PhD Brahim Selmaoui
email: brahim.selmaoui@ineris.fr

Soafara Andrianome

Rania Ghosn

René De Seze

Effect of acute exposure to radiofrequency electromagnetic fields emitted by a mobile phone (GSM 900 MHz) on electrodermal responsiveness in healthy human 1,2*

Keywords: Electrodermal activity (EDA), Galvanic skin response (GSR), Mobile phone, Skin conductance response (SCR)

Effect of acute exposure to radiofrequency electromagnetic fields emitted by a mobile phone (GSM 900 MHz) on electrodermal responsiveness in healthy human

Introduction

Mobile phones represent the most commonly used and studied source of electromagnetic fields (EMF) in recent decades. Due to the close proximity of the mobile phone to the head during its use, a particular concern has been raised with regard to the possible effects of EMF exposure on the brain.

Exploration of the autonomic nervous system (ANS) remains an interesting way to analyse the reaction of an organism to EMF exposure. Numerous provocation studies assessing the effects of EMF exposure (radiofrequency) on the physiology of organisms have been undertaken, and these typically explore brain electroencephalography (EEG), heart rate, heart rate variability (HRV) or respiration rate. Recording electrodermal activity (EDA), among other techniques, is a non-invasive and easy low-tech solution for measuring ANS changes. Electrodermal activity refers to changes in skin conductance on the surface, reflecting the autonomic innervations of eccrine sweat glands. Although EDA is also known as skin conductance (SC) or galvanic skin response (GSR), EDA is the currently used term. The EDA complex includes both background tonic (skin conductance level, SCL) and rapid phasic components (skin conductance responses, SCRs) that result from sympathetic neuronal activity.

As EDA is a sensitive index, it is also widely used in psychological studies [START_REF] Sequeira | Electrical autonomic correlates of emotion[END_REF][START_REF] Dawson | The skin conductance response, anticipation, and decision-making[END_REF]). An electrodermal response can be elicited by discrete stimuli, inducing rapid changes (phasic) that are in contrast to the slow variations associated with a basic level of EDA (tonic). Neuroimaging studies have shown that the basal measurement, represented by the SCL, varies in line with neural activity within the ventromedial prefrontal cortex and the orbitofrontal cortex, providing a task-independent representation of the autonomic state. On the other hand, several regions of the central nervous system were found to be activated for event-related activity or SCR [START_REF] Patterson | Task-independent functional brain activity correlation with skin conductance changes: an fMRI study[END_REF]Nagai et al., 2004).

Very few studies have examined EDA under mobile phone (MP) exposure; however, several studies on the effect of EMF on EEG have shown a consistent effect of MP on the EEG by increasing or decreasing alpha (8-13 Hz) power [START_REF] Reiser | The influence of electromagnetic fields on human brain activity[END_REF][START_REF] Croft | Acute mobile phone operation affects neural function in humans[END_REF][START_REF] Huber | Electromagnetic fields, such as those from mobile phones, alter regional cerebral blood flow and sleep and waking EEG[END_REF]Karamenko et al., 2003;[START_REF] Cook | Resting EEG is affected by exposure to a pulsed ELF magnetic field[END_REF][START_REF] Curcio | Is the brain influenced by a phone call? An EEG study of resting wakefulness[END_REF][START_REF] Maby | Short-term effects of GSM mobiles phones on spectral components of the human electroencephalogram[END_REF][START_REF] Perentos | Comparison of the effects of continuous and pulsed mobile phone like RF exposure on the human EEG[END_REF][START_REF] Regel | Pulsed radio frequency radiation affects cognitive performance and the waking electroencephalogram[END_REF][START_REF] Vecchio | Mobile phone emission modulates interhemispheric functional coupling of EEG alpha rhythms[END_REF][START_REF] Hinrikus | Effect of 7, 14 and 21 Hz modulated 450 MHz microwave radiation on human electroencephalographic rhythms[END_REF][START_REF] Vecchio | Mobile phone emission modulates event-related desynchronization of α rhythms and cognitive-motor performance in healthy humans[END_REF][START_REF] Perentos | The alpha band of the resting electroencephalogram under pulsed and continuous radio frequency exposures[END_REF][START_REF] Ghosn | Radiofrequency signal affects alpha band in resting electroencephalogram[END_REF][START_REF] Yang | Long-term evolution electromagnetic fields exposure modulates the resting state EEG on alpha and beta bands[END_REF].

With regard to EDA, one study showed a shorter latency of EDA under exposure to 900 MHz [START_REF] Esen | Effect of electromagnetic fields emitted by cellular phones on the latency of evoked electrodermal activity[END_REF]. In studies involving subjects hypersensitive to EMF, the results showed no effect of EMF exposure on the latency of the amplitude, the number of peaks of SCR [START_REF] Wilén | Psychophysiological tests and provocation of subjects with mobile phone related symptoms[END_REF] or on the level of skin conductance [START_REF] Eltiti | Short-term exposure to mobile phone base station signals does not affect cognitive functioning or physiological measures in individuals who report sensitivity to electromagnetic fields and controls[END_REF].

While these studies are informative for identifying the effects of EMF on general skin conductance, several techniques have been developed over recent decades to specifically analyse skin conductance [START_REF] Benedek | A continuous measure of phasic electrodermal activity[END_REF]. Development of mathematical models has enabled better reduction in noise and better estimation of SCR.

Using a mathematical-based model, we aimed to determine whether exposure to 26 min of EMF from a mobile phone with a GSM signal has an effect on numerous parameters of EDA, including slowly varying activity or skin conductance level (SCL) and the short-lasting skin conductance response (SCR), by analysing changes in several parameters (latency, amplitudes, the integral of SCR) in response to repeated auditory stimuli.

Materials and methods

Participants

Thirty participants were recruited for this study. Due to technical problems, data from two participants were excluded from the final analysis. Fourteen healthy males and 14 healthy females aged between 19 and 31 years (The mean age of participants was 24 ± 3 years) were enrolled in this study. All participants were healthy without disease. The average (mean ± SD) body mass index (BMI) of participants was 21.96 ± 2.1 kg/m², and systolic and diastolic blood pressure measures were 118 ± 8 and 72.8 ± 6.7 mmHg, respectively. There was no significant difference between male and female participants in terms of BMI (F = 2.074, not significant [ns]), blood pressure (F = 0.39, ns; F = 1.129, ns) and age (F = 3.089, ns). Participants were recruited through a website advertisement and were selected after a routine clinical examination. The inclusion criteria were regular sleep habits, the absence of medication, chronic disease, disability or acute illness, being a non-smoker, and the absence of neurological or psychiatric illness. Volunteers were asked to avoid eating or drinking alcohol and coffee, or any other stimulants 24 h prior to and during the experimental sessions. They were also instructed not to use a mobile phone for at least 2 h before the start of the experiment.

Participants received compensation, and the study was approved by the local ethics committee (CPP ref: N°ID-RCB: 2011-A01455-36).

Study design

Electrodermal activity, including SCR, was simultaneously recorded with electrocardiogram (ECG) and EEG parameters (results of the EEG were previously presented in [START_REF] Ghosn | Radiofrequency signal affects alpha band in resting electroencephalogram[END_REF] over two sessions in a randomised, double-blinded, counterbalanced study design. Neither the participant nor the researchers were aware of the exposure. Each subject was randomly assigned to receive two sessions of real or sham exposure. Each session included pre-exposure (run 1 and run 2), exposure (run 3, run 4, run 5) and post-exposure (run 6, run 7) periods. The total session duration was 61 min, including 26 min 15 s of real or sham exposure. Each session was separated by at least one week, and the order of exposure was counterbalanced. Each participant attended the two experimental sessions at the same time of the day to avoid any confounding effects related to circadian rhythm.

Exposure system

A commercial, dual-band, GSM mobile phone (Nokia 6650) was used for RF exposure. For the "real" exposure, participants were exposed to GSM-modulated exposure with the full power of the mobile phone (2W peak, 250 mW average, pulse modulated with 1/8 duty cycle) at 900 MHz for 26 min.

Maximum SAR, averaged for 10 g tissue, 1 g tissue or the peak value, was 0.49, 0.70 and 0.93 W/kg, respectively. The SARs during the "sham" exposure were <0.0001 W/kg. Before the exposure session, the mobile phone was placed on the left ear of the participants and secured using a tubular bandage (for more details see [START_REF] Ghosn | Effects of GSM 900 MHz on middle cerebral artery blood flow assessed by transcranial Doppler sonography[END_REF][START_REF] Ghosn | Radiofrequency signal affects alpha band in resting electroencephalogram[END_REF]. Skin temperature measured during the experiment in the right side (with no handset) did not significantly change in the cheek and under the ear. It was stable during sham and real exposure (about 24.5 °C).

Physiological measurements

The EDA and SCR were recorded using a BIOPAC MP 150 (Biopac systems, Goleta, CA, USA). Two Ag/AgCl electrodes were placed at the end of the second and third fingers of the participants' left hand (GSR 100C). The BIOPAC MP 150 was connected to a Dell laptop for acquisitions using Acqknowledge software. For all sessions, instructions were given through Omnistim at the beginning of the recordings. Omnistim also delivers a burst for SCR through loudspeakers, with signal tones of 60 dB and 1000 Hz lasting for 0.3 s. The interstimulus interval was 15 s. Each EDA recording had a duration of 2 min 45 s, and the inter-run duration was 6 min. Each stimulus delivered was recorded as a visual signal so that it could be synchronised with data for analysis. In order to obtain data of sufficient quality, participants were requested to take some sharp and deep breaths so that the quality of the response could be checked (positive control). The protocol for the EDA recording is shown in Figure 1.

Data analysis

The frequency of the sample was corrected and resampled from 10 to 1000 Hz. Artifacts were visually inspected, and a continuous decomposition analysis (CDA) was performed using Ledalab 3.3.4 [START_REF] Benedek | A continuous measure of phasic electrodermal activity[END_REF] for component analysis. In parallel, the software could also provide parameters using the trough to peak analysis method (TTP) and global measurement calculations. Skin conductance was first decomposed into tonic activity and phasic activity. The phasic and tonic amplitudes of each session for each run were recorded and analysed. The parameters of responses were then linked to an auditory stimulus, and the SCR studied was extracted using the CDA and TTP methods. This included a number of SCRs above the threshold, the latency of the response, SCR sum of amplitude and the integral of skin conductance response (ISCR). We defined the following parameters for SCR: a response window of 1 to 4 s, and a SCR threshold amplitude of 0.02 microSiemens (µS). The global measurement parameters were the global mean and global max deflection.

Statistical analysis

Statistical analysis was performed using R software and SPSS. After extraction of tonic and phasic activities, runs and sessions were analysed as factors and mean values for tonic and phasic activity during each run as the variables. For SCR-related parameters, the effect of habituation was analysed considering the event number as a factor. Then, two-way ANOVA was performed for each parameter, with the number of SCRs, latency, SCR, ISCR and global mean and global max deflection considered as variables, and runs and sessions considered as factors. A p-value <0.05 was considered statistically significant.

Results

Skin conductance (SC) and tonic (TA) and phasic activity (PA)

Tonic activity, extracted from skin conductance by CDA, was significantly different between the two sessions (F = 4.79, p < 0.0001), but was not significantly different between runs (F = 0.44, p = 0.9401; Figure 2a). Participants submitted to the "real" session of RF exposure had higher mean SC and TA during the session compared to those who received the "sham" session.

There was no difference between runs or sessions for PA, which was also extracted from the CDA, reflecting rapid changes in EDA (run: F = 1.04, p = 0.6756; session: F = 0.17, p = 0.4246; Figure 2b).

Proportion of significant responses (continuous decomposition method)

The number of significant responses or electrodermal responses, considering a peak detection threshold of 0.02 µS and a time window of 1 to 4 s, was determined for both sessions. A comparison of the proportion of responses to the auditory stimulus between sessions showed no difference between sessions (chi 2 = 2.283, p = 0.065), but there was a significant difference between runs (chi 2 = 35.144, p < 0.0001), independent of session (data not shown).

Habituation effect

Ten successive auditory stimuli were presented to the participants. All parameters were tested for a habituation effect using a block-test method considering the number of events as a factor. The majority of parameters were found to be influenced by the event number, except for latency which remained unchanged (Figures 3 and4).

Electromagnetic field effect

Parameters characterising SCR were analysed by two-way ANOVA with sessions and runs considered to be factors. The ANOVA for parameters determined by the three different methods is presented in Table 1. Differences between sessions were found using the TTP method (nSCR, latency and AmpSum), although this result was not confirmed by the CDA method (no effect of session; Figures 5 and6).

Discussion

The present study aimed to evaluate EDA, including SCR parameters, in response to brief auditory stimuli under exposure to RF EMFs. Electrodermal activity is a potential noninvasive marker of sympathetic activity as it can be used to evaluate changes in the function of the ANS. Most commonly, SCR is used as a peripheral index of a central state of sympathetic arousal [START_REF] Boucsein | Electrodermal Activity[END_REF][START_REF] Bach | Sympathetic nerve activity can be estimated from skin conductance responses -a comment on Henderson et al[END_REF]. Electrodermal acquisition is based on the innervations of the sweat glands by the sympathetic nervous system. Furthermore, sudomotor activity plays an important role in thermoregulation [START_REF] Wenger | Thermoregulation[END_REF] The method applied in this study, which involved the use of a model to analyse SCRs, allowed us to obtain less noisy results compared to conventional methods. Conventional analysis of SCR data consists of extraction of indices from an electrodermal recording using a method such as the peak detection method. These conventional methods are limited by the need for better temporal resolution, the need to reduce noise that cannot be attributed to sympathetic arousal, and the need for greater precision. In this model-based analysis, we used a mathematical model that considers the mechanism leading to a SCR, which results from sympathetic arousal and subsequent stimulation of the sudomotor nerve. A major problem with the conventional scoring technique is the difficult distinction between overlapping SCRs when there are short intervals between stimuli or between SCRs in quick succession. The decomposition procedure first proposed by [START_REF] Alexander | Separating individual skin conductance responses in a short interstimulus-interval paradigm[END_REF] was applied in this study. In order to remove between-subject variation, a z transformation is often advised. However, recent studies have shown that this transformation does not significantly improve the predictive validity of the data (Bach, 2014). More recently, newer model-based methods, similar to the model used in the present study, have shown higher sensitivity for distinguishing differences in sympathetic arousal between two different arousal responses under experimental conditions. This is due to the integrated removal of between-subject variance and methodological differences (Bach, 2014).

Our results showed no effect of RF EMF exposure for 26 min on EDA in healthy young people.

However, the tonic activity extracted from skin conductance measures was significantly different between the two sessions, but not different between runs. Participants had higher average SC and tonic activity during the real exposure session than the control session. These differences were seen before the exposure period as well as during the post-exposure period, meaning that these differences cannot be attributed to RF EMF exposure. Instead, they can be considered intraindividual variations from session to session. This difference is not due to diurnal variability [START_REF] Hot | Diurnal variations of tonic electrodermal activity[END_REF], as all experiments were performed at the same time of day.

With regard to latency, our results are not in accordance with those previously reported in a study by Esen and Esen, in which latency was found to be lengthened by approximately 200 ms under exposure to the EMFs of a mobile phone [START_REF] Esen | Effect of electromagnetic fields emitted by cellular phones on the latency of evoked electrodermal activity[END_REF]. It should be noted that there are differences in the methodological approach between our experiment and that of Esen and Esen. For example, the external stimulus in the present study consisted of an auditory stimulus, whereas Esen and Esen used a mechanical stimulus in the form of a "tap" on the patellar tendon. Furthermore, effects on electrodermal responses or SCRs are known to vary according to age, gender, ethnicity and skin physiology, and also depend on the nature of the stimuli and the environment of the experiment [START_REF] Boucsein | Electrodermal Activity[END_REF].

Another study by Johansson et al. on people with atopic dermatitis [START_REF] Johansson | No effect of mobile phone-like RF exposure on patients with atopic dermatitis[END_REF] reported no effect of exposure to a mobile phone-like RF in terms of the number of peaks per minute for the EDA.

In studies involving subjects who were hypersensitive to EMF, the results failed to show an effect of EMF exposure on average skin conductance [START_REF] Wilén | Psychophysiological tests and provocation of subjects with mobile phone related symptoms[END_REF][START_REF] Andrianome | An assessment of the autonomic nervous system in the electrohypersensitive population: a heart rate variability and skin conductance study[END_REF] or on EDA measured by the number of peaks per minute [START_REF] Eltiti | Short-term exposure to mobile phone base station signals does not affect cognitive functioning or physiological measures in individuals who report sensitivity to electromagnetic fields and controls[END_REF]. Moreover, a study on the effect of short-term exposure to magnetic fields did not show any effect on skin conductance, expressed as zscores, and further analyses showed that subjects showed different behavioural responses during exposure, ranging from reduced SC, increased SC or no change [START_REF] Stevens | Effects of 5 s exposures to a 50 microT, 20 Hz magnetic field on skin conductance and ratings of affect and arousal[END_REF].

In our study, we found that the number of SCRs was lower after placement of the mobile phone, regardless of whether exposure was "real" or "sham". This observation indicates that there is no specific activation or increase in sympathetic activity in response to exposure to the GSM signal or in response to the mobile phone. Furthermore, in response to successive identical stimuli, there was a significant difference between the first response and subsequent responses for all variables except for latency. Our data also showed a tendency toward a faster reduction in amplitude, which is in accordance with data in the literature [START_REF] Isen | Examining electrodermal hyporeactivity as a marker of externalizing psychopathology: a twin study[END_REF]. This is due, in part, to the habituation of subjects to the stimulus. Subjects become less responsive to an auditory stimulus under both conditions, as well as after the second presentation of the auditory stimulus. To analyse the overall responsiveness, the authors suggest that the number of SCRs in response to a series of stimuli, as shown above, are more reliable than studies of habituation [START_REF] Dawson | What does electrodermal activity tell us about prognosis in the schizophrenia spectrum?[END_REF].

Conclusions

The lack of effect of RF exposure on elicited SCRs and EDA indicates that sympathetic nervous activity, one of the main branches of the autonomic nervous system, was not affected by RF exposure, at least under the conditions of the present study.

The results of our study do not support the hypothesis that exposure to MP-like RF fields can affect the EDA system in healthy volunteers.

Figure 1 .

 1 Figure 1. Study protocol for physiological measurements during the pre-exposure, exposure and post-

Figure 2 .

 2 Figure 2. (a) Tonic activity (TA) during the sham and real sessions. Significant differences were

Figure 3 .

 3 Figure 3. Sum of amplitude of skin conductance responses (SCRs) for 10 consecutive auditory stimuli

Figure 4 .

 4 Figure 4. Latency of skin conductance responses (SCRs) during 10 successive auditory stimuli in

Figure 5 :

 5 Figure 5: Sum of amplitude of skin conductance responses (SCRs) for each run (from run 1 to run 7)

Figure 6 :

 6 Figure 6: Variations in latency of skin conductance responses (SCRs) for each run (from run 1 to run

  Figure 2 a)

Table 1 :

 1 Summary of analysis of variance Factors: Session and runs (run 1, run 2, run 3, run 4, run 5, run 6, run 7); variables: continuous decomposition analysis (CDA) method: nSCR, AmpSum, Latency, SCR, ISCR, trough to peak (TTP) method: n SCR; latency, AmpSum; global measurements: global mean, global max deflection. nSCR: number of significant skin conductance response, AmpSum: Sum of amplitudes of significant skin conductance response; SCR: Skin conductance response, ISCR: Integral of skin conductance response

	Session	Run	Interaction

Abbreviations: EDA: Electrodermal activity; EEG: Electroencephalogram; ECG: Electrocardiogram

Acknowledgements

This study was fully funded by Institut National de l'Environnement Industriel et des Risques (INERIS) and the program 190 'post-Grenelle' for the 'pôle applicatif en Toxicologie et Ecotoxicologie' of the French Ministry of Ecology and Sustainable Development. We would also like to thank Jean-François Lambert for his precious advice.

Declaration of interest statement

The authors report no conflicts of interest.


Louis-Gaëtan Giraudet
email: giraudet@centre-cired.fr

Céline Guivarch

Asymmetric impacts and over-provision of public goods

Keywords: D62, H41, Q54 public good, externalities, free riding, free driving, NIMBY, global warming

We elicit simple conditions for an old puzzle -over-provision of a public good. An asymmetric public good that benefits some contributors while harming others is subject to both free riding and free driving. Even though aggregate impacts are net positive, it can be over-provided if free drivers face provision costs that are sufficiently lower than free riders'. Asymmetric impacts further impose restrictions on Hicks-Kaldor improvements. We establish these results in a parsimonious model that can easily be applied to a variety of so-called NIMBY problems, for instance new public infrastructures and global warming mitigation.

Introduction

The ordinary prediction of the theory of public goods is that a non-rival, non-excludable commodity will be subject to free riding, hence under-provided in equilibrium [START_REF] Samuelson | The Pure Theory of Public Expenditure[END_REF][START_REF] Musgrave | Public finance in theory and practice[END_REF]. This prediction may fail in a variety of empiricallyrelevant, yet little-studied, contexts. In this note, we propose a simple model that provides a better understanding of the causes and welfare implications of an anomaly -over-provision of a public good, under-provision of a public bad -as well as a handy tool for its empirical identification.

Our model involves two key ingredients: heterogeneity in individual provision costs and, crucially, asymmetry in impacts from the public good. The latter is defining of so-called 'notin-my-backyard' (NIMBY) problems, in which some contributors benefit from the public good while others are harmed. The problem is typically posed by public infrastructures such as roads, airports, waste treatment plants, which locally generate aesthetic, air and noise pollution. One well-documented example is irrigation dams in India. [START_REF] Duflo | Dams[END_REF] show that while every farmer in the community may be willing to gain access to irrigation, the dam has side effects on agricultural production -negative upstream and positive downstream. Global warming is another important example. While virtually every individual on the planet enjoys energy services from the combustion of fossil fuels, it is increasingly documented that the ensuing increase in atmospheric temperature may produce benefits in high latitudes and damage in low latitudes, essentially through opposite effects on agriculture [START_REF] Mendelsohn | The Use of Cross-Sectional Analysis to Measure Climate Impacts on Agriculture: Theory and Evidence[END_REF] or heating and cooling expenditures [START_REF] Davis | Contribution of air conditioning adoption to future energy use under global warming[END_REF]. Heterogeneity in provision costs, the second ingredient of our model, is even more evident. Two communities differing in their exposure to impacts may well differ in other respects -wealth, technology endowment, etc. For instance, the contributors most vulnerable to global warming tend to be poorer, both within [START_REF] Hallegatte | Shock waves: managing the impacts of climate change on poverty[END_REF] and across countries [START_REF] Althor | Global mismatch between greenhouse gas emissions and the burden of climate change[END_REF].

Building on a parsimonious linear-state, two-player structure, our model produces two results. First, we find that when impacts are asymmetric, one player free rides in equilibrium while the other free drives. If the former player faces higher provision costs than her competitor, free driving can exceed free riding, thereby leading to excessive aggregate provision, even though impacts are net positive. Second, we find that under these conditions, net compensation, if needed to establish the social optimum, can flow from either player to the other. In the domain where aggregate provision is insufficient, in contrast, compensation systematically flows from the positively impacted player to the negatively impacted one.

Our contribution is two-fold. First, we make a connection between an early, somewhat forgotten literature on anomalous provisions of public goods and an emerging literature on free driving in greenhouse gas (GHG) emission abatement. The former was initiated by [START_REF] Buchanan | A note on public goods supply[END_REF] through a numerical counter-example involving immunization. Despite subsequent developments, 1 the anomaly-generating conditions remained poorly understood, let alone empirically investigated. More recently, several studies have pointed to geoengineering as a free-driving technology allowing countries to mitigate GHG emissions in excess of what would be privately optimal [START_REF] Weitzman | A Voting Architecture for the Governance of Free-Driver Externalities, with Application to Geoengineering[END_REF][START_REF] Manoussi | Cooperation and Competition in Climate Change Policies: Mitigation and Climate Engineering when Countries are Asymmetric[END_REF][START_REF] Heutel | Climate Engineering Economics[END_REF]. Our modeling structure allows us to reconcile both insights by characterizing in a simple manner how the interplay between free riding and free driving can cause anomalous provisions. Second, we propose a model flexible enough to be brought to the laboratory and the field in order to assess the empirical significance of anomalous provisions of public goods.

Model

Set-up

We consider provisions of private goods g ≥ 0 by two players, i and -i, which together produce a non-rival, non-excludable good G:

G = g i + g -i .
(1)

Player i derives some linear-quadratic value V i (g i ) from her provision of the private good:

V i (g i ) ≡ α i g i - β i 2 g i 2 ,
(2)

1 [START_REF] Williams | The Optimal Provision of Public Goods in a System of Local Government[END_REF] noted that the anomaly required some agents to over-contribute while others undercontribute, without further specifying the determinants of this dichotomy. [START_REF] Vincent | Reciprocal Externalities and Optimal Input and Output Levels[END_REF] pointed to economies of scale in external effects as a necessary condition for the anomaly, unless externalities have opposite signs -a condition deemed implausible by the author. [START_REF] Diamond | Aggregate Production with Consumption Externalities[END_REF] found sufficient conditions to rule out the anomaly which express own-consumption preferences and normality of demand. [START_REF] Buchholz | The overprovision anomaly of private public good supply[END_REF] later found these conditions to be too restrictive. Rather, they emphasized skewness in the distribution of incomes as a key determinant of the anomaly. Lastly, [START_REF] Sandmo | Anomaly and Stability in the Theory of Externalities[END_REF] noted coincidences between the anomaly and the notion of stability of the long-run aggregate demand function.

with parameter α i reflecting amenities and β i > 0 reflecting provision costs (which can be determined by wealth, technology endowment, etc.). In addition, she is impacted by the public good through an external effect X i (G):

X i (G) ≡ γ i G, (3)
with parameter γ i , of unspecified sign, reflecting individual (or local) impacts. The aggregate (or global) impact of the public good is γ i + γ -i . The players' utility functions are common knowledge and defined as:

U i (g i , G) ≡ V i (g i ) + X i (G). ( 4
)
The model has a very parsimonious structure, with separability of amenities, provision costs and impacts, and linearity of the latter. Importantly, we allow the sign of impact parameters γ to differ across players -an extreme form of heterogeneity we refer to as asymmetry. The problem is one of pure public good if g does not generate private amenities (∀i α i = 0) and of impure public good otherwise [START_REF] Cornes | Easy Riders, Joint Production, and Public Goods[END_REF][START_REF] Kotchen | Green Markets and Private Provision of Public Goods[END_REF].

In the case of a dam, g could reflect the services privately purchased from the dam (e.g., electricity, irrigation) and G the total capacity of the dam -a proxy of its impacts on agriculture, plausibly negative upstream and positive downstream. In the case of anthropogenic global warming, g could be a fossil fuel input to energy services (e.g., mobility, building weatherization) and G the build-up of greenhouse gases resulting from the combustion of fossil fuels -a proxy of atmospheric temperature, a modest rise of which plausibly has negative impacts in low latitudes and positive ones in high latitudes.

Taxonomy

Based on the sign and magnitude of impact parameters, we propose the following taxonomy of public goods (broadly speaking):

• γ i + γ -i > 0: G is a public good (strictly speaking). • γ i + γ -i < 0: G is a public bad. • γ i γ -i > 0: G is a universal public good or bad; -if γ i = γ -i , it is homogeneous, 4 -otherwise, it is heterogeneous. • γ i γ -i ≤ 0: G is a restricted public good or bad; -if γ i = -γ -i , it is symmetric, -otherwise, it is asymmetric.
While most analyses of impure public goods focus on the universal case, our more general model accommodates empirically relevant asymmetries in impacts.2 Figure 1 illustrates the different regimes in the (γ i , γ -i ) plane. In the remaining of the note, unless notified, we use 'public good' in its strict sense of a non-rival, non-excludable good generating positive impacts.

Predictions

Provisions

We start by comparing the Nash equilibrium of the game, N , to the social optimum, S. The former emerges through own-utility maximization and the latter through joint-utility maximization. First-order conditions lead to individual provision g N i = (α i + γ i )/β i in equilibrium and g S i = (α i + γ i + γ -i )/β i at the optimum. These provisions are positive as long as impacts are not too harmful (γ i > -α i and γ i + γ -i > -α i , respectively).

The players exert reciprocal externalities γ -i on one another (but internalize own-externalities γ i ). Player i is said to free ride, or under-contribute, if facing a positive externality, and free drive, or over-contribute, if facing a negative one (γ

-i > 0 ⇔ g N i < g S i )
. The provision of the aggregate G will be determined by impact and cost parameters:

Proposition 1 Provisions of the public good in the Nash equilibrium (N ) and the social optimum (S) compare as follows:

G N ≥ G S ⇔ β i γ i + β -i γ -i ≥ 0.
(5)

Remark 1 Let us call the equilibrium provision 'ordinary' if G S -G N is of the same sign as γ i + γ -i , and 'anomalous' otherwise. The provision is ordinary whenever impacts are universal (γ i γ -i > 0). Impacts need to be asymmetric (γ i γ -i < 0) and provision costs heterogeneous (β i = β -i ) for the anomaly to occur. These predictions do not depend on amenity parameters α i and hence equally apply to pure and impure public goods.

When impacts are asymmetric, one player free rides while the free drives. If provision costs β are identical, which effect dominates is directly given by the aggregate impact γ i +γ -i . Equilibrium provision is therefore ordinary. This implication no longer holds if relatively high impacts (in absolute value) are paired with relatively low provision costs. Assuming for instance γ i > 0, an asymmetric public good will be over-supplied if -(

β i /β -i ) -1 ≤ γ i /γ -i ≤ min(-1, -(β i /β -i ) -1
) and an asymmetric public bad will be under-supplied if

-1 ≤ γ i /γ -i ≤ max(-1, -(β i /β -i ) -1
). This is illustrated in Figure 2.

In the case of a dam, the upstream community will be willing to contribute too little to the construction of the dam while the downstream community will be willing to contribute too much. If, in addition, provision costs are lower downstream than upstream (perhaps because the downstream community is wealthier, or equipped with better technology), the dam will be over-sized; it will be under-sized with the reversed cost distribution. Likewise, global warming can be seen as an asymmetric public bad generating, at least in the short term, important damage in low latitudes and modest benefits in high latitudes. At odds with the ordinary prediction, our theory predicts that global warming could be under-supplied provided that emission reduction costs are lower in low latitudes than in high latitudes -a condition that however lacks empirical support. 3

Welfare

We now examine the welfare implications of moving from the Nash equilibrium to the social optimum. The reciprocal externalities can be internalized by a pair of prices (p i , p -i ) = (-γ -i , -γ i ) each imposed on every unit of the associated player's provision. 4 In a Pigovian 3 Emission reduction cost estimates are inherently difficult to obtain and therefore vary widely. Take India, a major emitter expected to severely suffer from global warming. According to some estimate, its emission reduction costs are among the lowest [START_REF] Nordhaus | Climate Clubs: Overcoming Free-Riding in International Climate Policy[END_REF] while another suggests it is among the highest [START_REF] Aldy | Comparing emissions mitigation efforts across countries[END_REF].

4 Player i's utility is then lessened by p i g i . Matching the ensuing equilibrium provision, (α i -

p i + γ i )/β i ,
with the socially optimal one yields p i = -γ -i . Unless the players face identical impacts (γ i = γ -i ), prices perspective, prices are implemented as domestic taxes and subsidies by a central authority.

In a Coasian perspective, implicit prices are self-imposed by cooperating players. Either way, prices may be accompanied by inter-player compensations, depending on conditions elicited below.

The variation of utility to player i is

U S i -U N i = -γ -i 2 /(2β i ) + γ i 2 /β -i . By construction, U S i +U S -i ≥ U N i +U N -i . Cooperation is said Pareto-improving if both U S i ≥ U N i and U S -i ≥ U N -i .
It is said Hicks-Kaldor-improving if only one of the latter two conditions holds, thus requiring the player whom cooperation makes better-off, say i, to transfer a positive amount τ to the worse-off player, say -i, such that

U N -i -U S -i < τ < U S i -U N i .
Working out the different cases, Proposition 2 Moving from equilibrium to the optimum is

• Pareto-improving if 1 √ 2β i /β -i ≤| γ i /γ -i |≤ 2 √ 2β i /β -i ,
• Hicks-Kaldor-improving otherwise. Specifically, -Player -i should compensate Player i if the left inequality is violated; -Player i should compensate Player -i if the right inequality is violated.

Remark 2 In the domain where the equilibrium provision of an asymmetric public good is ordinary (see Proposition 1), net compensation, if needed, systematically flows from the positively impacted contributor to the negatively impacted one (and vice versa with an asymmetric public bad). In all other regimes, net transfers can go either way.

To grasp the essence of the proposition and the associated remark, let us start by considering a public good provided by fully identical players, characterized by β i = β -i and γ i = γ -i > 0. Then, cooperation is Pareto-improving: it is in both player's interest to overcome the free-riding problem. This holds as long as cost and impact ratios are close to unity. If, now, Player i's marginal provision cost is less than half that of -i's, she free rides substantially less than her competitor, to a point where cooperation requires the former to compensate the latter. But if, with that same cost distribution, player i's impact is twice p i and p -i will differ.

that of -i's, the former is now the biggest free rider, and the direction of transfers is reversed. More generally, for any cost ratio, if transfers are needed, they can go either way, depending on the impact ratio; reciprocally, for any impact ratio, transfers can go either way, depending on the cost ratio. This is illustrated in Figure 3.

In asymmetric regimes, this two-way possibility is restricted to the domain of anomalous provisions. In the domain of ordinary provisions, in contrast, transfers only occur in one direction. Recall that for an asymmetric public good to be under-supplied, free riding must exceed free driving. Superimposing Figures 2 and3, we see that the domain where this is the case intersects with the domain in which cooperation either is Pareto-improving or requires i to compensate -i, but not with that commanding -i to compensate i. In contrast, in the domain where the public good is over-supplied, for free driving exceeds free riding, both directions of transfers are possible.

That compensation flows from those contributors exposed to positive impacts to those exposed to negative impacts might sound intuitive. Yet our analysis suggests this orientation should not be systematic. The disconnect between intuition and our prediction might stem from a natural tendency to characterize winners and losers in the light of external effects, instead of considering broader utility functions as economic reasoning would command.

Discussion

Our model offers a simple framework to better understand the causes and welfare implications of an economic anomaly -over-provision of a public good (and under-provision of a public bad). Its results are robust to pure and impure public-good regimes. It however rests on a parsimonious structure, of which we now question the generality and applicability. First, our model involves two players (or groups thereof) with marked differences in provision costs and impacts. In practice, the two-player setting seems relevant to the empirical analysis of NIMBY problems, which generally oppose two communities (e.g., upstream versus downstream, or central versus peripheral). It is less relevant to global warming, which involves a multitude of contributors that differ from one another in many other respects than their vulnerability to climate-change impacts. Still, our two-player model can be useful to study bilateral agreements between large GHG emitters, such as that between China and the United States, 5 which played an important role in the international negotiations leading to the Paris Agreement.

Second, the model features separable terms and linear impacts. Such a flexible structure conveniently lends itself to both empirical and experimental tests. Laboratory experiments testing the theory of public goods generally involve linear payoff functions and symmetric players. [START_REF] Dekel | Potential Pareto Public Goods[END_REF] have recently conducted novel experiments involving the kind of heterogeneity we consider in this note. Yet the authors do not examine the possibility of anomalous provision. Bringing our simple model to the lab could fill this gap. The model could also enrich empirical analyses with welfare assessment, for instance by eliciting whether, given the distribution of wealth upstream and downstream, dams such as those studied by [START_REF] Duflo | Dams[END_REF] are under-or over-sized. While those approaches seem fit for a variety of NIMBY problems, the model should be confined to the analysis of shortterm, moderate temperature variations when it comes to global warming, which is known to have non-linear impacts on economic productivity [START_REF] Burke | Global non-linear effect of temperature on economic production[END_REF].

Third, and relatedly, our results are established within a static framework. Again, this might be relevant to a variety of NIMBY problems but less so to global warming, an intrinsically dynamic problem. As long as the structure of the problem remains separable and linear, however, the predictions established in the static framework carry over to a dynamic framework [START_REF] Giraudet | Global warming as an asymmetric public bad[END_REF]. Introducing non-linearities would nevertheless raise new questions inherent in dynamic games.

Figure

  Figure 1: Taxonomy Notes: The origin corresponds to a pure private good.

Figure 2 :Figure 3 :

 23 Figure 2: Equilibrium provisions Notes: Assumption: γ i > 0. With γ i < 0, the picture would have a heterogeneous public bad on top, an asymmetric public good in the middle and an asymmetric public bad at the bottom. ANOMALY 1 corresponds to over-provision of a public good.ANOMALY 2 corresponds to under-provision of a public bad.

[START_REF] Dekel | Potential Pareto Public Goods[END_REF], to whom we borrow the term'universal,' use 'Potential Pareto Public Good' to designate what we refer to as 'asymmetric public good.' Our terminology is meant to avoid confusion between Pareto and Hicks-Kaldor improvements, which, as we show in Proposition 2, can both occur in asymmetric cases.

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