology content of the ETV4INNOVATION course.

-The implementation of a training and collaborative e-learning platform to ensure that all the materials developed is certainly deployed by creating an open online course: the main objective of this activity was to develop an e-learning platform in order to deploy all the training material developed in the previous activity by creating a Massive Open Online Course (MOOC).

RESULTS

After the implementation of the four activities defined in the methodology of the project, the general results expected were:

-From the key study activity, a report developed describing the current situation of the SMEs and the skills needs towards technological sectors prepared to foster ETV. This report has been used as a starting point to define the learning outcomes as well as to define the training material content which was developed in further activities.

-From the Joint Curriculum definition, a key document with the description of the training modules and units compiled in the ETV4INNOVATION course, as well as the final tool for knowledge verification.

-From the training material activity, the main results were the definitive version of all training materials necessary for the course of the project. All were previously harmonized to achieve a coherent structure of the content along the suggested curriculum.

-From the e-learning platform, the main result was the MOOC platform which contains the ETV4INNOVATION course (the training materials regarding ETV and their application in technological sectors such as waste, resources, etc.).

Key study activity and definition of ETV4INNOVATION Joint Curriculum

The overall aim of the Erasmus+ ETV4INNOVATION project is to design and develop a Joint Curriculum for filling gaps in relation with ETV for current and future managers and workers in technological sector as materials, waste & resources technologies, water technologies and energy technologies. ETV4INNOVATION is a two-and-a-half-year long EC funded VET Strategic Partnership project under Erasmus+ programme. It has been designed with the aim to support the development and the implementation of an innovative practice and a new training path in the field of ETV. The project is created to support the development and the implementation of an innovative practice and a new training path in ETV. ETV4INNOVATION Consortium has an objective to achieve one of the goals of the Europe 2020 strategy for smart, sustainable and inclusive economy. For this reason, a balance between technological innovation and efficiency in production process, as well as improvement of products have been promoted.

One of activities carried out was a Key Study on General Needs by SMEs towards ETV. In order to perform it, the "Cedefop's European skills and jobs (ESJ)" survey of 2014 in all EU Member States was analyzed. It revealed that about 30% of European employees possess qualifications that are not wellmatched to those required by their jobs, while about 45% of EU adult workers believe that their skills can be better developed or utilized at work. According to the conclusions obtained by Cedefop in this study, ETV4INNOVATION project considered the relevancy of looking deeper into a complete analysis of the current skills and needs to improve the technological sectors towards ETV, with high level of competences, to boost the competitiveness of the eco-technology sectors and the green innovative requirements for today and tomorrow. The study was elaborated through a two-stage methodology, consisting of a desk research and a survey followed by an endorsement on the key finding by relevant stakeholders.

The desk research was carried out to identify the future and current needs related to eco-technology and ETV, as well as the main needs and requirements to face these challenges by Europe technology industry and other stakeholders.

As a result of this desk research the ability, capacity, skills and competences need set towards ETV and SMEs were established and detected. The endorsement of the researched needs was undertaken though a planned survey conducted with 90 participants from SMEs and related stakeholders, including regional administrations, universities, development agencies, research and innovation entities, VETs, etc.

According to the conclusions obtained from the desk research and the endorsement survey, it has been defined and analyzed the most suitable curriculum having as a basis both the skills and competences required for achieving green and innovative technologies and the key elements in which ETV are based:

-Basic aspects of verification of environmental technology -The ETV programme as a commercial tool on domestic and international markets -Practical aspects of the ETV verification

ETV4INNOVATION Training Material

One of the main results of the ETV4INNOVATION project is the creation of the training materials that will provide users the opportunity to expand your knowledge on how to apply for ETV and the added value of achieving this verification from technical and commercial perspectives. In each lesson, ETV4INNOVATION partnership has prepared different interactive materials: ▪ Presentations to summarise and facilitate the knowledge acquisition ▪ Interactive tools for some main training materials (infographics, videos, etc.) ▪ A coursebook with all the information included in the modules ▪ Additional readings in order to deep in particular training topics ▪ Quizzes to assimilate the acquired knowledge

The course is divided in 3 modules, each once composed by different lessons:

Module 1. Basics aspects of Verification of Environmental Technology

In this module, the student will have a first approach to the necessary concepts of ETV. This Module is divided in six lessons:

1. Overview of ETV 2. ETV in the world 3. Main objectives of ETV 4. Targeted Technologies 5. Verification process 6. Main features of ETV/Summary

Module 2. The ETV programme as a commercial tool on domestic and international markets

In this module, the student will have a complete view of the ETV programme as a tool for boosting the commercialisation of a product, process or service in domestic and international markets. This Module has five lessons:

1. ETV objectives in terms of commercial exploitation 2. Technology Readiness Level 3. The integration of marketing plan for ETV sellers 4. Which are the benefits of clients regarding ETV? 5. Success case of ETV

Module 3. Environmental Technology Procedure

In this module, student will have a complete training on the practical aspects of an ETV verification which some success cases. This Module has four lessons: Fig. 1 shows the global architecture of ETV4INNOVATION platform, based on Opigno, the specific elearning module of the Content Management System (CMS). This module was selected because it is open source and makes possible to create interactive contents. [START_REF] Egido | Opigno 2.0, un LMS basado en drupal 8. Documentaciónhoy[END_REF] This e-learning platform could work as a template for future training content and will allow uploading and designing web VET curriculum regarding others technology fields. In addition, it is compatible and fully working on smartphones and table systems.

CONCLUSIONS

Within the European Commission's European Green Deal, one of the measures they seek to adopt to become the first climate-neutral continent is to get manufacturers to provide reliable, comparable and verifiable information about their products so that buyers can make more sustainable decisions and reduce the risk of "green washing".

To this end, tools such as ETV are essential to provide a protocol for testing and verifying the performance of innovative environmental technologies. In this context, ETV4INNOVATION training course has designed and developed a Joint Curriculum for filling gaps related to ETV for current and future managers and workers in technological sectors. This course is free and is available at ETV4INNOVATION e-learning platform. In general, project has strengthened the education and training path of target users and stakeholders.

3. 3 ETV4INNOVATION

 3 Training and Collaborative PlatformConsortium has developed an e-learning platform in order to deploy all the training materials prepared during the project by creating a massive open online course (MOOC).

Figure 1 .

 1 Figure 1. ETV4INNOVATION e-learning platform.

ACKNOWLEDGEMENTS

The authors would like to thank the Irish National Agency, Léargas, for funding the ETV4INNOVATION project -Development of an online resource to train enterprises with the final objective to foster them into environmental technology verification (Project Reference no. 2017-1-IE01-KA202-025703) under the Erasmus+ Programme, and specifically under the Key Action "Cooperation for innovation and the exchange of good practices" and the Action "Strategic Partnership for vocational education and training", thanks to which this project has been implemented.