Kolia -Diafouka

Keywords:

INTRODUCTION

Cervical cancer is the third most common female cancer worldwide and the fourth leading cause of cancer death in women. Persistent infection of the uterine cervix with oncogenic high-risk types of human papillomavirus (HR-HPV), particularly HPV16, is recognized as the cause for the development of precancerous cervical lesions that may progress to invasive cancer. [START_REF] Bosch | The causal relation between human papillomavirus and cervical cancer[END_REF] However, only a minority of women infected with HR-HPV develops premalignant and malignant lesions, suggesting that cofactors are involved in HR-HPV-associated carcinogenesis.

Polyomaviridae is a continuously growing family of viruses with up to 11 new human representatives recently added to the previously described BK polyomavirus (BKPyV) and JC polyomavirus (JCPyV). [START_REF] Dalianis | Human polyomaviruses in disease and cancer[END_REF] Human polyomaviruses (HPyVs) subclinically infect the general population and may induce disease in immunocompromised patients. [START_REF] Dalianis | Human polyomaviruses in disease and cancer[END_REF] Moreover, these viruses have the potential capacity to induce tumors through their tumor antigen (T-Ag) oncoproteins. [START_REF] Dalianis | Human polyomaviruses in disease and cancer[END_REF] The oncogenic capabilities of HPyVs were recently illustrated with the discovery of the Merkel cell polyomavirus (MCPyV), which is now commonly accepted as the etiologic agent of the Merkel cell carcinoma, a rare but aggressive skin cancer. [START_REF] Feng | Clonal integration of a polyomavirus in human Merkel cell carcinoma[END_REF] Several studies have investigated the presence of HPyVs in different tissues, and their presence in the female genital tract may suggest an association between HPyVs and precancerous or cancerous cervical lesions, as illustrated by the detection of BKPyV, JCPyV and MCPyV genomic sequences in cervical cancer cases. [START_REF] Imajoh | Detection of Merkel cell polyomavirus in cervical squamous cell carcinomas and adenocarcinomas from Japanese patients[END_REF][START_REF] Comar | High prevalence of BK polyomavirus sequences in human papillomavirus-16-positive precancerous cervical lesions[END_REF][START_REF] Alosaimi | Increased prevalence of JC polyomavirus in cervical carcinomas from women infected with HIV[END_REF][START_REF] Salehi-Vaziri | Merkel cell polyomavirus and human papillomavirus infections in cervical disease in Iranian women[END_REF] In order to study more deeply the relationships between HPyVs, cervical lesions and immune status, we have investigated the presence of recently described HPyVs including MCPyV, HPyV6, HPyV7 and the Trichodysplasia spinulosa polyomavirus (TSPyV) in cervical samples from African and French women, in relation to HR-HPV co-infection, cervical disease stage, and HIV serostatus. with the aim to evaluate cervical cancer screening approaches in HIV-1 infected African women. In addition, 50 samples were collected from randomly selected HIV-seronegative women with a median age of 40.0 years (range, 25-68 years) attending a gynaecological outpatient department at the Montpellier University Hospital, France, who were tested for HR-HPV infection. Cervical biopsies were obtained from all of the African women who had a positive HR-HPV test and/or abnormal cervical cytology and/or abnormal visual inspection using acetic acid and Lugol's iodine, and these women were selected to provide similar proportions of women with high-grade lesions (CIN2+) and women with normal histological findings or low-grade lesions (<CIN1). In the group of French women, biopsies were performed only in case of abnormal cytological findings.

Written informed consent was obtained from the women. primers. Quantification of MCPyVs DNA was normalized to the amount of whole genomic DNA as previously described. [START_REF] Du-Thanh | Detection of Merkel cell and other human polyomavirus DNA in lesional and non lesional skin from patients with Kaposi sarcoma[END_REF] For the other viruses, semi-quantification was based on the cycle threshold value (Ct) and samples with Ct>40 were considered as negative.

Proportions were compared using the chi-square test or Fisher's exact test, and the MCPyV DNA levels were compared using the Mann-Whitney U test. MCPyV copy number of 80 copies/10 6 cells (range 14 to 210 copies/10 6 cells), the lower limit of detection (LLD) being 10 copies/10 6 cells. Median MCPyV DNA load was 79 copies/10 6 cells in women without CIN2+ lesions and 83 copies/10 6 cells in women with CIN2+ lesions (p=0.75). HPyV6 and HPyV7 DNA levels were always weakly detected in the positive samples, with cycle thresholds ranging from 34 to 40.

As shown in Table 1, there was no association between the detection of MCPyV DNA and HIV serostatus, the detection of HR-HPV, or the presence of CIN2+ cervical lesions. HPyV6 (n=6) and HPV7 (n=4) were detected in 4 and 4 HIV-infected African women and 2 and 0 HIV-negative French women, respectively (p=1.00). The small sample size did not allow for sufficient statistical power to validly analyse the association of HPyV6 and HPyV7 detection with HR-HPV infection and cervical lesions.

DISCUSSION

This study, involved two very different populations of women in terms of geographic origin, HIV status, prevalence of HR-HPV infection and precancerous cervical lesions. Overall, MCPyV DNA was frequently detected at low levels in cervical specimens, whereas HPyV6 and HPyV7 were rarely detected, and TSPyV was not detected. MCPyV DNA detection was not associated with HIV status, HR-HPV infection or precancerous cervical lesions.

Moreover, MCPyV DNA levels were not different between women with or without precancerous lesions. The sample size of HPyV6/HPyV7 was too small for useful statistical comparisons.

These results, which do not support a role of these human polyomaviruses in the development of cervical lesions, are in agreement with a recent study reporting a high frequency of MCPyV detection in cervical carcinomas but no statistically significant association between

MCPyV infection and cervical cancer. [START_REF] Salehi-Vaziri | Merkel cell polyomavirus and human papillomavirus infections in cervical disease in Iranian women[END_REF] These observations argue against a possible role of MCPyV as a cofactor of HPV-induced carcinogenesis. Interestingly, the detection rates of MCPyV, HPyV6, HPyV7 and TSPyV in cervical specimens observed in this study are similar to those reported in normal skin specimens, these viruses being considered as belonging to the skin microbiota. [START_REF] White | The rapidly expanding family of human polyomaviruses: recent developments in understanding their life cycle and role in human pathology[END_REF]10 This observation suggests that MCPyV and, to a lesser extent, HPyV6 and HPyV7 might belong to the female genital tract microbiota. The transmission routes of these viruses and their mode of propagation to the female genital tract remain to be elucidated, and the hypothesis of genital mucosa contamination by skin viruses via hands or sexual activity cannot be excluded.

Key messages

MCPyV is frequently detected at low levels in cervical samples, whereas HPyV6, HPyV7 and TSPyV are rarely or not detected.

MCPyV detection is not associated with HIV status, HR-HPV infection or precancerous cervical lesions.

Polyomaviruses might belong to the female genital tract microbiota.

Licence for Publication statement

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	METHODS
	A total of 190 cervical samples collected with the Digene cervical sampler and placed in the
	specimen transport medium (STM) (Qiagen, Gaithersburg, MD) were analysed. One hundred
	and forty samples were collected from a subset of women with a median age of 35.4 years C o (range, 21-47 years) enrolled in the HARP (HPV in Africa Research Partnership) study, n d i f which is conducted in two Sub-Saharan African countries, South Africa and Burkina Faso,
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Table 1

 1 Associations between MCPyV detection and geographic origin/HIV serostatus, HR-

	Sexually Transmitted Infections

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  10. Foulongne V, Sauvage V, Hebert C, et al. Human skin microbiota: high diversity of DNA viruses identified on the human skin by high throughput sequencing. PLoS One 2012;7 :e38499.

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Acknowledgements:

Other contributing members of the HARP study group included: A. Chikandiwa, T. Omar and P. Michelow (Johannesburg, South Africa); O. Goumbri-Lompo and N. Meda (Ouagadougou, Burkina Faso); C. Gilham and H.A. Weiss (London, UK); S. Doutre, V.

Costes, M.N. Didelot and N. Nagot (Montpellier, France).


Anne-Laure Ligozat

Brigitte Grau

Anne Vilnat

Isabelle Robba

Arnaud Grappy

Lexical validation of answers in Question Answering

Question answering (QA) aims at retrieving precise information from a large collection of documents, typically the Web. Different techniques can be used to find relevant information, and to compare these techniques, it is important to evaluate question answering systems. The objective of an Answer Validation task is to estimate the correctness of an answer returned by a QA system for a question, according to the text snippet given to support it. In this article, we present a lexical strategy for deciding if the snippets justify the answers, based on our own question answering system. We discuss our results, and show the possible extensions of our strategy.

Introduction

Question answering (QA) aims at retrieving precise information from a large collection of documents, typically the Web. To be considered as reliable by users, a question answering system must be able to give them elements of justification to evaluate the answer whithout having to read the whole document. Here is an example of such a justification: Question: When was the Berlin wall demolished? Answer: in 1989 Justification (supporting snippet): the Berlin wall divided East and West Berlin for 28 years, from the day construction began on August 13, 1961 until it was dismantled in 1989.

Answer validation

In 2006, an Answer Validation Exercise (AVE) 1 was introduced in the evaluation campaign CLEF, which aimed at automatically validating the correctness of the answers given by QA systems according to their supporting snippets. The goal of this exercise is to improve the performance of QA systems by developing methods for automatic evaluation of answers, and to make answer assessment semiautomatic. The organizers provided answers of QA systems with their supporting snippets, and the participants had to decide if each answer was correct or not according to the snippet. For example, the following couple (hypothesis, snippet) can be given: The hypothesis is the reformulation of the question "Who was Yasser Arafat?" containing the answer given by a system "Palestine Liberation Organization Chairman".

In AVE, the corpus of about 3,000 pairs hypothesissnippet was built semi-automatically from answers of a question answering evaluation campaign. AVE participants were evaluated on their capacity to predict the correctness of the answer (assessed by human validators), and had to return a value of implication (YES or NO) for each input pair. The precision, recall and f-measure of participants to AVE were calculated by the following formulas:

precision = #predicted as Y ES correctly #predicted as Y ES recall = #predicted as Y ES correctly #Y ES pairs f -measure = 2 * precision * recall precision+recall

Validating answers with a question answering system

In 2006, after several participations to question answering campaigns (in monolingual French and English tasks and cross-lingual ones), we decided to participate to AVE. In this campaign, our objective was to use our own question answering system for French, FRASQUES. The relevance of a justification with respect to a hypothesis is evaluated according to the information about the answer deduced from the question, and to the answer given by our system.

Our answer validation system

The answer validation system uses three out of the four modules of the question answering system as figure 1 shows. The input of the answer validation is a pair hypothesis-snippet, along with the original question Q and the answer to judge A1.

Fisrt, the question is analyzed by the Question analysis module, which processes a syntactic analysis of the question to detect some of its characteristics such as its keywords, the expected answer type (which can be a named entity like person, country, date... or a general type like conference or address), the focus of the question (which is defined as the entity about which a characteristic is required).

Then, the Document processing module is used, but on the snippet to judge instead of the output of the search engine. This module uses Fastr 3 to recognize linguistic variants of the question terms: for example, "Europe's currency" will be recognized as a variant of "European currency". Then the named entities of the documents are tagged with around 20 named entity types.

The Answer extraction module extract the anwer(s) A2 that is found by our system in the snippet. The extraction strategy depends on the expected type of the answer. If the answer is a named entity, the named entity of the expected type which is closest to the question words is selected. Otherwise, patterns of extraction are used. These patterns were written in the Cass 4 format, a syntactic parser used here for answer extraction instead of syntactic analysis. These rules express the possible position of the answer with respect to the question characteristics such as the focus or the expected type of the answer. Cass thus tags the answers in the candidate sentences.

Finally, the answer A1 is evaluated, and the system returns YES if the answer is considered as justified or NO otherwise, with a confidence score.

The decision algorithm proceeds in two main steps. During the first one, we try to detect quite evident mistakes, The second step proceeds to more precise verifications. In an ideal case, a correct justificative snippet should contain a declarative reformulation of the question plus the answer, all the terms of the question should be present in the snippet, linked by the same relations.

Question terms

Among question terms, some play a more important role, for instance, the object of the question, that we named focus. The focus is the entity about which the question is asked and either a characteristic or a definition of this entity may be searched. In "Which is the political party of Lionel Jospin?", the focus is "Lionel Jospin".

Four particular roles are distinguished in the questions: focus, expected type of the answer, main verb, and proper names.

In order to confirm our intuitions about these roles, we conducted a corpus study after the evaluation: we measured the presence of each term in correct and incorrect snippets of the AVE corpus. centage of correct and incorrect snippets in which it was found (second and third columns). For example, the first line can be interpreted as follows: for the questions for which we detected a focus, 89% of the correct snippets contained this focus, while only 49% of the incorrect snippets did. This corpus study confirmed our intuition concerning the importance of a good extraction of the focus, or a good proper names recognition.

Relations between terms

Concerning relations between terms, to control them it is necessary to parse the whole sentence. Since in many cases, such parses are difficult to obtain whith sufficient confidence, we chose to control only the relations which are likely to associate the answer and some particular terms like the focus or the general type. This control is done using extraction patterns, which are written as Cass parser rules.

Answer type

Another verification concerns the expected type of the answer. When this type is a named entity, the system can determine if the answer is of the correct type. In the other cases, either the type is present in the snippet near the answer or it is implicit and does not appear in the snippet. This last possibility is illustrated in the question "Which animal lays blue eggs?" which answer is "Ameraucana chickens lay blue eggs". The use of external resources, such as Wikipedia in which many categories and definitions can be found, would be necessary to infer the link between the specific answer and the type to which it belongs.

Score computing

All these criteria lead us to calculate 2 scores. The first of them concerns the correlation between the answer extracted from the hypothesis A1 and the answer found by our system FRASQUES: A2. When both answers are completely different the hypothesis is refuted. When they are similar, or when FRASQUES did not obtain any answer, the decision is made by taking into consideration the presence/absence of particular terms. This first score is positive, either when both answers are similar or when the important terms are present in the snippet; it is negative in the other cases.

The second score takes into consideration the number of terms present in the snippet and their value. It is positive when at least one of the term is present, negative otherwise.

Finally a snippet is considered to be an acceptable justification if:

• if A2 is empty and the score determined by the terms is positive, and in this case, this score is the final score,

• if A1 and A2 are similar and the second score is positive, and then the final score is the highest of both scores,

• if both scores are opposite, we choose the highest which it is positive.

Results

The evaluation corpus contained 3,266 pairs, but among them, 202 were not judged. Since hypotheses were generated automatically, they contain many syntax errors, hence we used only the question plus the answer extracted from the hypothesis.

During our participation to AVE, many errors remained in our programs that have since been corrected. AVE organizers gave the values expected for each pair of the corpus so we could reevaluate our system. A precise examination of these values enabled us to see that the jugements were sometimes incorrect: some of the positive values are erroneous because the snippet does not contain any justification of the answer. We changed 82 positive values into negative ones.

Table 2 does not present our official results but our new results after diverse improvements of our programs. The first line gives the number of justified pairs (YES), and not justified pairs (NO) evaluated by the human judges on the corpus. The second line contains our results and the third one our correct results. The three last lines are the precision, recall and f-measure of these results following the formula given section 2. Among our NO answers, we distinguish sure ones from the others. A NO answer is sure if a criteria of the first step is not satisfied (see the preceding section); such NO answers receive the highest confidence score. We found 1035 pairs of "sure NO". Among them, 995 were correctly judged, so the precision for these answers is 0.96.

Improvement of the lexical criterion

For the above evaluation, the parameters given for each criterion were set manually. In order to improve the use of Precision Recall f-measure 0.55 0.55 0.55 Table 2. FRASQUES results at AVE 2006 lexical information, we decided to train our system on part of the AVE corpus. In this corpus, we put aside the "sure NO" pairs, and divided the remaining pairs to have a learning corpus and a training one. For the training, we used the machine learning tool Weka 5 . We considered as criteria the presence of the following terms of the question: focus, general type of the answer, main verb, and multiword expressions. We obtained the following results: precision 0.43, recall 0.84 and f-measure 0.57. These results show that the criteria can be improved by adjusting the scores of each parameter. We are presently implementing theses changes.

Related works

Penas et al. ([4]) give an overview of the first Answer Validation Exercice in 2006 6 . The approaches can be divided into two main categories: logical proofs of the entailment and lexical comparison of hypotheses and snippets.

In the first category, Tatu et al. ( [START_REF] Tatu | Automatic Answer Validation using COGEX[END_REF]) use a named entity recognizer, a syntactic parser and a semantic parser to transform hypotheses and snippets into a rich logic representation. Then both representations are submitted to COGEX, a natural language logic prover, that decides whether the text entails the hypothesis or not and also gives a justification of this decision. This system obtained the best results for English (with an f-measure of 0.46) and Spanish (f-measure: 0.60).

In the second category of systems, Herrera et al. ( [START_REF] Herrera | UNED Submission to AVE 2006[END_REF]) developed an approach based on an SVM (Support Vector Machine) classification. They apply lemmatization and entity recognition on both snippet and hypothesis. Then they determine the entailment between the numeric entities of the hypothesis and those of the snippet, and also the entailment between the named entities. The model of their classifier is then trained on all these features plus the percentage of word, unigrams, bigrams, trigrams of the hypothesis present in the snippet. Both their runs obtained the second and the third place in the Spanish task with an f-measure of 0.57 and 0.56).

AVE exercise is strongly connected to Textual Entailment ( [START_REF] Bar-Haim | The Second PASCAL Recognising Textual Entailment Challenge[END_REF][START_REF] Dagan | The PASCAL Recognising Textual Entailment Challenge[END_REF]). In Textual Entailment, the approaches are more diverse than in AVE; yet, the best systems also use extensive linguistic and background knowledge, or a very large training corpus.

Conclusion

In this article, we have presented a strategy for answer validation in question answering. This strategy is based on our own question answering system: the hypothesis and the text snippet are analyzed by the question answering system, and we have defined several criteria which enable us to detect whether the snippet justifies the answer. In our evaluation of hypothesis-snippet pairs, we distinguish with reasonable precision and recall the cases for which the snippet is most likely to justify the answer. We have also presented the possible extensions of our strategy, by using external resources to acquire additional knowledge. Since the answer has to be entirely justified by the snippet, it is important to respect the notion of justification.

This first experiment in answer validation constitutes a step towards semi-automatic validation of answers in question answering. It also helped us improve our system, since some of the criteria we used for answer validation had not been implemented in our question answering system.

  Hypothesis: Yasser Arafat was Palestine Liberation Organization Chairman 2 Snippet: President Clinton appealed personally to Palestine Liberation Organization Chairman Yasser Arafat and angry Palestinians on Wednesday to resume peace talks with Israel

3Figure 1 .

 1 Figure 1. Answer validation system

Table 1

 1

	shows for each term, the per-

Table 1 . Presence of the important terms ex- tracted from the question in the positive and negative answers

 1

http://nlp.uned.es/QA/AVE/

In our AVE examples, the answer will be written in bold.

http://www.cs.waikato.ac.nz/ ml/index.html

Cross Language Evaluation Forum, http://clef-qa.itc.it/CLEF-2006.html


Jules Sadefo Kamdem
email: jules.sadefo-kamdem@umontpellier.fr

I R R A Except

WILLINGNESS TO PAY OF AN EXPO-POWER UTILITY DECISION MAKER TO LIMIT CLIMATE CHANGE

Keywords: Willingness to Pay, Climate Change, Expo-Power Utility; I.A.R

, not only because they hinder infinite future welfare losses (for an exponential utility function), but because it is easy to change some moments of the distribution (jointly or holding the others fixed) while studying how uncertainty influences the willingness to pay as explained in Pindyck [1].

.

Introduction

Climate change is often described as the biggest environmental problem of the 21st century (Stern et al., 2006), although evidence is considered insufficient for some. According to the Intergovernmental Panel on Climate Change (IPCC), greenhouse gas (GHG) emissions from human activities, including the burning of fossil fuels such as coal, are "most likely" Temperature, changes in precipitation, and increasing variability in weather conditions. Without a coordinated policy around the world, advanced climate models predict that the average temperature in the United States for instance will increase by about 3 to 8 degree Faraday (1.7 to 4.4 degrees Celcius) by the end of the century, compared with today (IPCC 2007). In addition, the distribution of daily temperatures could be expected to increase in order to pose serious problems of individual or societal well-being, resulting in conflicts often due to starvation or, more generally, to the sharing of scarce resources diminished by climate change. Several direct and indirect economic consequences of climate change (climate damage) are: changes in crop yield, loss of land and capital due to rising sea levels, changes in fish catches, material damage caused by hurricanes, changes in labor productivity and health expenditures due to illness and heat stress; changes in tourism flows and energy demand for heating and cooling. Other important impacts of climate change are studied outside the modeling framework. One of the obvious consequences of climate change is the occurrence of more severe natural disasters, which are causing loss of human life not to mention their increasingly significant financial costs. To limit the negative consequences of climate change, scientists estimate that global warming can not exceed two degrees Celcius above the pre-industrial temperature. However, we are moving towards the critical threshold of 1.5 degrees Celcius . To limit global warming, we must reduce emissions of greenhouse gases by 80% to 95% by 2050 compared to the amount emitted in 1990. Many studies already show that this objective is achievable, but there is a need for urgent action at all levels. One must change the way a whole society works is anything but obvious. This usually requires a long-term effort on the part of the many sectors and actors that influence the way of life of this society. To implement meaningful changes, incentives are generally required. It is therefore essential that the political authorities take steps in this direction. Thus, several countries or insurance companies are asking the question: what is the willingness to pay today of an individual or society to avoid or be resilient faced with a temperature increase of 2 degrees Celcius compared to that of today ? To this effect, as indicated in Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] (which is our main reference for this paper), economic analyze of climate change policies often focus on "probable" scenarios -those within a confidence interval d About 66 -90% -for emissions, temperature, economic impacts and abatement costs. It would be difficult to justify the immediate adoption of a rigorous reduction policy in the light of these scenarios and the consensual estimates of discount rates and other relevant parameters. Indeed,Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] wonders: 1) whether a rigorous policy can be justified by a cost-benefit analysis that distributes all possible results; And 2) whether the demand for reduction depends more on expected results or uncertainty about outcomes ? Since, recent climate science and economic impact studies allow one to at least roughly estimate the distributions for temperature change and its economic impact. Pindyck (2012) show how these distributions can be incorporated in and affect conclusions from analyses of climate change policy. In its framework for policy analysis, Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] estimates a simple measure of "willingness to pay" (WTP) that is the fraction of consumption w * (τ ) that an society would be willing to sacrifice, now and throughout the future, to ensure that any increase in temperature at a specific horizon H is limited to τ . Note that Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] has focused only on the "demand side" of climate policy. After a drawing on the current state of knowledge of global warming and its impact and the use of information on temperature change distributions based on scientific studies compiled by the IPCC, as well as information On the economic impacts of the recent "integrated assessment models" (IAMs), Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] focuses on the gamma distribution for these variables. It models the economic impact as a relationship between temperature variation and the growth rate of GDP as opposed to its level. This distinction is justified by theoretical and empirical considerations and implies that global warming can have a permanent impact on future GDP. It then examines whether reasonable values for the remaining parameters (eg, initial growth rate and risk aversion index) can give values of w * (τ ) greater than 2% or 3% for Low values of τ , which could favor a strict abatement. It also shows how w * (τ ) depends on the mean with respect to the standard deviation of the future temperature, and it calculates the iso-WTP-combinations of mean and standard deviations for which the WTP Is constant. This gives a further insight into how uncertainty drives WTP.

In the vein of Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] and some of his references, we address the problem of calculating the willingness to pay to avoid an increase in the temperature of the tau order. Note that Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] works under certain assumptions of which the two main ones we retain are that: 1) the variation of temperature over time follows an asymmetric gamma probability distribution, and 2) the society has a utility function of the CRRA type.

Our contribution is to say that a legal person, a society or a country does not necessarily have preferences in the face of climate change characterized by a type CRRA utility function. In fact, it is well-known that using CRRA framework imposes a specific restriction. At the end of the COP 21 in Paris, we noted that after the US presidential elections, the United States wanted to question its accession to the Paris agreement. It should be noted that more years ago, during previous conferences from Lima to Paris organized against climate change, several emerging countries (India, China, ... etc) were reluctant. This shows that climate change aversion can not easily be measured by constant risk aversion. With EU preferences, resistance to intertemporal substitution is constrained to be equal to relative risk-aversion. Note that resistance to substitution controls the attitude toward variations in consumption across time, under certainty. Also, risk-aversion controls the attitude toward variations in consumption across states of the world, at a given date. Moreover, we can mention that the modification of the curvature of the utility function makes substitution resistance and risk aversion move. As an alternative, we adopt the expo-power utility function of Saha Saha [START_REF] Saha | Expo-power utility: A 'flexible'form for absolute and relative risk aversion[END_REF], which is quite flexible and whose CRRA, DRRA, HRRA, CARRA, IRRA, DARA preferences can be identified according to the choice of the parameters. We arrive under our new assumption, all things equally otherwise as in Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF], highlighting a propensity to pay w * (τ ) which depends on the three parameters of Expo-Power utility function. . Usually, when an economic analysis of climate change policy is conducted, it is made in 5 steps2 which lead to reforms -even a stringent policy of abatement. In uncertainty, authorities sometimes face the dilemma of spending funds almost unnecessarily on programs of climatic reduction or do nothing and undergo the irreversible effects of climate change. In spite of this uncertainty (uncertainty on predictions or available forecasts), it remains nevertheless wise to adopt a perfect timing of preventive policy faced with irreversible risks. Although there are different views on the quickness in adopting a policy of reduction, (see, for example, for two period models,Kolstad [START_REF] Charles D Kolstad | Fundamental irreversibilities in stock externalities[END_REF] Fisher and Narain [START_REF] Anthony | Global warming, endogenous risk, and irreversibility[END_REF], and for multi-period models or with continuous time; Kolstad [START_REF] Charles D Kolstad | Learning and stock effects in environmental regulation: The case of greenhouse gas emissions[END_REF], Pindyck [START_REF] Robert S Pindyck | Irreversibilities and the timing of environmental policy[END_REF][START_REF] Robert S Pindyck | Optimal timing problems in environmental economics[END_REF] and Newell and Pizer [START_REF] Richard | Regulating stock externalities under uncertainty[END_REF]) for a consideration of irreversible effects, Pindyck [START_REF] Robert S Pindyck | Uncertainty in environmental economics[END_REF] undertakes a study of the interaction between uncertainty and irreversibility in climate change policy.

Another approach of economic analysis (instead of the 5 conventional stages) is to calculate a measure of the willingness to pay3 , like Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF], the fraction of consumption w * (τ ), that the society would be ready to sacrifice4 in order to ensure that any increase in temperature at a future time be limited at τ , this, besides helping us solely focus on the influence of uncertainty and the ramifications thereof, will save us from having to make estimations of future concentrations of gases and costs of abatement. This paper builds on the work of Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] and takes into consideration different classes of utility of economic agents. Pindyck considers social utility through a C.R.R.A utility function and thanks to the expo-power utility function of Saha [START_REF] Saha | Expo-power utility: A 'flexible'form for absolute and relative risk aversion[END_REF], we will consider, by a variation of parameters governing this function, all5 the utility range except C.R.R.A . Besides covering the other utility classes that a C.R.R.A omits, this function permits us to discern if under the other behavioral regimes of economic agents, the willingness to pay remains more affected by uncertain outcomes than certain outcomes when we vary the expectation and standard deviation of the temperature distribution. Throughout the dissertation, unless otherwise specified, we will have the following abbreviations:

Throughout this paper, we maintain the small-tailed gamma distributions of temperature and economic impact of Pindyck, not only because they hinder infinite future welfare losses (for an exponential utility function), but because it is easy to change some moments of the distribution (jointly or holding the others fixed) while studying how uncertainty influences the willingness to pay as explained in Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF].

We calculate the W.T.P using an exponential utility function. Taking δ=0 and g 0 =0.020, H=100, and time t going up to 500 years, we obtain values of w * (0) close to those obtained by Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] ; about 2.5% for the DARA/DRRA class, 0.3% for the DARA/IRRA class, 0.025% for the CARA/IRRA and 0.015% for IARA/IRRA classes. In the averse classes, an increase of the utility discount rate leads to an increase of the W.T.P, even more, we have the two scenarios in the DARA/IRRA class for an α parameterization threshold (β held constant). For the least averse class, DARA/DRRA, we have the opposite and an increase of the utility discount rate produces a reduction of the W.T.P. Uncertainty and climate policy; which, between uncertain and certain outcomes, have a greater influence on the W.T.P ?, to answer this question, we calculate the W.T.P for changes in expectation and the standard deviation of the temperature distribution (the parameter of economic impact held constant at its predicted value of 0.0001363). For the same value of W.T.P, we also calculate the corresponding expectation/standard deviation couple; as well, we calculate the sensitivity of the W.T.P with respect to changes in E(T ) and SD(T ) (∂w * /∂E(T ) and ∂w * /∂SD) and for all these different cases, treating expectation and standard deviation as economic goods, we calculate the marginal rate of substitution, M.R.S=-(∂w * /∂SD)/(∂w * /∂E(T )), to unveil social preferences.

In what follows, in section 2, we explain the methodology and its relationship with that of Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF]. Section 3 recalls the work of Pindyck and the inherent conclusions. Section 4 considers the work of Pindyck from another angle, changing the C.R.R.A utility function for the Expo-power of Saha [START_REF] Saha | Expo-power utility: A 'flexible'form for absolute and relative risk aversion[END_REF] and shows the values of W.T.P and its parameterization obtained for this change. Section 5 summarizes the findings and provides future prospects. In section 6, we give the conclusion of our paper.

Abbreviation

M eaning W.T.P Willingness to pay.

w * (τ ) fraction of consumption that society would be willing to sacrifice to ensure that any increase in temperature at a future point be limited to τ ; Also W.T.P.

E

L(T )

Loss function (of G.D.P due to temperature T.) .

Table 1.1: List of abbreviations.

Pindyck (2012) Background Revisited and methodology

We resume the modeling of Pindyck (see Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF],291 -293), and we introduce the exponential utility function:

U (C t ) = θ -e -βC α t , θ > 1, α = 0, β = 0, αβ > 0.
that we use to calculate the fraction of consumption w * (τ ), that society would be willing to sacrifice to ensure that any increase in temperature at a future point be limited to τ ; this fraction, w * (τ ), is a measure of the willingness to pay, W.T.P .

employing the consumption model of Pindyck,

C t = exp - 2γHT H ln(1/2) + (g 0 -2γT H )t + 2γHT H ln(1/2) (1/2) t/H . (2.0.1)
If, T H and γ were known, social-welfare would be given by;

W = ∞ 0 U (C t )e -δt dt. (2.0.2)
where,

U (C t )e -δt = θe -δt -e -δt-βexp - 2αγHT H ln(1/2) +(g 0 -2γT H )αt+ 2αγHT H ln(1/2) (1/2) t/H . (2.0.3)
If we assume that society sacrifices a fraction w * (τ ) of the present and future consumption to ensure that T H ≤ τ , with uncertainty at time t = 0, social welfare would be given by;

W 1 (τ ) = [θ -e -β(1-w * (τ )) α ]E 0,τ ∞ 0 U (C t )e -δt dt. (2.0.4)
where, E 0,τ is the conditional expectation at t = 0 using the distributions of T H and γ, conditioned by T H ≤ τ .

If no sacrifice of consumption is made, social welfare would be given by;

W 2 = [θ -e -β ]E 0 ∞ 0 U (C t )e -δt dt. (2.0.5)
where, E 0 like before, is the expectation at t = 0 using the distributions of T H and γ, but without any conditioning on T H . The W.T.P to keep T H ≤ τ , is calculated as the quantity w * (τ ), which will assert equivalence between W 1 (τ ) et W 2 .

The case of a C.R.R.A utility function: the work of R.S. Pindyck

Professor Robert S. Pindyck conducts a study of uncertainty in climate change policy, he demonstrates how the W.T.P is more affected by uncertain outcomes6 (standard-deviation instead of expected mean temperature from temperature distribution) and suggests that a better understanding of the uncertainty in temperature distributions and economic impact (in the I.P.C.C studies and others) could lead to stronger W.T.P predictions, more reliable and possibly a better climate policy. Throughout he uses a C.R.R.A utility function. In what follows, we resume the key points in Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] and later we will modify the C.R.R.A utility function.

According to his model7 , temperature follows;

T t = 2T H [1 -( 1 2 ) t H ] (2.1.1)
the G.D.P growth model is,

g t = g 0 -γT t (2.1.2)
or, by (2.1.1), we have,

g t = g 0 -γ2T H [1 -( 1 2 ) t H ] (2.1.3)
at the horizon H, the G.D.P is G.D.P H , but as we suppose slight losses (for a negative growth rate g t ), it is nevertheless updated and indeed is L(T )G.D.P H , where L(T ) is the loss function (such that at the horizon H, for time t, the percentage of G.D.P lost due to excess temperature T is 1-L(T )). Throughout we will use the loss function L(T ) = e -βT 2 and the consumption C t 8 , is given by,

C t = e t 0 g(s)ds = exp - 2γHT H ln(1/2) + (g 0 -2γT H )t + 2γHT H ln(1/2) (1/2) t/H . (2.1.4)
and the relation γ/β 9 becomes γ = ln( 2)

2ln(2)-1 βT H .
The utility function is

U (C t ) = C η t 1-η
, such that welfare is calculated as:

W = ∞ 0 U (C t )e -δt dt = 1 1 -η ∞ 0 e ω-ρt-ω( 1 2 ) t H dt (2.1.5) with ρ = (η -1)(g 0 -2γT H ) + δ (2.1.6) ω = 2(η -1)γHT H /ln(1/2) (2.1.7)
The I.P.C.C [START_REF] Solomon | Ipcc, climate change 2007: The physical science basis[END_REF] study foresaw temperature increases of 2 to 4.5 with an expectation of 3.0 • C, and the values predicted in more recent studies like the I.P.C.C [START_REF] Rajendra K Pachauri | Climate change 2014: Synthesis report[END_REF] maintain predictions of the same order due to reforms already in place to limit greenhouse gases. Weitzman [START_REF] Martin L Weitzman | Additive damages, fat-tailed climate dynamics, and uncertain discounting[END_REF], has percentile estimations of 17% probability that the doubling of CO 2 concentration would lead to a 4.5 • C growth of mean temperature, probability of 5% for a growth of ≥ 7.0 • C, and a probability of 1% for an increase of ≥ 10.0 • C. In using these 3 percentiles, Pindyck fits a shifted gamma temperature distribution.

This shifted gamma distribution, is the θ-shifted gamma distribution:

f (x; r, λ, θ) = λ r Γ(r) (x -θ) r-1 e -λ(x-θ) , x ≥ θ. (2.1.8) with Γ(r) = ∞ 0 s r-1 e -s
9 Ct = e t 0 g(s)ds = exp -2γHT H ln(1/2) + (g0 -2γTH )t + 2γHT H ln(1/2) (1/2) t/H but in fact, Ct = L(Tte t 0 g 0 ds , since L(T ) = e -βT 2 , at horizon H, CH = exp - 2γHTH ln(1/2) + (g0 -2γTH )H + γHTH ln(1/2) = e -βT 2 H e g 0 H
, which implies that γ = ln( 2) Now that we have the parameters for the temperature and economic-impact distributions, everything makes sense and (2.1.5) is well defined. W.T.P is calculated as the quantity w * (τ ) which will assert equivalence between the constrained welfare expectation (under a consumption reduced of w * (τ )%, temperature distribution censured 11 en τ and that of economic-impact unchanged) and the unconstrained welfare expectation (under total consumption, of 100%, and the distributions as estimated above). 11 we divide by τ θ T f (T )dT so that it remains a density.

2ln(2)-1 βT H .
We have, which implies that w * (τ ) is given by;

[1 -w * (τ )] 1-η 1 -η ∞ 0      1 F (τ ) τ θ T +∞ θγ e ω-ρt-ω( 1 2 ) t H f (T )g(γ) dT dγ      dt = 1 1 -η ∞ 0      +∞ θ T +∞ θγ e ω-ρt-ω( 1 2 ) t H f (T )g(γ) dT dγ      dt. (2.1.9) with F (τ ) =
w * (τ ) = 1 -       ∞ 0 +∞ θ T +∞ θγ e ω-ρt-ω( 1 2 ) t H f (T )g(γ) dT dγ dt ∞ 0 1 F (τ ) τ θ T +∞ θγ e ω-ρt-ω( 1 2 ) t H f (T )g(γ) dT dγ dt       1 1-η . (2.1.10)

Certainty on temperature and its economic impact

Supposing that temperature follows with certainty (2.1.1), and that the economic parameter is constant at its predicted mean of E(γ) = 0.0001363, we have;

[1 -w * (τ )] 1-η 1 -η ∞ 0 e ωτ -ρτ t-ωτ ( 1 2 ) t H dt = 1 1 -η ∞ 0 e ω-ρt-ω( 1 2 ) t H dt. (2.1.11)
where ρ and ω are given by (2.1.6) and (2.1.7), and ρ τ and ω τ (due to w * (τ )% of sacrificed consumption) are given by;

ρ τ = (η -1)(g 0 -2τ E(γ)) + δ (2.1.12) ω τ = 2τ H(η -1)E(γ)/ln(1/2) (2.1.13)
and the W.T.P is calculated as;

w * (τ ) = 1 -      ∞ 0 e ω-ρt-ω( 1 2 ) t H dt ∞ 0 e ωτ -ρτ t-ωτ ( 1 2 ) t H dt      1 1-η .
(2.1.14)

Uncertainty on temperature and its economic impact

In the case of uncertainty, we use the temperature distribution, f (T ), and economic-impact distribution g(γ), obtained higher to calculate the W.T.P as in (2.1.10). Temperature follows

T t = 2T H [1-( 1 2 )
t H ], varying the W.T.P for values of T H at the horizon H = 100, we have in the figure, a calculation of w * (0), the willingness to pay to keep T t nil, with the parameters; η = 2, g 0 = 0.015, 0.020, 0.025, δ = 0. W.T.P increases proportionally to T H and inversely to g 0 , which means that a smaller initial G.D.P growth rate would imply a greater W.T.P . According to Pindyck, the more g 0 is small, the more we get a low consumption discount rate, which will produce a greater W.T.P .

The case of uncertainty: W.T.P and τ .

In the case of uncertainty, everything is calculated as an expectation using the distributions of temperature and economic-impact, and the W.T.P is given by ( 2 According to the figure, consumers (with an index of risk aversion, η of 2, and with an initial growth-rate, g 0 of 0.020) would be ready to sacrifice about 2% of consumption to ensure that the increase of temperature be nil.

2.2.3

The case of uncertainty: W.T.P and η.

Figure 2.5 shows the relation between w * (3) and η. W.T.P is inversely proportional to η and δ, although under another temperature distribution with a higher mean, the W.T.P would be greater. We have the parameters; τ = 3, g 0 = 0.020, δ = 0 and 0.01. Similarly (but in this case, η 12 and not g 0 ) as above, the more η is small (nearing 1), we have greater W.T.P values, but as soon as δ increases (be it just of 0.01), W.T.P is considerably reduced to less than 2%. In Figure 2.6, the mean of the temperature distribution is increased to, E(T H ) = 5.0 • C, and we have, for the same parameters as before, greater W.T.P values.

Discussion and comments

To justify 13 his choice of the G.D.P growth rate in the modeling of the economic impact, taking into consideration the relation between temperature and the rate, instead of the growth level of the G.D.P, Pindyck in Pindyck [START_REF] Robert S Pindyck | Modeling the impact of warming in climate change economics[END_REF], conducts a comparative study of the two considerations, and obtains almost similar values of W.T.P . It could also be the case that the low weight of distribution tails underestimate the probability of catastrophic events (strong temperature increases beyond its expectation or great G.D.P losses beyond predictions of the economic impact distribution) and thus underestimating the W.T.P, but in Pindyck [START_REF] Robert S Pindyck | Fat tails, thin tails, and climate change policy[END_REF], Pindyck shows that so far as the marginal utility is bounded, these events disappear and a small-tailed distribution could produce greater W.T.P values than a thick-tailed distribution.

When he gets small W.T.P values (even for an utility discount rate of zero, δ=0.) This is due to the low weight of the distribution tails and the fact that the corresponding consumption discount rate (ηg 0 ) is also low, and losses at the horizon H=100 years, are very low in the early years. For the C.R.R.A agent, if we want to have strong W.T.P values, we must be limited to small values of the index of risk aversion (η<1.5), small G.D.P growth rate (g 0 <0.020) and a nil utility discount rate, δ=0.

For a variation of parameters 14 , Pindyck obtains values of w * (0) (the willingness to pay to limit the increase of temperature at 0) around 2%, and far below 2% (around 1%) in the case of limiting the increase of temperature to the predicted value of the I.P.C.C of 3 • C. The mean and the standard deviation of the temperature distribution vary, and one at a time (holding the other constant each time), he calculates the W.T.P . He also calculates the corresponding mean/standard deviation couple producing the same W.T.P .

All through, the marginal rate of substitution obtained is larger for small values of standard deviation and considerably larger means; which means that uncertainty on temperature (from the economic agents' point of view) is more important than certainty in terms of welfare; is a greater driving force of W.T.P, and thus of climate policy. A W.T.P value of 2% is small but does not mean that an abatement policy should not ensue; considering only 2% of the G.D.P for a country in terms of monetary value, if not 1% to limit the increase to its predicted value of 3 • C, this is a good investment.

The Model with the Saha Expo-Power Utility Function

Most economic agents do not exhibit a constant utility. The utility function we use is at the same time D.R.R.A, I.R.R.A, D.A.R.A, C.A.R.A and I.A.R.A (by a variation of parameters); except C.R.R.A 15 , and as such is a complement to the work of Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF]. We use the utility function introduced by Saha [START_REF] Saha | Expo-power utility: A 'flexible'form for absolute and relative risk aversion[END_REF],

U (C t ) = θ -e -βC α t , θ > 1, α = 0, β = 0, αβ > 0. (3.0.1)
We have, by a variation of parameters,

DRRA CRRA IRRA DARA α < 0,β < 0 | 0 < α < 1, β > 0 CARA | | α = 1,β > 0 IARA | | α > 1,β > 0 Table 3.1:
Classification of the exponential utility function in terms of α, β. . lets put, for U (C t )e -δt given by (2.0.3), 14 See Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF],pages 299 and 300. 15 We note that R (w) = A(w) + wA (w), and if our utility function is already C.A.R.A, we have A (w) = 0, and R (w) = A(w)= constant = 0. So an utility function cannot be at the same time C.A.R.A and C.R.R.A .

M τ (t) = 1 F (τ ) τ θ T +∞ θγ U (C t )e -δt f (T )g(γ) dT dγ. (3.0.2) M ∞ (t) = +∞ θ T +∞ θγ U (C t )e -δt f (T )g(γ) dT dγ. (3.0.3) with F (τ ) = τ θ T f (T )
dT , θ T , θ γ the lower limits of the distributions of temperature and economic-impact.

Proceeding like before, to find the willingness to pay,

[θ -e -β(1-w * (τ )) α ] ∞ 0 M τ (t)dt = [θ -e -β ] ∞ 0 M ∞ (t)dt [θ -e -β(1-w * (τ )) α ]G τ = [θ -e -β ]G ∞ . (3.0.4)
and the W.T.P is given by;

w * (τ ) = 1 -- 1 β ln θ -[θ -e -β ] G ∞ G τ 1/α . (3.0.5)

Certainty on temperature and its economic impact

In the case of certainty, temperature follows

T t = 2T H [1 -( 1 2 )
t H ], and the parameter of economic impact is constant at γ = E(γ) = 0.0001363. As a result of w * (τ )% of consumption sacrificed, at time t, at the horizon H (because T H ≤ τ ), C t takes the form,

C t = exp -2γHτ ln(1/2) + (g 0 -2γτ )t + 2γHτ ln(1/2) (1/2) t/H , put, ξ τ = 2γHτ ln(1/2) , ( 3.1.1
)

ζ τ = g 0 -2γτ. (3.1.2)
to calculate the W.T.P, we have;

[θ -e -β(1-w * (τ )) α ] ∞ 0 (θe -δt -e -δt-βexp{-αξτ +αζτ t+αξτ (1/2) t/H } )dt, = [θ -e -β ] ∞ 0 (θe -δt -e -δt-βexp{-αξ+αζt+αξ(1/2) t/H } )dt, [θ -e -β(1-w * (τ )) α ]I τ = [θ -e -β ]I T H . (3.1.3) with ξ = 2γHT H ln(1/2) , ζ = g 0 -2γT H under a full consumption regime.
We then have in this case that w * (τ ) is given by;

w * (τ ) = 1 -- 1 β ln θ -[θ -e -β ] I T H I τ 1/α . (3.1.4)

Uncertainty on temperature and its economic impact

In the case of uncertainty, like in Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF], everything is calculated as an expectation using the distributions of temperature and economic impact, and the W.T.P is given by (3.0.5) .

Influences on W.T.P : the case of certainty.

In the case of certainty, temperature follows

T t = 2T H [1 -( 1 2 )
t H ] and varying the W.T.P for values of T H at the horizon H = 100, we have in the figures which follow w * (0), the willingness to pay to keep T t nil according to the different utility classes. We vary the parameters α, β, θ 16 , g 0 and δ, and i study how these affect the W.T.P .

W.T.P and T H : the case D.A.R.A and D.R.R.A:

α < 0, β < 0.
This category represents the least risk averse agents, and with parameter values of α=-0.3, β=-3.55, θ=2.33, g 0 =0.015, 0.020, 0.025, δ=0 we have: We have that, the W.T.P remains proportional to temperature increases, such that at the horizon H, a greater value of T H would produce a greater willingness to pay; similarly, the W.T.P remains inversely proportional to the initial G.D.P growth rate, g 0 . For T H = 10 • C, g 0 = 0.015, Figure 3.1 tells us that, for the less averse person w * (0) = 7%.

W.T.P and T H : the case D.A.R.A and I.R.R.A:

0 < α < 1, β > 0.
This category represents the average risk averse agents, and with parameter values of α=0.42, β=1.2, θ=2.33, g 0 =0.015, 0.020, 0.025, δ=0 we have: The W.T.P remains proportional to temperature increases, and inversely proportional to the initial G.D.P growth rate. For T H = 10 • C, g 0 = 0.015, Figure 3.2 tells us that for the averagely less averse person w * (0) = 2%. This category represents the averagely more risk averse agents, and with parameter values of α=1, β=1.2, θ=2.33, g 0 =0.015, 0.020, 0.025, δ=0, we have:

W.T.P and T H : the case I.A.R.A and I.R.R.A:

α > 1, β > 0.
This category represents the most risk averse agents, and with parameter values of α=1.2, β=1.2, θ=2.33, g 0 =0.015, 0.020, 0.025, δ=0, we have that W.T.P remains proportional to temperature increases, and inversely proportional to the initial G.D.P growth rate. For T H = 10 • C, g 0 = 0.015, Figure 3.4 tells us that for the most averse individual w * (0) = 0.09%, and significantly below 1%.

Influences on W.T.P: the case of uncertainty.

In the case of uncertainty, temperature and the economic-impact parameters follow the distributions of Figure 2.1 and Figure 2.2 respectively. Varying the W.T.P for values of τ at the horizon H = 100, in the following charts, we have the calculation of w * (τ ), the willingness to pay to keep T H ≤ τ , for different values of τ . We vary the parameters α, β, g 0 and δ, and study how these affect the W.T.P. For parameters values of α=-0.2, -0.3, β=-3.55, θ=2.33, g 0 =0.019, 0.020, 0.021 and δ=0, we have: In Figure 3.5, for α=-0.3, β=-3.55 and g 0 =0.019, we have that the less averse individual would be willing to sacrifice w * (0) = 2.3%. If we take the values of α=-0.2 and g 0 =0.020, the value of w * (0) increases to 2.9%. We have that, for agents characterized by a DARA/DRRA utility, the W.T.P is proportional to α and inversely proportional to g 0 and τ . In fact, for the D.A.R.A/D.R.R.A class, w * (0) is terribly sensitive17 to a decrease in g 0 , and if g 0 falls to 0.018, w * (0) increases significantly to 16.4% in Figure 3.6.

W.T.P and τ : the case D.A.R.A and I.R.R.A:

0 < α < 1, β > 0.
For values of α=0.35, 0.42, β=1.2, θ=2.33, g 0 =0.019, 0.020, 0.021 and δ=0, we have: In Figure 3.7, with parameter values of α=0.42, β=1.2 and g 0 =0.019, w * (0)=0.3%, and if α is reduced to 0.35, for g 0 =0.020 and the same value of β, w * (0) inceases to 0.44%. We have, in this utility class, that the W.T.P is inversely proportional to α, τ and g 0 .

W.T.P and τ : the case C.A.R.A and I.R.R.A:

α = 1, β > 0.
For values of α=1, β=1.2, θ=2.33, g 0 =0.019, 0.020, 0.021 and δ=0, we have in Figure 3.8, for g 0 =0.020 and β=1.2, that w * (0)=0.025%. Since α is constant in this case, to know how the W.T.P behaves with respect to increases in α, we shall see, in the next figure, how for α > 1, the W.T.P decreases; So the W.T.P remains inversely proportional to α, τ and g 0 . For the variation with respect to β, we shall see later that W.T.P is also inversely proportional to β. In this case, we progressively fall into the most averse utility class, IARA/IRRA ; and we get the lowest values of W.T.P . For values of α=1.07, 1.2, β=1.2, θ=2.33, g 0 =0.019, 0.020, 0.021 and δ=0, we have: and we see in Figure 3.9 that if α is reduced from 1.2 to 1.07, with the same value of β, w * (0) increases from 0.015% to 0.018%. We therefore have in this case as seen above, that the W.T.P is inversely proportional to α, τ and g 0 . Above, in the DARA/DRRA frame, we said that the W.T.P is directly proportional to α, but this is not always the case. In fact, for some small values of α, the W.T.P is inversely proportional, and then there is bifurcation, and the W.T.P becomes directly proportional. In Figure 3.10, for a variation of values of α, with β constant at -3.55, g 0 =0.020, for δ=0, the W.T.P is inversely proportional to α until α=-0.31, then proportional thereafter (for δ=0.01, until α=-0.33, then proportional thereafter).

Moreover, for the case DARA/DRRA, just as in the C.R.R.A frame, an increase in the utility discount rate, δ, would reduce the W.T.P; we will see that this is not always the case for other utility classes.

If we increase E(T ) to 5, with the same parameter values, we can see in Figure 3.11 that the two curves of w * (3) have risen; from 3.4% to 6.4% for δ=0.01, α around -0.34 (for δ=0, α around -0.325). We see that an increase of E(T ) leads to an increase of w * (3). For the case DARA/IRRA, with E(T )=3, the averagely least averse class, we obtain values of w * (3) around 1%. For values of α in (0,1), with β constant at 1.2, g 0 =0.020, δ=0 and 0.01, we have: Figure 3.12: W.T.P and τ =3, the case D.A.R.A and I.R.R.A, β=1.2, θ=2.33, g 0 =0.020, δ=0, 0.01.

Unlike the case DARA/DRRA, here, the W.T.P is always inversely proportional to alpha and an increased utility discount rate, δ, can either increase or decrease the W.T.P; in Figure 3.12, although For the CARA/IRRA frame, with E(T )=3, the averagely more averse class, α is constant at 1 and β varies for positive values. For β varying in [0.2,1.2], g 0 =0.020, δ=0 et 0.01, we have: We find that the W.T.P is inversely proportional to β but directly proportional to δ because in Figure 3.14, an increase in the utility discount rate, δ, increases the W.T.P .

If we increase E(T ) to 5, with the same parameter values, we have: and likewise, we see that w * (3) is proportional to E(T ). When we take values of α>1, for β positive, we fall into the most averse utility class, IARA/IRRA, and we get the least values of w * (3) . For α varying in [1.001,1.8], g 0 =0.020, δ=0 et 0.01, we get: For this case we find that W.T.P is inversely proportional to α, and like before, directly proportional to δ because in Figure 3.16, an increase of the utility discount rate, δ, increases W.T.P .

If we increase18 E(T ) to 5 with the same parameter values, we get that w * (3) is proportional to E(T ).

W.T.P, E(T ) and SD(T ).

In order to discover between certain and uncertain events, which affect the most the W.T.P for each subclass, in what follows, we calculate the W.T.P for changes in expectation and standard deviation of temperature19 ; expectation, which is our grasp of certainty, and the standard deviation, of uncertainty, will vary one in turn (holding the other constant each time), and we shall observe the W.T.P produced. We shall also calculate, for the same value of W.T.P, the corresponding expectation/standard-deviation couple; as well, we shall calculate the sensitivity of the W.T.P with respect to changes in E(T ) and SD(T ) (∂w * /∂E(T ) and ∂w * /∂SD). For all these different cases, we shall calculate the marginal rate of substitution20 , M.R.S, between expectation/standard-deviation, treating expectation and standard-deviation as economic goods, and we would finally be able to decide which of the two influences the most the W.T.P of economic agents. In Table 3.2, we calculate w * (0) (the willingness to pay to prevent any increase of temperature) for changes in the mean and standard-deviation of temperature distribution. In the first part, the standard deviation is maintained at 2.12 and the mean varies, then it is the mean that is held constant at 3.0 and the standard deviation varies; finally, the third part shows combinations of mean and standard deviation which produce the same W.T.P of 0.0179. Apart from extreme values of E(T ) and SD(T ), ∂w * /∂E(T ) is contained in the range 0.0018-0.0064 and ∂w * /∂SD in 0.0022-0.0240, in such a way that the M.R.S lies between -1 and -3.75. We can see that the M.R.S oscillates between small and large values; in Figure 3.18 from -5.81 for (1.55,4.92) to -2.13 for (2.12,3.0) and -3.73 for (4.12,0.11)21 , incrementing the expectation of the temperature distribution to 4.0 • C, we have the iso-W.T.P curves in Figure 3.18 and we can see how the W.T.P is directly proportional to changes in the mean. The fact that the M.R.S is large in absolute value for small values of standard deviation shows us how major changes in the mean are equivalent in terms of welfare, to small changes in the standard deviation (The average doubles from 2.5 to 5, the standard deviation reduces only by 0.8 but still producing the same W.T.P of 0.0179). The top curve shows combinations of mean and standard deviation for which the W.T.P is 0.0221, this value of the W.T.P results in keeping the standard deviation at 2.12 • C, but incrementing the mean to 4 • C. We have, for E(T )=3 and parameter values of α=0.42, β=1.2, θ=2.33, g 0 =0.020 and δ=0, as before in Table 3 The fact that the M.R.S is large in absolute value for small values of the standard deviation shows us how major changes in the mean are equivalent, in terms of welfare, to small changes in the standard deviation (The mean doubles from 2.5 to 5, the standard deviation reduces only by 1.5 but producing the same W.T.P of 0.0027). We have the corresponding iso-W.T.P curve, For parameter values of α=1, β=1.2, θ=2.33, g 0 =0.020 and δ=0, we are in the CARA/IRRA class. In Table 3 For parameter values of α=1.2, β=1.2, θ=2.33, g 0 =0.020 and δ=0, we are in the IARA/IRRA class. In Table 3.5, ∂w * /∂E(T ) lies within -0.0016-0.0008 and ∂w * /∂SD within -0.0266-0.0341, in such a way that the M.R.S lies within -17 and -47. We can see occasional jumps in the M.R.S in Figure 3.21 for large values of standard deviation, which explains why sometimes averse people do not care about the certainty in the predictions of economic analysis. Although we get this result, in general the slope increases more significantly for small values of standard deviation, as above.

Discussion and Conclusions

We resume the work of Pindyck [START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] with the exception of the utility function and the ramifications generated from such a consideration. For a choice of conservative parameters 22 , we obtain values of w * (0) around 2.5% for the DARA/DRRA class, 0.3% for the DARA/IRRA class, 0.025% and 0.015% for the CARA/IRRA and IARA/IRRA classes. In the case of w * (3), limiting the temperature increase at the I.P.C.C's predicted value of 3 • C, these percentages are reduced; we get values of w * (3) around 2% for the DARA/DRRA class , 0.2% for the DARA/IRRA class, 0.020% and 0.007% for the CARA/IRRA and IARA/IRRA classes respectively.

After this analysis, the remarkable result obtained is that, unless we are in the least averse class, increased utility discount rate does not necessarily lead to a W.T.P reduction; in fact, in the averse classes23 , an increased utility discount rate increases the W.T.P, and much more, we have both scenarios in the DARA/IRRA class for an α parameterization threshold (β held constant).

It is rather intuitive that the less averse individual would sacrifice more and the more averse less, and this, without having to estimate the W.T.P in the different subclasses, gives a decreased W.T.P as aversion increases. Returning to the title of this paper, uncertainty and climate change policy; which, between uncertain and certain outcomes, have a greater influence on the W.T.P ?, the answer is uncertain outcomes. As Pindyck, we get decreasing iso-W.T.P curves with strong M.R.S values for small standard deviation values (and corresponding large means). Although there are occasional jumps of the M.R.S in averse classes for large standard deviation values (and corresponding small means), overall (as shown in the iso-W.T.P curves), the slope increases for small standard deviation values and corresponding large means, which is why we get strong M.R.S values in these regions. We can therefore conclude that, whatever be the class of aversion,uncertain outcomes affect more the W.T.P, are a greater enhancer of W.T.P, and thus, of climate policy. The W.T.P values obtained in this paper are consistent with a moderate climate policy. It is true that 2.5% of the G.D.P translates into a considerable monetary amount, but is justified if indeed we can limit any increase in temperature.

What proportion of consumption, if sacrificed, will be enough to limit future temperatures at a desired τ ?, or, what parameterization (for α, β and θ in the exponential utility function) is a perfect representation of the different aversion sub-classes ?. An answer to these questions will bring further clarifications to the modeling of this paper and lead to effective climate-policy reforms.

  Parameters giving index of risk aversion in the exponential utility function. W Social welfare. T Parameter of temperature distribution. γ Parameter of economic-impact distribution (temperature on G.D.P.) .

  ds. Obtaining r = 3.8, λ = 0.92, and θ = -1.13. Also, for the distribution of economic-impact (impact of temperature on G.D.P), we first find a distribution for β in the loss function L(T ) = e -βT 2 , and we use the relation γ = ln(2) 2ln(2)-1 βT H , to get the γ distribution. According to the I.P.C.C [11], global G.D.P losses average 1-5%, for a 4 • C warming, according to Dietz and Stern [15] 0.5-2% of G.D.P lost due to T = 3 • C, and 1-8% for T = 5 • C. Standing on 8 with C0 = 1.

Figure 2 . 1 :

 21 Figure 2.1: Temperature distribution (E(T ) = 3.0 • C, r = 3.8, λ = 0.92)

Figure 2 . 2 :

 22 Figure 2.2: Economic-impact distribution (E(γ) = 0.0001363, r = 4.5, λ = 21, 341)

10 γ = ln( 2 )

 2 2ln(2)-1 βT H , which implies that βT 2 = 2ln(2)-1 ln(2) HT γ. Since for L% of lost G.D.P, 1 -L = exp[-βT 2 ]= exp[-0.557HT γ], for H = 100, T = 4 • C, 0.97 = e -223.5E(γ) , 0.99 = e -223.5γ 1 , 0.95 = e -223.

  5γ

2

 2

  .

  )dT , θ T , θ γ the lower limits of the temperature and economic-impact distributions, ρ and ω as in (2.1.6) and (2.1.7).

Figure 2 . 3 : 2 . 2

 2322 Figure 2.3: W.T.P and temperature change T H , η = 2, g 0 = 0.015, 0.020, 0.025, δ = 0.

  .1.10). The following figure calculates W.T.P with the values; η = 1.5 et 2, g 0 = 0.015, 0.020, 0.025 and δ = 0. W.T.P is inversely proportional to τ , η and g 0 .

Figure 2 . 4 :

 24 Figure 2.4: W.T.P, T and γ uncertain, η = 1.5 and 2, g 0 = 0.015, 0.020, 0.025, δ = 0.

Figure 2 .

 2 Figure 2.5: W.T.P and η for τ = 3, g 0 = 0.020, δ = 0 and 0.01.

Figure 2 . 6 :

 26 Figure 2.6: W.T.P and η for τ = 3, E(T H ) = 5 • C, g 0 = 0.020, δ = 0 and 0.01.

Figure 3 .

 3 Figure 3.1: W.T.P and temperature change T H ,the case D.A.R.A and D.R.R.A, α=-0.3, β=-3.55, θ=2.33, g 0 =0.015, 0.020, 0.025, δ= 0.

Figure 3 . 2 :

 32 Figure 3.2: W.T.P and temperature change T H , the case D.A.R.A and I.R.R.A, α=0.42, β=1.2, θ=2.33 , g 0 =0.015, 0.020, 0.025, δ=0.

3. 3 . 3 W

 33 .T.P and T H : the case C.A.R.A and I.R.R.A: α = 1, β > 0.

Figure 3 . 3 :

 33 Figure 3.3: W.T.P and temperature change T H , the case C.A.R.A and I.R.R.A, α=1, β=1.2, θ=2.33 , g 0 =0.015, 0.020, 0.025, δ= 0. The W.T.P remains proportional to temperature increases, and inversely proportional to the initial G.D.P growth rate. For T H = 10 • C, g 0 = 0.015, Figure 3.3 tells us that for the averagely more averse individual w * (0) = 0.17%.

Figure 3 . 4 :

 34 Figure 3.4: W.T.P and temperature change T H , the case I.A.R.A and I.R.R.A, α=1.2, β=1.2, θ=2.33 , g 0 =0.015, 0.020, 0.025, δ=0.

Figure 3 . 5 :

 35 Figure 3.5: W.T.P and τ , the case D.A.R.A and D.R.R.A, α=-0.2, -0.3, β=-3.55, θ=2.33, g 0 =0.019, 0.020, 0.021, δ=0.

Figure 3 . 6 :

 36 Figure 3.6: W.T.P and τ , the case D.A.R.A and D.R.R.A, α=-0.3, β=-3.55, θ=2.33 , g 0 =0.01801, 0.01802, 0.01803, δ=0.

Figure 3 . 7 :

 37 Figure 3.7: W.T.P and τ , the case D.A.R.A and I.R.R.A, α=0.35, 0.42, β=1.2, θ=2.33, g 0 =0.019, 0.020, 0.021, δ=0.

Figure 3 . 8 :

 38 Figure 3.8: W.T.P and τ ,the case C.A.R.A and I.R.R.A, α=1, β=1.2, θ=2.33, g 0 =0.019, 0.020, 0.021, δ=0.

Figure 3 . 9 :

 39 Figure 3.9: W.T.P and τ , the case I.A.R.A and I.R.R.A, α=1.07, 1.2, β=1.2, θ=2.33, g 0 =0.019, 0.020, 0.021, δ=0.

3. 4

 4 .5 W.T.P and α: E(T )=3 and 5: the case D.A.R.A and D.R.R.A: α < 0, β < 0. For the DARA/DRRA class, with E(T )=3, the least averse class (if not at all), we get values of w * (3) close to a 100% for values of α<-0.35, for δ=0.01 (α<-0.33, for δ=0.).

Figure 3 .

 3 Figure 3.10: W.T.P and τ =3, the case D.A.R.A and D.R.R.A, β=-3.55, θ=2.33, g 0 =0.020, δ=0, 0.01.

Figure 3 .

 3 Figure 3.11: W.T.P and τ =3,the case D.A.R.A and D.R.R.A, E(T )=5, β=-3.55, θ=2.33, g 0 =0.020, δ=0, 0.01.

Figure 3 .

 3 Figure 3.13: W.T.P and τ =3, the case D.A.R.A and I.R.R.A, E(T )=5, β=1.2, θ=2.33, g 0 =0.020, δ=0, 0.01.

3. 4 . 7 W

 47 .T.P and α: E(T )=3 and 5: the case C.A.R.A and I.R.R.A: α = 1, β > 0.

Figure 3 .

 3 Figure 3.14: W.T.P and τ =3, the case C.A.R.A and I.R.R.A, α=1, θ=2.33, g 0 =0.020, δ=0, 0.01.

3. 4 . 8 W

 48 .T.P and α: E(T )=3 and 5: the case I.A.R.A and I.R.R.A: α > 1, β > 0.

Figure 3 .

 3 Figure 3.15: W.T.P and τ =3, the case C.A.R.A and I.R.R.A, E(T )=5, α=1, θ=2.33, g 0 =0.020, δ=0, 0.01.

Figure 3 .

 3 Figure 3.16: W.T.P and τ =3, the case I.A.R.A and I.R.R.A, β=1.2, θ=2.33, g 0 =0.020, δ=0, 0.01.

Figure 3 .

 3 Figure 3.17: W.T.P and τ =3, the case I.A.R.A and I.R.R.A, E(T )=5, β=1.2, θ=2.33, g 0 =0.020, δ=0, 0.01.

3. 4 .

 4 10 W.T.P, E(T ) and SD(T ): E(T )=3 and 4 : the case D.A.R.A and D.R.R.A: α < 0, β < 0. For values of E(T )=3, α=-0.3, β=-3.55, θ=2.33, g 0 =0.020 et δ=0, we are in the DARA/DRRA class.

Figure 3 . 18 :

 318 Figure 3.18: Iso-W.T.P curves, w * (0)=0.0179 (E(T )=3) and w * (0)=0.0221 (E(T )=4.0 • C); α=-0.3, β=-3.55, θ=2.33, g 0 =0.020 and δ=0.

3. 4 .

 4 12 W.T.P: E(T ) and SD(T ): E(T )=3: the case C.A.R.A and I.R.R.A: α = 1, β > 0.

Figure 3 .

 3 Figure 3.19: Iso-W.T.P curve, w * (0)=0.0027 (E(T )=3.0 • C); α=0.42, β=1.2, θ=2.33, g 0 =0.020 and δ=0.

Figure 3 .

 3 Figure 3.20: Iso-W.T.P curve, w * (0)=0.0002 (E(T )=3.0 • C); α=1, β=1.2, θ=2.33, g 0 =0.020 and δ=0.

3. 4 .

 4 13 W.T.P: E(T ) and SD(T ): E(T )=3: the case I.A.R.A and I.R.R.A: α > 1, β > 0.

Figure 3 .

 3 Figure 3.21: Iso-W.T.P curve, w * (0)=0.00018 (E(T )=3.0 • C); α=1.2, β=1.2, θ=2.33, g 0 =0.020 and δ=0.

  .3 results for the DARA/IRRA class. ∂w * /∂E(T ) lies within 0.0008-0.0134 and ∂w * /∂SD within 0.0002-0.0603, in such a way that the M.R.S should lie within -0.3 and -5. We can notice that the M.R.S oscillates between small and large values; in Figure3.19 from -16.81 for (1.54,4.78) to -2.27 for (2.12,3.0) and -12.27 for (3.81,0.67).

Table 3 . 3

 33

	1.2 • C	2.12 • C	0.0018	0.0026	0.0017	-0.67
	3.0 • C 2.12 • C 0.0027	0.0011	0.0026	-2.27
	5.2 • C	2.12 • C	0.0042	0.0017	0.0020	-1.15
	5.8 • C	2.12 • C	0.0046	0.0023	0.0023	-1
	3.0 • C	1.53 • C	0.0018	0.0134	0.0108	-0.81
	3.0 • C 2.12 • C 0.0027	0.0011	0.0026	-2.27
	3.0 • C 2.117 • C 0.0027	0.0011	0.0037	-3.30
	4.78 • C 1.54 • C	0.0027	0.0008	0.0145	-16.58
	4.8 • C	1.53 • C	0.0027	0.0019	0.0002	-0.14
	4.2 • C	1.66 • C	0.0027	0.0008	0.0037	-4.22
	3.0 • C 2.12 • C 0.0027	0.0011	0.0026	-2.27
	0.6 • C	3.85 • C	0.0027	0.0064	0.0414	-6.46
	0.67 • C 3.81 • C	0.0027	0.0049	0.0603	-12.27

.4, ∂w * /∂E(T ) lies within -0.0165-0.0055 and ∂w * /∂SD within -0.0876-0.0087, in such E(T ) SD(T ) w * (0) ∂w * /∂E(T ) ∂w * /∂SD M.R.S : W.T.P, changes in E(T ) and SD(T ). γ=E(γ)=0.0001363, E(T )=3. α=0.42, β=1.2, θ=2.33, g 0 =0.020 and δ=0.

Table 3 . 5

 35

	E(T ) ∂w 1.2 • C 2.12 • C SD(T ) w * (0) 0.0002 0.0008	0.0001	-0.13
	3.0 • C 2.12 • C 0.00018	0.0006	0.00009	-0.16
	2.0 • C 2.12 • C 0.00015	-0.0004	-0.0018	-4.39
	7.0 • C 2.12 • C	0.0003	-0.0006	-0.0008	-1.33
	3.0 • C 1.53 • C 0.00016	-0.0016	-0.0266	-16.31
	3.0 • C 2.12 • C 0.00018	0.0006	0.00009	-0.16
	3.0 • C 2.52 • C 0.00019	0.0008	0.0004	-0.62
	3.0 • C 2.30 • C 0.00019	0.0006	0.0001	-0.32
	3.57 • C 1.65 • C 0.00018	0.0008	0.0008	-9.125
	3.6 • C	1.6 • C	0.00018	0.00018	0.0006	-3.71
	1.8 • C 3.16 • C 0.00018	0.00018	0.0008	-4.29
	1.0 • C 3.75 • C 0.00018	0.0008	0.0341	-42.18
	3.0 • C 2.12 • C 0.00018	0.0006	0.00009	-0.16

* /∂E(T ) ∂w * /∂SD M.R.S : W.T.P, changes in E(T ) and SD(T ). γ=E(γ)=0.0001363, E(T )=3. α=1.2, β=1.2, θ=2.33, g 0 =0.020 and δ=0.

According to Pindyck[START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] : (1) Projections of future emissions of CO2 and equivalents (greenhouse gases) under a usual scenario of reduction policy, and estimations of future atmospheric concentrations of CO2. (2)Average or regional projections of temperature change which will result from strong concentrations of CO2. (3) Projections of G.D.P and consumption lost due to the high temperatures. (4) Estimations of the cost of greenhouse gas dejection for diverse concentrations. (5) Considerations on the social utility and the preferential time rate, in such a way of weighing the consumption lost by reduction face to benefits of consumption which will follow a reduced warming.

Throughout this paper will carry the abbreviation W.T.P .

That this proportion of consumption be enough to limit the future temperatures at τ is not the purpose of the analysis, but to discern the willingness of the agents.

C.A.R.A, D.A.R.A, I.A.R.A, D.R.R.A and I.R.R.A except C.R.R.A .

He generates a table of mean/standard-deviation couples which produce the same W.T.P, and even for high expected mean temperature, he finds the corresponding standard deviation low, altogether producing the same W.T.P, which demonstrates how certainty on the predicted mean temperature is more useful to society than the mean temperature in itself.

This model allows a doubling of temperature as t → ∞, which is in accordance with the I.P.C.C[START_REF] Solomon | Ipcc, climate change 2007: The physical science basis[END_REF][START_REF] Wilbanks | Impacts, Adaptation, and Vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF][START_REF]Mitigation of climate change. Summary for Policymakers[END_REF].

In Ramsey's growth model, the consumption discount rate is given by Rt = ηgt + δ, so a small G.D.P growth rate, g0 (or index η, for a given δ) will produce a small consumption discount rate and thus consumers would be more inclined to sacrifice a fraction of their present and future consumption.

This choice might underestimate G.D.P losses .

We keep θ constant at 2.33, since the exponential utility function is unique but for an affine transformation, the parameter θ does not affect the characterization of social utility classes.

if for example g0 decreases to 15% we have values of w * (0) close to 100%, which doesn't translate into a climate policy, reason why we took the initial value of g0 at 0.019, which is very close to its expected value of 0.020 .

Why increase the expectation value?: Because a recent study like Sokolov et al.[START_REF] Ap Sokolov | Probabilistic forecast for 21st century climate based on uncertainties in emissions (without policy) and climate parameters[END_REF], suggests expected temperature values around 4-5 • C at the horizon 2100.

The parameter for economic impact held constant at its expected value of 0.0001363, as in Pindyck[START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF] .

M.R.S=-(∂w * /∂SD)/(∂w * /∂E(T )) .

for T =4.0 • C, from -4.9 for (1.628,5.511) to -1.45 for (2.12,3.0) and -0.45 for (4.839,0.327).

We take δ=0 and g0=0.020 as in Pindyck[START_REF] Robert S Pindyck | Uncertain outcomes and climate change policy[END_REF].

DARA/IRRA, CARA/IRRA and IARA/IRRA.