Yohan Sahraoui

Samy Youssoufi

Jean-Christophe Foltête

A COMPARISON OF IN SITU AND GIS LANDSCAPE METRICS FOR RESIDENTIAL SATISFACTION MODELING

Introduction

Residential satisfaction is an important subject of research in varied domains such as environmental psychology [START_REF] Hur | Neighborhood satisfaction, physical and perceived naturalness and openness[END_REF][START_REF] Amerigo | A Theoretical and Methodological Approach To the Study of Residential Satisfaction[END_REF], sociology [START_REF] Kahana | Person, Environment, and Person-Environment Fit as Influences on Residential Satisfaction of Elders[END_REF], and urban planning [START_REF] Ellis | Retail land use, neighborhood satisfaction and the urban forest: an investigation into the moderating and mediating effects of trees and shrubs[END_REF][START_REF] Kweon | Landscape components, land use, and neighborhood satisfaction[END_REF]. Residential satisfaction may be thought of as the match between an individual's ideal residential environment and certain attributes of her actual environment, including its psychological, social, and spatial components. For [START_REF] Amerigo | A Theoretical and Methodological Approach To the Study of Residential Satisfaction[END_REF] residential satisfaction arises from the evaluation of objective attributes of the residential environment through the filter of personal characteristics related to the individual's expectations, experiences, aspirations, or values. Each individual expresses a degree of residential satisfaction that is peculiar to her and that arises from the confrontation between a cognitive process and objective, physical, or social attributes of her environment [START_REF] Weidemann | A Conceptual Framework for Residential Satisfaction[END_REF]. However, some similarities exist in landscape preferences, explained by evolutionary theories such as biophilia (Wilson, 1984), restorative environments [START_REF] Ulrich | Visual landscapes and psychological well-being[END_REF], prospect-refuge theory [START_REF] Appleton | The Experience of Landscape[END_REF] or by the landscape preference matrix proposed by Kaplan and Kaplan (1989). These theories hold that landscape preferences are formed by the common evolutionary history of humans who react positively to landscape features that supposedly ensure their survival and improve their well-being.

Of the various components of an individual's residential satisfaction, satisfaction with the landscape around her residence has been much studied [START_REF] Hur | Factors That Influence Residents' Satisfaction With Neighborhoods[END_REF][START_REF] Kweon | Landscape components, land use, and neighborhood satisfaction[END_REF][START_REF] Lee | Relationship between landscape structure and neighborhood satisfaction in urbanized areas[END_REF]. Much work has been done on identifying the landscape criteria to which individuals are sensitive to varying degrees [START_REF] Cavailhès | GIS-Based hedonic pricing of landscape[END_REF][START_REF] Cheshire | On the Price of Land and the Value of Amenities[END_REF][START_REF] Cho | Amenity values of spatial configurations of forest landscapes over space and time in the Southern Appalachian Highlands[END_REF][START_REF] Kong | Using GIS and landscape metrics in the hedonic price modeling of the amenity value of urban green space: A case study in Jinan City, China[END_REF][START_REF] Lee | Relationship between landscape structure and neighborhood satisfaction in urbanized areas[END_REF][START_REF] Tyrväinen | The economic value of urban forest amenities: An application of the contingent valuation method[END_REF]. Generally research can be split into two main bodies depending on the objective pursued. The first body of work endeavors to better understand individuals' landscape preferences by referring to the monetary value individuals are ready to attribute to landscape [START_REF] Cavailhès | GIS-Based hedonic pricing of landscape[END_REF][START_REF] Cho | Spatial analysis of the amenity value of green open space[END_REF][START_REF] Gao | Effect of urban landscapes on land prices in two Japanese cities[END_REF][START_REF] Jim | Impacts of urban environmental elements on residential housing prices in Guangzhou (China)[END_REF][START_REF] Tyrväinen | The amenity value of the urban forest : an application of the hedonic pricing method[END_REF] or by focusing on the various cognitive processes at work in the human-environment relationship [START_REF] Kaplan | Environmental Preference: A Comparison of Four Domains of Predictors[END_REF][START_REF] Sevenant | Cognitive attributes and aesthetic preferences in assessment and differentiation of landscapes[END_REF][START_REF] Zube | Landscape Perception: Research, Application and Theory[END_REF]. The second body of studies addresses more operational issues by looking to include landscape in decisions about regional planning (Domingo-Santos, de Villarán, Rapp-Arrarás, & de Provens, 2011; [START_REF] Kaplan | Out in the country: Sprawl and the quest for nature nearby[END_REF], environmental management [START_REF] Bell | Landscape pattern, perception and visualisation in the visual management of forests[END_REF] or urban planning [START_REF] Lee | Relationship between landscape structure and neighborhood satisfaction in urbanized areas[END_REF]. This involves evaluating landscape quality.

There are two main approaches to evaluating landscape quality [START_REF] Daniel | Whither scenic beauty? Visual landscape quality assessment in the 21st century[END_REF][START_REF] Vouligny | An assessment of ordinary landscapes by an expert and by its residents: Landscape values in areas of intensive agricultural use[END_REF]. The first 'perceptive' approach is based on individual preferences and looks at landscape through the eye of the beholder [START_REF] Lothian | Landscape and the philosophy of aesthetics: is landscape quality inherent in the landscape or in the eye of the beholder?[END_REF]. The aim is usually to ask an individual what he thinks of a given landscape or a specific landscape criterion. These methods borrowed from the cognitive sciences can be used to explore the affective relationship between the individual and the landscape rather precisely by concentrating on the subject's different mental and psychological processes [START_REF] Daniel | Methodological Issues in the Assessment of Landscape Quality[END_REF][START_REF] Zube | Landscape Perception: Research, Application and Theory[END_REF]. However, they are limited by the necessarily subjective character of the evaluation of the landscape by individuals and so cannot be readily applied in a context of regional planning decision-making.

The second approach brings together what are called expert methods [START_REF] Daniel | Whither scenic beauty? Visual landscape quality assessment in the 21st century[END_REF]. They start from the postulate that a landscape can be evaluated objectively and that its intrinsic quality depends on various visual attributes. By this approach, landscape exhibits inherent qualities that can be evaluated as neutrally as possible, although a strict objectivity may be not totally reached. The approach generally entails making a descriptive inventory of the various components of the landscape and capturing them in the form of variables describing the landscape content, shapes, colors, diversity, texture, harmony, and coherence [START_REF] Vouligny | An assessment of ordinary landscapes by an expert and by its residents: Landscape values in areas of intensive agricultural use[END_REF]. This landscape description may be based on (1) a field approach whereby an expert attempts to describe the landscape as seen in situ, using a grid to evaluate the landscape [START_REF] Arriaza | Assessing the visual quality of rural landscapes[END_REF][START_REF] Otero Pastor | Landscape evaluation: comparison of evaluation methods in a region of Spain[END_REF][START_REF] Vouligny | An assessment of ordinary landscapes by an expert and by its residents: Landscape values in areas of intensive agricultural use[END_REF]; (2) a cartographic or GIS approach using digital modeling whereby landscape information is based on spatial data describing altitude and land use [START_REF] Bastian | Environmental amenities and agricultural land values: A hedonic model using geographic information systems data[END_REF][START_REF] Bishop | Assessment of visual qualities, impacts, and behaviours, in the landscape, by using measures of visibility[END_REF][START_REF] Cavailhès | GIS-Based hedonic pricing of landscape[END_REF]Domingo-Santos et al., 2011;[START_REF] Horst | A prototype method to map the potential visual-amenity benefits of new farm woodlands[END_REF]. The in situ approach provides a wealth of information but that is limited to the survey sites; by contrast, digital modeling can be used to evaluate the landscape systematically and continuously in space, the quality of the evaluation being closely tied to the precision of the initial data. Otero [START_REF] Otero Pastor | Landscape evaluation: comparison of evaluation methods in a region of Spain[END_REF] compare a cartographic and an in situ assessment method. They show that there are no significant differences between the two and that a cartographic method with accurate information can be used in landscape assessment to make savings compared with field observation.

Within the approaches based on spatial data, GIS visibility analyses are particularly relevant for landscape assessment (de la Fuente [START_REF] De La Fuente De Val | Relationship between landscape visual attributes and spatial pattern indices: A test study in Mediterranean-climate landscapes[END_REF][START_REF] Sang | Landscape metrics and visual topology in the analysis of landscape preference[END_REF]. Two main bodies of GIS computation methods exist for modeling the visible landscape: the sight-line method [START_REF] Fisher | Extending the Applicability of Viewsheds in Landscape Planning[END_REF]Joly, Brossard, Cavailhès, Hilal, Tourneux, Tritz, Wavresky, 2009) involves characterizing the visible landscape from a count of pixels seen from virtual viewpoints whereas the solid angles method (Domingo-Santos et al., 2011) uses trigonometric calculations to define the visual exposure of visible landscape elements [START_REF] Llobera | Extending GIS-based visual analysis: the concept of visualscapes[END_REF]. The latter method measures the visual salience of landscape characters on the retina of a potential observer.

There are currently no studies comparing approaches based on in situ metrics and GIS visibility metrics to explain residential satisfaction related to landscape. The purpose of this paper is to compare the capacity of each group of landscape metrics to explain residential satisfaction of individuals by: (1) a GIS-based approach that involves generating spatial visibility metrics; (2) an in situ approach based on collecting landscape information by means of a field audit. Our main hypothesis is that the field audit is expected to explain more about individuals' landscape satisfaction because its descriptors are qualitatively richer. If this hypothesis is confirmed, it shall call into question the value of GIS visibility metrics based on land cover data. Conversely, if GIS visibility metrics explain more about individuals' landscape satisfaction, they could be substituted with interest for in situ observations. If both approaches have similar explanatory powers, then we would need to examine whether the methods complement or duplicate one another. This study has been conducted in a suburban area near Besançon (eastern France). The choice of study area means we can re-examine the results of a residential survey reported in [START_REF] Youssoufi | Determining appropriate neighborhood shapes and sizes for modeling landscape satisfaction[END_REF].

Material and methods

Study area

Besançon is a medium-sized city of about 120 000 inhabitants around which a suburban ring has developed since the 1970s and now counts some 60 000 inhabitants. The study area chosen lies to the north-west of Besançon (Fig. 1). It is made up of a group of 11 administrative districts (communes) where substantial suburbanization has occurred. The relief is characterized by low limestone hills separated by marl depressions. Land cover is mostly woodland (40%), farmland (34%), and villages (14.5%) with varying degrees of detached suburban housing.

Measuring the degree of landscape satisfaction

The degree of landscape satisfaction was evaluated by a survey in June 2009 and presented in [START_REF] Youssoufi | Determining appropriate neighborhood shapes and sizes for modeling landscape satisfaction[END_REF]. Here we set out the most important points of the survey. A sample of 1057 people was stratified by population of the administrative districts. Each respondent answered a series of questions about their residential environment and was GISreferenced based on the postal address. Four main themes were covered by the survey: (1) evaluation of the surrounding landscape, (2) access to various urban amenities, (3) perception of the residential built environment in terms of aesthetics and density, and (4) evaluation of the individual and public transport conditions. Individuals were invited to respond to a series of assertions by saying whether they agreed with them.

A multiple correspondence analysis (MCA) was performed [START_REF] Tenenhaus | An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data[END_REF] to synthesize the responses in the form of factorial axes (see Appendix A and B for details). The interpretation made it possible to identify the first factor (the only one considered in the present article) as the landscape satisfaction gradient. As in [START_REF] Youssoufi | Determining appropriate neighborhood shapes and sizes for modeling landscape satisfaction[END_REF], the coordinate of individuals on this axis is the target variable of the analysis. The second factor concerns the level of satisfaction for accessibility to urban amenities.

In the present study, the constraints of the field survey for producing in situ metrics (see section 2.4.) mean that work has to be done on a subsample of individuals. A random draw of 60 individuals (5.7%) was made from the initial sample of 1057, maintaining the spatial stratification by administrative district. The Fig. 2 shows the level of satisfaction and dissatisfaction for these 60 individuals.

Field audit and in situ metrics

The field audit consisted in examining the landscape on the ground by a descriptive inventory of landscape scenes [START_REF] Arthur | Scenic assessment: An overview[END_REF]. This involved focusing on landscape elements that could be seen from the residential location of each of the 60 individuals in the sample, by authorizing movements of a few tens of meters along streets in the neighborhood. This survey was used to capture the landscape elements of the relevant residential zone using systematic field audit rules.

The in situ evaluation was conducted with a landscape analysis grid (Tab. 1) defined from the literature [START_REF] Otero Pastor | Landscape evaluation: comparison of evaluation methods in a region of Spain[END_REF][START_REF] Vouligny | An assessment of ordinary landscapes by an expert and by its residents: Landscape values in areas of intensive agricultural use[END_REF] and some variables that are not present in the literature but seemed to us appropriate to add after a preliminary field audit. It is the case for example of the distinction between the different forested elements (forest, isolated trees, copses, hedges, etc.) or to the composition of residential spaces such as walls and fences, advertising hoardings, etc. This field audit was conducted over five days in early autumn 2014 in identical weather conditions by two researchers together on all 60 individuals, to ensure inter-rater reliability by discussing the classifications and metrics for each case. The grid was composed of 32 metrics pertaining to landscape composition, configuration, and aesthetics and referring to the three themes set out below.

Natural elements and green areas (12 variables)

Previous studies noted the influence of natural elements and green areas on landscape satisfaction, such as for example agricultural areas (e.g. [START_REF] Hietala-Koivu | Agricultural landscape change : a case study in Yläne, southwest Finland[END_REF][START_REF] Howley | Landscape and Urban Planning Exploring public preferences for traditional farming landscapes[END_REF][START_REF] Lindemann-Matthies | Aesthetic preference for a Swiss alpine landscape: The impact of different agricultural land-use with different biodiversity[END_REF], vegetation [START_REF] Ellis | Retail land use, neighborhood satisfaction and the urban forest: an investigation into the moderating and mediating effects of trees and shrubs[END_REF], water [START_REF] Faggi | Water as an appreciated feature in the landscape: a comparison of residents' and visitors' preferences in Buenos Aires[END_REF] and gardens [START_REF] Kendal | Plants traits link people's plant preferences to the composition of their gardens[END_REF]. This description allowed for the features not included in the land-cover data presented such as copses or isolated trees, gardens, orchards, and kitchen gardens.

Residential spaces (11 variables)

These metrics were used to characterize the anthropic composition of landscapes by several elements such as the visual preponderance of built environments (Arriazza et al., 2004), the type of architecture characteristics, street and main roads, or the visual impact of more pointlike features. Under this heading, we also took into account the upkeep of residential spaces, which is noted as an important element affecting neighborhood satisfaction by [START_REF] Hur | Physical upkeep, perceived upkeep, fear of crime and neighborhood satisfaction[END_REF]. Most of the variables have been defined here in field audits so as to be adapted to the description of a suburban residential zone.

General configuration of the landscape (9 variables)

These metrics derived in part from theories of landscape aesthetics [START_REF] Ode | Capturing Landscape Visual Character Using Indicators: Touching Base with Landscape Aesthetic Theory[END_REF] provide the possibility of allowing for more general landscape atmospheres. The texture metric, for example, can be used to determine the diversity of depths of view of landscape scenery through the arrangement of landscape features. Coherence provides information in a complementary way about the level of harmony of landscape features composing the scenes. Metrics specifying the degree of openness of the landscape, the level of enclosure, and the general topography may also be important in establishing the overpowering or uplifting effects of landscape [START_REF] Rogge | Perception of rural landscapes in Flanders: Looking beyond aesthetics[END_REF][START_REF] Appleton | The Experience of Landscape[END_REF]. [START_REF] Stamps | Fractals, skylines, nature and beauty[END_REF] also showed the importance of the shape of the skyline on preferences.

A modality was attributed to each variable of each observation site surveyed making it possible (1) to determine the level of visual presence of landscape elements or (2) to characterize certain elements of landscape scenes.

Tab. 1. In situ metrics considered by the landscape analysis grid

Land cover data and GIS visibility metrics

The land-cover data were defined mainly from a vector data base (BD Topo IGN, 2009) used for characterizing buildings, artificial areas, transport infrastructures, forests, and water bodies. The BD Agreste (2010) was used for characterizing agricultural areas, by distinguishing grassland and cropland These data were digitized at 5 m resolution. The resulting map was composed of 14 land cover categories (Fig. 2). Two other data layers at 5 m resolution were used to compute visibility metrics: a digital elevation model (DEM) and a digital surface model (DSM). A 50 m spatial resolution DEM was provided by the French National Geographic Institute (IGN). This DEM was re-sampled and interpolated by the regularized splines method to give the same resolution as on the land cover data. The DSM was defined by attributing a given height to land-cover categories corresponding to objects that might hide the view. Accurate information about the height of each building was obtained from the BD Topo (IGN), while a random height between 15 m and 25 m was assigned to the forest pixels. A height of 0 was assigned to all other land-cover categories.

The visible landscape modeling was based on the solid angles method (Domingo-Santos et al., 2011) for quantifying the landscape visible on the retina of a potential observer. These analysis were performed with the PixScape software (see https://sourcesup.renater.fr/pixscape/en.html ).

From each residential location, the landscape was characterized by 22 metrics.

Composition metrics quantified each visible element from the 14 land-cover categories to characterize the visual content of landscapes [START_REF] Joly | A Quantitative Approach to the Visual Evaluation of Landscape Daniel[END_REF][START_REF] Miller | A method for estimating changes in the visibility of land cover[END_REF]. For a landscape category i, P(i) is the proportion of angular surface given by:

𝑃(𝑖) = 𝑎 𝑖 ∑ 𝑎 𝑗 𝑛 𝑗=1
where ai is the number of pixels corresponding to the category i and n is the number of landscape categories.

In addition, configuration metrics allowed to measure the diversity of the visible landscape (S), its degree of aggregation or fragmentation (CONTAG and IJI), its openness (DISTave), its texture (SD and DL), and the shape of the skyline (SL), which can be more or less rough according to the relief or the presence of elevated elements (see appendix C for details).

From the residential location of each of the 60 individuals, we defined a reticular environment of 500 meters, corresponding to a walkable neighborhood around residential locations. Within this neighborhood, the GIS visibility metrics were computed every 10 m, at a height of 1.75 m. The resulting values were finally averaged by residential location to represent the day-today landscape around each of them. The appendix D presents the descriptive statistics of all the variables.

Statistical analysis

Several statistical analyses were conducted to evaluate the effect of landscape elements on individuals' satisfaction. As the satisfaction was dependent on the combination of several landscape characteristics, we used multivariate statistical models in which landscape satisfaction was the target variable and indicators describing landscape were the explanatory variables. Two different models were put in place by taking account of the statistical nature of the metrics: (1) OLS multiple regression to evaluate the influence of GIS visibility metrics which are quantitative, and (2) multiple analysis of variance (MANOVA) to estimate the role played by in situ metrics which are qualitative. Both models were applied with a variableselection procedure based on minimizing the Akaike Information Criterion (AIC). This procedure was applied to all possible combinations including 1 to 10 explanatory variables.

Cross-classification analysis of metrics

The Moran index was calculated from a neighborhood of from 200m to 2000m every 200m for the residual variables derived from the two models, to check for any spatial autocorrelation, which would indicate a geographical gap in the explanatory variables.

To assess the shared contribution of each set of metrics, the common explained variance between the two models was quantified by calculating the r2 coefficient between the two series of predictions of individuals' satisfaction. Model residuals were then analyzed to identify locations where satisfaction is well explained by in situ metrics (weak MANOVA residuals) but less well explained by GIS visibility metrics (large residuals of OLS regression) or vice versa. To do this, each residual variable was discretized into three classes (strongly negative, around zero, and strongly positive) using the values -0.5 𝜎 and +0.5 𝜎 as class limits. The residual variables discretized in this way were then cross-classified so as to define nine cases. The more specific cases were subjected to a qualitative ground survey to find ways to better understand how the models deviated.

Results

Tab. 2 sets out the results of the statistical analyses. All the VIF values are less than 7 for each of the three models, proving the absence of collinearity among variables. The results are presented in two lots for each of the models.

Tab

Model based on GIS visibility metrics

The selection of the better model has led to the selection of seven explanatory variables among the 21 candidates. The explanatory power of the model is significant (r2 = 0.466; Adjusted R2 = 0.394; p < 0.001; AIC = -156.56). Two configuration metrics are selected: the texture (DL) which affects residential satisfaction positively and the contagion index which plays a negative role. Five composition metrics also selected are negative with respect to residential satisfaction: residential and retail buildings, agricultural areas (cropland and grassland), and water.

Model based on in situ metrics

Of the 32 in situ metrics, four have been selected from the MANOVA. They describe the presence of croplands, color contrasts, landscape openness, and general topography. The proportion of variance explained is lower overall than in the previous model (r2 = 0.463; Adjusted R2 = 0.379; p < 0.0001; AIC = -154.27). The absence or limited presence of croplands and landscapes with weak color contrasts influence individuals' satisfaction positively. Conversely, closed and overlooked landscapes affect satisfaction negatively.

Measuring spatial autocorrelations

The z-scores associated to the Moran index calculated at each step of distance are between -1.23 and 0.69 concerning the OLS regression, and between -0.36 and 0.38 concerning the MANOVA. The p-values calculated in each case are always higher than 0.1 concerning the OLS multiple regression, and always higher than 0.2 concerning the MANOVA. These values are not significant at the 5% level, which shows that no spatial autocorrelation of residuals occurs.

Cross-classification analysis

The common explained variance between the two models is low (10.4%) and leads us to explore the model residuals more closely.

The cross-classification analysis of residuals is dependent on the classification of residential locations in nine cases (Fig. 3). -Case D: residuals around zero in the in situ model and strongly negative in the GIS-derived model. This case applies to situations where a favorable feature identified in in situ metrics is absent from GIS visibility metrics.

-Case F: residuals around zero in the in situ model and strongly positive in the GIS-derived model. This case suggests that a landscape element described by the in situ variables (but overlooked in the GIS visibility metrics) unfavorably influences satisfaction.

-Case G: residual strongly negative for both models. This case suggests that a feature perceived as a nuisance in the residential environment is ignored in both approaches.

To illustrate the value of the crossing analysis, one location meeting the criteria D and F and G were randomly defined and surveyed in the field.

The residence representing the criterion D is located on a site providing a degree of intimacy, overlooking the surrounding residential space. The opposite situation is observed for the criterion F. The residential location is located on a very visible site overlooked by the surrounding residential space. These elements, not characterized by GIS visibility metrics, are well revealed by in situ metrics and appears to have an effect on landscape satisfaction. For the criterion G, residential location concerns a more built-up pattern, with the presence of an apartment building adjacent to an unused semi-public residential space lending an impersonal feel to this location. We also observed significant road traffic in the neighborhood.

Discussion

The purpose of this article was to compare two approaches of landscape evaluation for residential satisfaction modeling: (1) an approach based on landscape metrics collected in situ and (2) an approach based on the use of GIS visibility metrics constructed from spatial data.

To achieve this objective, the analysis consisted in explaining the residential satisfaction of a sample of 60 individuals by using the metrics provided by the two approaches as explanatory variables. Comparison of the explanatory powers of the models reveals a slight difference that GIS visibility metrics explain slightly better individual satisfaction (r2 = 0.466; Adjusted R2 = 0.394; AIC = -156.56) than in situ metrics (r2 = 0.463; Adjusted R2 = 0.379; AIC = -154.27). This result suggests that some of the information contained in the landscape and to which individuals are sensitive can be captured equally well by a GIS evaluation as by a field audit, thus infirming our first initial hypothesis. We can note that the sample of 60 individuals could contribute to reduce the reliability of the models compared to the initial number of 1057 individuals. However, we argue that this sample was a good tradeoff between the valuable time to the in situ analysis and the reliability of the models.

Regarding the metrics selected by the models, all composition metrics have a negative effect on landscape satisfaction. This is particularly true of the presence of water, which is counterintuitive given the many studies showing water to be the most attractive element of a landscape (e.g. [START_REF] Faggi | Water as an appreciated feature in the landscape: a comparison of residents' and visitors' preferences in Buenos Aires[END_REF][START_REF] Kaltenborn | Association between environmental value orientations and landscape preferences[END_REF], producing beneficial psychophysiological effects (e.g. [START_REF] Hartig | Restorative Effects of Natural Environment Experiences[END_REF][START_REF] Parsons | The potential influences of environmental perception on human health[END_REF] or influencing the amenity value of residential locations [START_REF] Mahan | Valuing urban wetlands: A property price approach[END_REF][START_REF] White | Influence of Natural Amenities on Residential Property Values in a Rural Setting[END_REF]. However, the water elements present in study area are not rivers and large expanses of water such as lakes. They are small streams whose unprepossessing character (lack of maintenance, unhealthiness) has been emphasized in the field. The negative effect of the view of the built elements appears consistent with [START_REF] Arriaza | Assessing the visual quality of rural landscapes[END_REF] and with studies about restorative environments claiming that the exposure to urban environments (in contrast to natural environments) has a negative role on affective states [START_REF] Ulrich | Visual landscapes and psychological well-being[END_REF] and attention restoration (Kaplan & Kaplan, 1989). In our study, agricultural areas also have a negative influence on landscape satisfaction, whether grassland or cropland. It exists contradictions in the literature about the landscape satisfaction associated to cropland and grassland. For example, [START_REF] Howley | Landscape and Urban Planning Exploring public preferences for traditional farming landscapes[END_REF], [START_REF] Stumse | Environmental attributes and the prediction of visual preferences for agrarian landscapes in Western Norway[END_REF] and Hietala-Koivu (1999) highlighted that grassland (or extensive agricultural landscapes) are generally preferred than cropland (or intensive agricultural areas), and Arriazza et al. ( 2008) claims the opposite idea proving in its study that people prefer intensive and homogeneous agricultural areas. Several studies focusing on grasslands have showed that their intrinsic characteristics influence the people preferences, for example the species richness or plants [START_REF] Lindemann-Matthies | Aesthetic preference for a Swiss alpine landscape: The impact of different agricultural land-use with different biodiversity[END_REF] or the presence of dispersed trees [START_REF] Ives | Values and attitudes of the urban public towards peri-urban agricultural land[END_REF]. In order to understand why in our case grasslands play a negative role on residential satisfaction, we think that more accurate and detailled information on their intrinsic composition and configuration could help to better explain it. The positive effect for people of landscape openness is recognized by many studies (e.g. [START_REF] Kaplan | The analysis of perception via preference: A strategy for studying how the environment is experienced[END_REF][START_REF] Rogge | Perception of rural landscapes in Flanders: Looking beyond aesthetics[END_REF], but depending on the type of landscape seen [START_REF] Coeterier | Dominant attributes in the perception and evaluation of the Dutch landscape[END_REF]. Along the same lines, the positive role played by the overlooking position of the residential location is consistent with the prospect-refuge theory [START_REF] Appleton | The Experience of Landscape[END_REF] explaining that people prefer places where they could have a wide view (prospect), while going unseen (refuge).

It is worth noting that the distances covered in the field around residential locations differ from the reticular distance for calculating GIS visibility metrics. Here, GIS calculations characterized the landscape within a relatively wide residential environment whereas field observations were limited to the near residential environment. Moreover, although the study bears on two ways of evaluating the same landscapes, the two groups of variables compared are not equivalent because certain variables are specific to the in situ approach. This is the case of the variables for aesthetics and the general configuration of the landscape, which seem to have an effect on satisfaction. To put this bias aside, the analysis could be supplemented by comparing variables that can be strictly transposed from the in situ to the spatial approach. For example, the forest variable that occurs in both groups has a better explanatory power in GIS visibility metrics (r2 = 0.139) than in in situ metrics (r2 = 0.071).

Conversely for openness, the variable computed from spatial data gives an r2 of 0.006 whereas the corresponding in situ variable brings an r2 of 0.086. However, all of equivalent variables in both approaches were selected only by one model (e.g. texture in the GIS evaluation, and openness in the in situ evaluation), confirming the complementarity of these approaches.

The return to the field on three residential locations (Fig. 4) allows us to better understand the limitations of GIS-visibility metrics. Locations D and F confirms the prospect-refuge theory of [START_REF] Appleton | The Experience of Landscape[END_REF]. It may therefore be assumed that the shortage of information in the model based on spatial metrics contributes to overestimating the individual's predicted satisfaction value. Location G features an apartment building, although such housing is very scarce in the suburban areas of the study area and was not included in the architectural characteristics in the analysis grid. Spatial data also failed to differentiate between houses and apartment buildings. Concerning the unused semi-public residential area, the feeling of appropriation and of place attachment of a residential area are essential features in the way individuals perceive and enjoy space [START_REF] Bonaiuto | Indexes of perceived residential environment quality and neighbourhood attachment in urban environments: a confirmation study on the city of Rome[END_REF]. However, noise pollution associated with road traffic in this residential area is a substantial explanatory factor of the degree of individuals' well-being and residential satisfaction [START_REF] Gidlöf-Gunnarsson | Noise and well-being in urban residential environments: The potential role of perceived availability to nearby green areas[END_REF][START_REF] Skånberg | Adverse health effects in relation to urban residential soundscapes[END_REF]. It is important here to recall that the in situ observations were made off the premises of the individuals surveyed and were limited to consideration of the quality of their private space and their level of privacy. Moreover, it should be observed at this stage that the share of uncertainty observed in the models may also be related to personal variations in landscape appreciation.

Fig. 4. Photographs of selected residential locations

These three cases attest to the need for closer consideration of land use rather than mere land cover. Landscape and the residential environment more generally cannot be reduced to their material character alone. They are also interpreted and meaningful in terms of the use of the objects that compose them and those uses may be perceived as amenities or disamenities.

Allowance for the quality of soundscapes of suburban areas might also be a way to improve models. To supplement this study, it would be instructive to test again the worth of GIS visibility metrics by using very high resolution data (e.g. Lidar) and by comparing the same variables over several levels of spatial resolution. The point that small landscape elements (pylons, power lines, advertising hoardings, walls, fences) that are not usually shown on land cover maps are not selected in the model based on in situ metrics is a further argument. However, this approach runs up against the problem of the cost of acquiring and processing very-high resolution data, which limits its potential generalization. In this perspective, it was said earlier that the use of high-resolution spatial data would make it possible to capture additional landscape features or to reveal a more complex landscape pattern. Another possibility would be to improve the descriptive quality of GIS visibility metrics, especially by transposing certain aesthetic landscape descriptors in the form of spatial indicators [START_REF] Ode | Capturing Landscape Visual Character Using Indicators: Touching Base with Landscape Aesthetic Theory[END_REF].

Ultimately, it is possible to discuss the reliability of residential satisfaction measures as response variables for the assessment of landscape quality. Indeed, surveys based on landscape photographs constitute another commonly used approach for landscape quality assessment [START_REF] Natori | Differences in rural landscape perceptions and preferences between farmers and naturalists[END_REF][START_REF] Wherrett | Creating Landscape Preference Models Using Internet Survey Techniques[END_REF]. But the advantage of residential satisfaction measures is that they take into account ordinary landscapes that are the living environment of people. However, the main shortcoming is that the landscape is only one of the components of residential satisfaction, and that taking into account other attributes of the residential area is likely to minimize the importance of the landscape. The same considerations may apply to econometric modeling based on housing values.

Conclusion

This study shows that residential satisfaction associated with landscape is explained just as well by GIS evaluation as by in situ evaluation. These results contradict our main hypothesis that richer and more subtle information of in situ metrics provides a greater explanatory power of residential satisfaction. However, they demonstrate the quality of the GIS visibility metrics created from spatial data, even if certain information about the landscape, such as that relating to landscape ambiance, can only be captured through field audits. The findings show, however, that each of the two approaches yields specific information that is complementary rather than redundant. One of the challenges of this study is to better explain satisfaction by refining spatial metrics. Several avenues have been contemplated for achieving this. Several ways could be followed to improve the assessment of individual satisfaction: the resort to high-resolution spatial data, the construction of metrics describing land use rather than land cover, and the development of metrics for visible landscape patterns to describe various aesthetic aspects of landscape. The aim is eventually to establish a method for evaluating landscape quality that can be generalized and could serve as a decision-making tool in regional planning and urban development. Even if our results are particular to a context of suburban areas, we argue that our protocol could be reproduced in other urban and landscape contexts with a step of calibration through conducting another residential satisfaction survey.

Fig. 1 .

 1 Fig. 1. Study area

Fig. 2 .

 2 Fig. 2. Landscape neighborhood satisfaction level. The satisfaction and dissatisfaction levels correspond to the coordinates of individuals on the first factor of the MCA. The satisfaction level corresponds to the positive values and the dissatisfaction level to the negative values.

Fig. 3 .

 3 Fig. 3. Cross-classification analysis of residuals. Residential locations are positioned on the scatter-plot given the residual values of each statistical model.

Appendix B. The assertions used to evaluate landscape satisfaction. F1 and F2 correspond to the coordinates of the assertions in the MCA factorial space.

		Assertions	MCA Factors F1 F2
		In your neighborhood, you feel that your home is
		completely hidden	-0.36	-0.45
		a little hidden	-0.14	0.03
		rather visible	-0.02	0.13
		highly visible	0.30	-0.07
		You think that the environment around your home is
		very rural	-0.37	-0.20
		rather rural	-0.01	0.33
	Assessment of daily landscape	rather urban very urban Around your home, you feel that the view is	0.66 0.18	0.05 -0.44
		completely blocked	0.75	0.04
		rather blocked	0.26	0.31
		rather unobstructed	-0.17	-0.09
		completely unobstructed	-0.56	-0.31
		Around your home
		you do not appreciate the landscape	0.82	0.24
		you rather appreciate the landscape	-0.08	0.04
		you appreciate the landscape a lot	-0.68	-0.29
		There are enough shops in your neighborhood
		Yes	0.09	-0.48
		No	-0.19	1.05
		Retail shops are within walking distance
		Yes	0.14	-0.45
	Accessibility to shops	No	-0.31	0.98
	and services	The opening hours of shops match your needs
		Yes	0.03	-0.18
		No	-0.26	1.52
		There are enough standard services in your neighborhood
		Yes	0.13	-0.54
		No	-0.22	0.87
		You feel that your neighbors are too close
		Yes	0.96	-0.18
		No	-0.35	0.06
		In your home, the proximity of the roads causes discomfort
		Yes	0.43	0.15
		No	-0.08	-0.02
		You enjoy the aesthetics of the houses in your neighborhood
	Perception of aesthetics and density of built surroundings	Yes No In your neighborhood, you feel that the houses are too close to each other -0.17 0.98 Yes 1.39	-0.08 0.27 0.07
		No	-0.38	-0.02
		From your home, some features spoil the view
		Yes	0.78	0.40
		No	-0.21	-0.10
		You think that there are too many houses in your neighborhood
		Yes	1.74	0.23
		No	-0.28	-0.04
		For your daily movements, the bus takes more time than the car
		Yes	0.05	0.08
		No	-0.04	-0.30
	Conditions of	The supply of public transport is sufficient
	transportation and daily	Yes	0.04	-0.27
	mobility	No	-0.01	0.18
		It is easy to move in the neighborhood by using soft modes (walking, cycling)
		Yes	-0.01	-0.06
		No	0.10	0.50

Appendix C. Details of configuration metrics computation

Metrics

Code Description Values

Skyline SL

Obtained by dividing the total length of the horizon by a straight line whose length corresponds to the width of the view.

Close to 1 when the horizon is planar, and highest (unbounded) in the case of a very jagged skyline.

Openness DISTave

Obtained by the mean of the length of sight lines.

Higher when the view is broader (in meters).

Contagion CONTAG

Based on pixel adjacencies in the image [START_REF] O 'neill | Indices of landscape pattern[END_REF]. It corresponds to the probability of finding a pixel of the landscape category i next to a pixel of the landscape category j.

Close to 100 if a same landscape category are clustered, close to 0 for images in which all pixels are dispersed.

Interspersion and Juxtaposition Index

IJI

Based on patch adjacencies of the landscape [START_REF] Mcgarigal | FRAGSTATS: spatial pattern analysis program for quantifying landscape structure[END_REF].

Each category is evaluated according to its adjacency with all other categories.

Close to 100 if no regularity is observed and close to 0 if landscape categories are arranged in a highly structured manner.

Shannon diversity index S

Based on the notion of entropy [START_REF] Shannon | The mathematical theory of communication[END_REF]. It is used to measure the diversity of the visible landscape.

Close to 0 if there is a unique landscape category, and close to 1 if a significant number of categories are present in equal proportions.

Shannon distance SD

Texture of the visible landscape, i.e. the greater or lesser variation of sight lengths, evaluated using DL (spatial variation of the view depth) and SD (normalized Shannon based on the ratio of sight lines in defined length classes). In the case of SD, five length classes were defined: less than 10 m, from 11 to 100 m, from 101 to 1000 m, from 1001 to 10 000 m, and more than 10 000 m.

Close to 0 when only a few variations in length are present and close to 1 when many variations occur.

Depthline DL


David Sohr
email: david.sohr@univ-lyon1.fr

Jens Ulrich Thomas

Stefan Skupin

Using Airy beams for combined glass cutting and edge shaping

Keywords: Airy beams, Borosilicate glass, Diffractive optical elements, Laser beam shaping, Laser materials processing, Numerical analysis

When glass is cut to shape, ultra short laser pulses with an elongated, usually straight, focal volume (line focus) can be used to modify the workpiece throughout its entire depth with one single laser shot. At the same time, processed glass is often required to have a seamed or round edge, which usually requires an extra grinding step. Alternatively, curved line foci can be used to combine cutting and edge shaping of glass sheets in one laser process. We reconsider the Airy-Gauss beam for this purpose. Plasma ignition in the side lobes of the Airy beam and surface damage provoke unwanted effects, in particular an asymmetric laser modification of the glass sheet. We provide numerical results on origins of the asymmetry of the volume modification and show that with rather simple optical adjustments a symmetric convex edge can be created in a 920 µm thick glass sheet.

INTRODUCTION

Ultra short laser pulses have been widely adapted as tool for industrial glass cutting throughout the past years. These cutting processes make use of the possibility of creating weakened zones within the glass volume along which the sheet can be separated in a subsequent process. The geometry of the weakened zone and the extent of damage that is created by a single laser pulse depends on the focal intensity distribution of the laser beam and on the choice of laser parameters for a given substrate. The focal intensity distribution of the optical Bessel beam is quasi invariant over propagation distances far greater than its transversal extent and exhibits a high contrast between the main lobe and the side lobes. For these reasons this beam has been widely studied and optimized and is probably the most commonly used one for glass cutting nowadays.

In recent work, a larger emphasis on the other hand has been put on focus geometries that produce a rounded or bevelled edge. [START_REF] Flamm | Ultrafast laser cutting of transparent materials: the trend towards tailored edges and curved surfaces[END_REF] Indeed, briefly after we demonstrated a rounded edge using an Airy-Gauss beam in single pass and an etching step for separation, 2 a mechanically cleaved edge was produced with a modified Airy-Gauss beam. [START_REF] Ungaro | Single-pass cutting of glass with a curved edge using ultrafast curving bessel beams and oblong airy beams[END_REF] Not long afterwards a strongly curved, etched edge was demonstrated employing a multispot approach for edge shaping. [START_REF] Flamm | Protecting the edge: Ultrafast laser modified c-shaped glass edges[END_REF] A characteristic feature of the non linear propagation of the unmodified Airy-Gauss beam in glass is the preferential energy deposition in front of the linear focus. [START_REF] Sohr | Shaping convex edges in borosilicate glass by single pass perforation with an airy beam[END_REF] Crucially this meant that we had to position the vertex of the parabolic Airy beam below the center of the glass sheet to be able to perforate the glass sheet completely, thus producing an asymmetric edge.

Here, we first present more numerical results that help to identify the origin of this preferential energy deposition. Subsequently we propose how a simple adjustment of the optical setup can be used to compensate for this effect of preferential absorption and demonstrate a curved, symmetric edge after etching.

PREFERENTIAL ABSORPTION

The asymmetric longitudinal damage distribution along the parabolic Airy trajectory 2 that is observed in experiments [also see Fig. 1 (a)] can be well reproduced in non linear simulations. Kerr self-focussing seems a likely candidate to explain such a shift towards the laser source, which increases with peak power. This effect is usually included in our non linear model. [START_REF] Bergé | Ultrashort filaments of light in weakly ionized, optically transparent media[END_REF] Simulations without the Kerr term however indicate, that while self-focussing leads to a slightly higher lateral confinement of the focal region for long pulse durations, it is otherwise negligible. The asymmetry and shift towards the laser source in the distribution of deposited energy along the propagation are practically exclusively caused by plasma effects. This explanation is corroborated by the observation, that the asymmetry in the simulated energy density strongly depends on the choice of the electron-electron collision time τ c , as shown in Fig. 1 (b). Typical literature values of τ c are in the low fs regime, [START_REF] Eppelt | Diagnostic and simulation of ps-laser glass cutting[END_REF] with a rather large uncertainty. In fact, already assuming a constant electron-electron collision time is a rather crude approximation. Nevertheless, the sensitivity of the deposited energy density on the collision time τ c in our model can be used to contrain this parameter. We compared the experimentally observed morphology of the Airy beam in combination with our non-linear simulations and found a rather good agreement for τ c = 20 fs.

SYMMETRIC EDGE WITH AN AIRY BEAM

In previous work, it has beem demonstrated that lateral offsets of the Gaussian input beam and the cubic phase mask can be used to reposition the Airy focal line . When the beam remains centered on the mirror plane of the 2D cubic phase, the symmetry of the focal distribution is conserved. To compensate for the effect of preferential absorption described in the section before, we adjust the Airy parabola by controlling the offset x 0 of the Gaussian input beam on the cubic phase mask (DOE) by adding a small tilt with a mirror that is placed early in the beam path of our previous optical setup. [START_REF] Sohr | Shaping convex edges in borosilicate glass by single pass perforation with an airy beam[END_REF] The modified setup is shown in Fig. 2. For our choice of parameters, the resulting tilt of the Airy parabola in the glass is in good approximation proportional to x 0 /f , where f is the focal length of the microscope objective (MO). By fully illuminating the DOE (9 mm clear aperture diameter) with a 10 mm diameter Gaussian input beam we increase the uniformity of the intensity distribution along the propagation compared to previous experiments. [START_REF] Sohr | Shaping convex edges in borosilicate glass by single pass perforation with an airy beam[END_REF] We use an increased burst energy of 300 µJ to achieve sufficiently strong modifications while the other laser parameters remain the same. For this set of parameters we find that a lateral offset x 0 of 350 µm compensates the asymmetric energy deposition. Such offset corresponds to a tilt of the Airy beam by 2 • . We place the resulting modifications, that have a length of ca. 800 µm, at the centered of a sheet of borosilicate glass (SCHOTT Borofloat 33) that is 920 µm thick. This way we avoid complications due to the lower damage threshold at the surface. [START_REF] Boling | Laser induced surface damage[END_REF] Additionally, we choose a pitch that is suffiently large to avoid overlap between neighbouring modifications, see Fig. 3.

In a subsequent etching step, uniform removal of the unmodified covers at both surfaces occurs before the increased etching rate of the laser modified glass 8 leads to a separation when the etching radius exceeds the modification pitch at each depth within the sample, see Fig. 4. The resulting convex edge is symmetric to the accuracy of the relative focus position within the glass.

CONCLUSIONS

The permanent modifications that are produced by a focussed Airy beam in borosilicate glass exhibit an asymmetric distribution of damage. This asymmetry increases with the pulse peak intensity and can be attributed to plasma ignition in the side lobes and a resulting shielding of the parts of the beam that feed the lower part of the focal line. This asymmetry can be compensated by tilting the Airy beam which we achieve by using an input beam with a lateral offset on the phase mask. By taking care to avoid unwanted surface or interaction effects we obtain a symmetric edge as a results of etching after Airy beam laser structuring of a 920 µm thick SCHOTT Borofloat 33 sheet.

Figure 1 .

 1 Figure 1. Longitudinal cross section of volume modification (a) for f = 20 mm, tp = 10 ps and Ep = 342 J with the ends and the maximum indicated by yellow dots. These characteristic points are also marked on the graphs of the energy loss or line density of deposited energy (b) for various pulse energies and two values of the assumed collision time. The error of determining the position on the microscope image is shown as a broad bar. The longer collision time τc = 20 fs yields good agreement, with the energy density at the ends being nearly constant across all pulse energies and the energy density maxima occuring at the positions of maximum damage.

Figure 2 .

 2 Figure 2. Optical setup as in Sohr et al., 2 using a diffractive optical element (DOE) and microscope objective (MO) to create the Airy-Gauss beam. A small tilt early in the optical path (d ≈ 3 m) effectively introduces a lateral offset x0 of the Gaussian input beam with respect to the DOE and MO. Note that the offset beam is still centered on the mirror plane of the cubic phase mask.

Figure 3 .Figure 4 .

 34 Figure 3. Lateral cross sections of the permanent modifications as used for glass cutting at three different depths within the glass. Positions of the cross sections within the glass are indicated on the left.


Yann Bramoullé
email: yann.bramoulle@univ-amu.fr

Christian Ghiglino

Loss Aversion and Conspicuous Consumption in Networks †

Keywords: Loss Aversion, Conspicuous Consumption, Social Networks

We introduce loss aversion into a model of conspicuous consumption in networks. Agents allocate their income between a standard good and a status good to maximize a Cobb-Douglas utility. Agents interact over a connected network and compare their status consumption to their neighbors' average consumption. Loss aversion has a profound impact. If loss aversion is large enough relative to income heterogeneity, a continuum of Nash equilibria appears and all agents consume the same quantity of status good. Otherwise, there is a unique Nash equilibrium and richest agents earn strict status gains while poorest agents earn strict status losses.

Introduction

We often compare ourselves to others. Following [START_REF] Veblen | The Theory of the Leisure Class: An Economic Study in the Evolution of Institutions[END_REF], a vast literature has documented the importance of conspicuous consumption. The consumption of goods like clothing and cars, appears to be driven, in part, by considerations of social status. 1 Comparisons are rooted in the social structure. People generally have different reference groups, which may include family, friends, and colleagues. Consumption in our reference group defines a reference level to which we compare our own consumption. Do we then react in symmetric ways to status gains and losses? The literature on loss aversion stresses the importance of asymmetries in the response to deviations from reference points. People appear to place more weight on relative losses than on relative gains. 2 The literature on conspicuous consumption has largely ignored loss aversion, however.

In this paper, we introduce loss aversion into the network model of conspicuous consumption of [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF]. Agents allocate their income between a standard good and a status good to maximize a Cobb-Douglas utility. Agents are embedded in a social network and compare their own status consumption to the average consumption among their network neighbors. Under loss aversion, status losses from a negative difference to the reference level are larger in magnitude than status gains from a positive difference of the same size. We analyze the network game induced by these social preferences.

We find that loss aversion has a profound impact on outcomes. Our main result establishes the existence of two mutually exclusive domains. In the conformism domain, there is a continuum of Nash equilibria where all agents consume the same quantity of status good.

This domain only appears under loss aversion. In the differentiated domain, there is a unique Nash equilibrium where agents consume different quantities of the status good. We provide a simple, explicit condition characterizing the boundary between these two domains and valid for any connected network. Indeterminate, conformist equilibria appear if and only if loss aversion is large enough relative to income heterogeneity.

1 See, e.g., [START_REF] Charles | Conspicuous consumption and race[END_REF]. 2 Kahneman and [START_REF] Kahneman | Prospect theory: an analysis of decision under risk[END_REF] introduced prospect theory to help explain decisions under risk, see also [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF]. A key ingredient in prospect theory is loss aversion with respect to a given reference point. Loss aversion also appears to play an important role in explaining riskless choices, see [START_REF] Thaler | Toward a positive theory of consumer choice[END_REF] and [START_REF] Barberis | Thirty years of prospect theory in economics: A review and assessment[END_REF] for a survey.

1 Intuitively, the kink in the utility functions induced by loss aversion means that agents are extra motivated to avoid status losses. They really want not to consume less than the average consumption of their network neighbors. When the comparison network is connected and income heterogeneity is not too high, the interplay of these motives across the network leads all agents to consume the same amount of status good. This pure conformism becomes untenable, however, when incomes are too different. In that case, we show that consumption of the status good for the richest agents always lies strictly above their reference levels, while it lies strictly below for the poorest agents.

An important first step in our analysis is to compute agents' best responses. Without loss aversion, the best response is linear in the average consumption among network neighbors with slope strictly lower than 1, see [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF]. By contrast, we show that with loss aversion, the best response is piecewise linear with three pieces. Crucially, we show the emergence of an intermediate domain where the best response is precisely equal to the reference level -and hence with a slope equal to 1. The best response is also continuous and strictly increasing overall, implying that the consumption game is supermodular.

In a last step, we assess the robustness of our results. We show that our main result extends to a setup with heterogeneity in concerns for status and in loss aversion. This heterogeneity tends to reduce the emergence of conformism. We then provide preliminary results, and conditions under which the emergence of a continuum of conformist equilibria is guaranteed, when agents maximize some general increasing and quasi-concave utility function. We find that agents may still act as pure conformists over some intermediate range.

Our analysis contributes to the literature on conspicuous consumption. Our model builds on the early literature on status games, as formalized by [START_REF] Frank | The demand for unobservable and other nonpositional goods[END_REF] and developed by [START_REF] Hopkins | Running to keep in the same place: Consumer choice as a game of status[END_REF]. As in these papers, we assume that agents allocate expenditures between two goods, a standard good and a status good. [START_REF] Frank | The demand for unobservable and other nonpositional goods[END_REF] and [START_REF] Hopkins | Running to keep in the same place: Consumer choice as a game of status[END_REF] assume that an agent cares about their rank in the overall consumption distribution, i.e., about the proportion of agents with a status consumption lying below own consumption. By contrast and following the literature on keeping up with the Joneses, we assume that utilities depend on a reference level of consumption. 3 We further assume that each agent may have a different reference group, defining a network of comparisons as in [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF]. We provide the first analysis of the impact of loss aversion and find that it can lead to pure conformism in status good consumption. Closer to our work, [START_REF] Friedman | Conspicuous consumption dynamics[END_REF] consider a continuum of identical consumers also allocating income between a standard good and a status good. In their framework, an agent cares about the difference between their status consumption and the status consumption of everyone else, with possibly different weights on positive and negative differences. By contrast, we tackle the more realistic -and more technically challenging -case of a finite society, where the action of one agent may have non-negligible impacts on others. Further, we characterize outcomes for any distribution of incomes and any connected network, without imposing the network to be complete. We show that pure conformism on status consumption only emerges when loss aversion is large enough relative to income heterogeneity.

Our analysis also contributes to the literature on games played on networks, surveyed in [START_REF] Bramoullé | Games played on networks[END_REF] and [START_REF] Jackson | Games on networks[END_REF]. To our knowledge, the only papers looking at status games on networks are [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF] and [START_REF] Immorlica | Social status in networks[END_REF]. [START_REF] Immorlica | Social status in networks[END_REF] consider a game where agents take a costly action which confers both private benefits and social status. The utility depends on a weighted sum of the differences between own action and the actions of neighbors taking a higher action. There is no weight on positive differences, an assumption akin to extreme loss aversion. 4 By contrast, we analyze the choices of consumers allocating income across two categories of goods. The relative weights on positive and negative differences can vary, covering situations of no loss aversion (equal weights), extreme loss aversion (zero weights on positive differences), and intermediate cases. We uncover the existence of two domains: one with a continuum of conformist equilibria, and one with a unique differentiated equilibrium.

As described above, we introduce loss aversion in the model of [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF]. 5

4 [START_REF] Immorlica | Social status in networks[END_REF] assume that an agent compares their action with the action of every other agent and then aggregates these pairwise comparisons by premultiplying with exogenous network weights. By contrast, comparisons operate through an network-based reference level in our framework. 5 Ghiglino and Goyal (2010) develop a general equilibrium analysis with exchange, endogenizing the price of the goods. By contrast, we consider exogenous prices, as in [START_REF] Frank | The demand for unobservable and other nonpositional goods[END_REF] and [START_REF] Hopkins | Running to keep in the same place: Consumer choice as a game of status[END_REF]. Leaving the offer unspecified allows the results to be relevant for general market structures. For example, production of the goods could be competitive, monopolistic, or include rigidities or incompleteness.

In the absence of loss aversion, best responses are linear and the equilibrium is unique. A remarkable implication of our main result is that equilibrium uniqueness is non-generic. Even an arbitrarily small wedge between the weights on negative and positive differences gives rise to a continuum of equilibria, and this holds even with arbitrarily small concerns about others' consumption. Loss aversion can lead to pure conformism in status consumption, a feature absent from [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF].

Finally, our analysis contributes to the literature on loss aversion. Introduced by Kahneman and [START_REF] Kahneman | Prospect theory: an analysis of decision under risk[END_REF] to analyze decisions under risk, loss aversion has also been proved useful to help explain riskless choices, see [START_REF] Thaler | Toward a positive theory of consumer choice[END_REF] and [START_REF] Barberis | Thirty years of prospect theory in economics: A review and assessment[END_REF] for a survey.

A central challenge in this literature is to develop a better understanding of the formation of reference points. 6 In a context of relative concerns, the ideas that agents have different reference groups and that an agent's reference level is built from consumption in their reference groups are very natural, and have empirical support. Across different contexts, people appear to compare themselves to their neighbors, family, and colleagues. 7 We provide the first analysis of the impact of loss aversion on conspicuous consumption in a network context.

The Model

We introduce loss aversion into a network model of conspicuous consumption. We consider a society of n consumers. Each agent i allocates her budget w i > 0 between the consumption of a standard good, x i ≥ 0, and of a status good, y i ≥ 0. The price of the standard good is normalized to 1 and let p denote the relative price of the status good. The budget constraint

is x i + py i ≤ w i .
Agents are embedded in a directed social network, describing comparison relationships.

Denote by N i the comparison group of agent i, of size |N i |. We consider a connected network; any agent can be reached from any other agent through an indirect path in the network. This 6 "The central idea in prospect theory is that people derive utility from "gains" and "losses" measured relative to a reference point. But in any given context, it is often unclear how to define precisely what a gain or loss is, not least because Kahneman and Tversky offered relatively little guidance on how the reference point is determined.", see [START_REF] Barberis | Thirty years of prospect theory in economics: A review and assessment[END_REF], p.178. For decisions under risk, [START_REF] Kőszegi | A model of reference-dependent preferences[END_REF] propose a framework where an agent's reference point is their expectation held in the recent past about outcomes.

7 See, e.g., [START_REF] Clark | Who compares to whom? the anatomy of income comparisons in europe[END_REF], [START_REF] Luttmer | Neighbors as negatives: Relative earnings and well-being[END_REF], [START_REF] Neumark | Relative income concerns and the rise in married women's employment[END_REF].

implies that no agent is socially isolated, ∀i, N i = ∅. Each agent compares his consumption of the status good, y i , to the average consumption in her comparison group, ȳi = j∈N i y j

|N i |
.

Denote by G the interaction matrix such that

g ij = 1 |N i | if j ∈ N i and g ij = 0 if j / ∈ N i . Thus, ȳi = j g ij y j .
Agents' preferences are described by the following Cobb-Douglas utility function, which depends on own consumption of the standard good and of own and peers' consumption of the status good.

u i (x i , y i , y -i ) = x σ i ϕ(y i , y -i ) 1-σ with ϕ(y i , y -i ) = y i + α + (y i -ȳi ) if y i ≥ ȳi ϕ(y i , y -i ) = y i + α -(y i -ȳi ) if y i ≤ ȳi
where σ ∈ (0, 1) represents the consumption elasticity of the standard good and α -≥ α + ≥ 0 capture how much agents compare their consumption of the status good to others' consumption. Note that this utility function is well-defined, and greater than or equal to zero, when ϕ(y i , y -i ) ≥ 0. For completeness, we assume that u i = -L < 0 when ϕ(y i , y -i ) < 0.

This means that agent i wants to consume at least α - 1+α -ȳi units of status good before starting to consume the standard good. This formulation nests well-known cases. When α -= α + = 0, there is no social comparison. Agents have standard Cobb-Douglas preferences and x i = σw i and y i = (1 -σ) w i p . When α -= α + = α > 0, there is social comparison without loss aversion. This is the benchmark case analyzed in [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF]. An agent's consumption depends on her peers' consumption, defining a simultaneous, complete information network game.

The budget constraint binds, implying x i = w i -py i . The utility as a function of status consumption only is u

i (y i , y -i ) = (w i -py i ) σ ((1 + α)y i -αȳ i ) 1-σ with y i ∈ [0, w i p ]. This yields, ∂u i ∂y i = (-pσ w i -py i + (1+α)(1-σ) (1+α)y i -αȳ i )u i . And ∂u i ∂y i = 0 ⇔ y i = σ α 1+α ȳi + (1 -σ) w i p when u i > 0.
When α is not too high, individual best response is linear and there exists a unique Nash equilibrium to the consumption game. Denote by I the identity matrix. The unique equilibrium is interior and equal to

y = 1 -σ p (I -σ α 1 + α G) -1 w
Under income homogeneity, in particular, this reduces to

y i = 1-σ 1-σ α 1+α w
p . An increase in the importance of social comparison, as measured by α, unambiguously increases the consumption of the status good and decreases the consumption of the standard good.

Our main contribution is to introduce the possibility of loss aversion in status seeking.

When α -> α + , the social losses from having a consumption level of the status good below peers' average are larger in absolute value than the social gains from having a status consumption above peers' average. This introduces a kink in the utility function, which is not differentiable around the reference level y i = ȳi . Our main objectives are then to characterize the Nash equilibria of the consumption game under social comparison and loss aversion, and to understand how loss aversion affects equilibrium behavior.

Results

We develop our analysis in several steps. We first derive the individual best response of an agent under loss aversion. We show that the best response is an increasing, piecewise linear function with three pieces. A key implication of loss aversion is to induce pure conformist behavior over an intermediate range. We then present our main result on Nash equilibria.

We uncover the existence of two mutually exclusive and qualitatively different domains. In a first domain, every agent plays the same action and there is a continuum of Nash equilibria.

In a second domain, there is a unique Nash equilibrium and agents play different actions.

Individual best response

As a preliminary remark, note that the budget constraint implies that y i ∈ [0, w i p ] and hence that ȳi ∈ [0, wi p ]. Therefore, an agent can afford the minimal level of consumption of status good for every possible consumption levels of her peers if and only if w i ≥ α - 1+α -wi and we maintain this assumption in what follows.

We next derive the individual best response in the consumption game. Denote by f i (y -i ) the best response of agent i, i.e., the solution to the problem of maximizing u i (y i , y

-i ) = (w i -py i ) σ ϕ(y i , y -i ) 1-σ under the constraint that y i ∈ [0, w i p ].
Note that u i is continuous, and hence admits a maximum on the compact interval [0, w i p ]. Further, we show that ln(u i ) is strictly concave on this interval's interior, and hence u i admits a unique maximum. Let

a -= α - 1+α -and a + = α + 1+α + , such that 0 ≤ a + ≤ a -≤ 1.
Proposition 1. The individual best response of agent i in the consumption game is equal to:

f i (y -i ) = σa + ȳi + (1 -σ) w i p if ȳi ≤ 1-σ 1-σa + w i p f i (y -i ) = ȳi if 1-σ 1-σa + w i p ≤ ȳi ≤ 1-σ 1-σa - w i p f i (y -i ) = σa -ȳi + (1 -σ) w i p if ȳi ≥ 1-σ 1-σa - w i p
Proof. We first show that ln(u i ) is strictly concave over ]0, w i p [. When i consumes at least the minimal amount of status good, we have:

ln(u i ) = σln(w i -py i ) + (1 -σ)ln((1 + α -)y i -α -ȳi ) if y i ≤ ȳi and ln(u i ) = σln(w i -py i ) + (1 -σ)ln((1 + α + )y i -α + ȳi ) if y i ≥ ȳi . This yields ∂ln(u i ) ∂y i = -pσ w i -py i + (1-σ)(1+α -) (1+α -)y i -α -ȳi if y i ≤ ȳi and ∂ln(u i ) ∂y i = -pσ w i -py i + (1-σ)(1+α + ) (1+α + )y i -α + ȳi if y i ≥ ȳi . Therefore, ∂ln(u i ) ∂y i
is continuous and strictly decreasing until y i reaches ȳi from the left and then, again, continuous and strictly decreasing when y i increases from ȳ+ i . Moreover, ln(u i ) is left and right differenntiable at y i = ȳi and

∂ln(u i ) ∂y i (ȳ - i ) = - pσ w i -pȳ i + (1 -σ)(1 + α -) ȳi > ∂ln(u i ) ∂y i (ȳ + i ) = - pσ w i -pȳ i + (1 -σ)(1 + α + ) ȳi
The left-derivative at y i = ȳi is larger than the right-derivative, and hence ln(u i ) is strictly concave.

Since ln(u i ) is a strictly concave function over ]0, w i p [, tends to -∞ at both extremes, and has a kink at ȳi , it has a unique interior maximum and there are two possible cases.

Either ∂ln(u i ) ∂y i = 0 and y i = ȳi . Or ∂ln(u i ) ∂y i (ȳ - i ) ≥ 0 and ∂ln(u i ) ∂y i (ȳ + i ) ≤ 0, and the maximum lies precisely at the kink, y i = ȳi . If y i < ȳi , then ∂ln(u i )

∂y i = 0 ⇒ y i = σa -ȳi + (1 -σ) w i p . This is a valid solution only if σa -ȳi + (1 -σ) w i p < ȳi . If y i > ȳi , then ∂ln(u i ) ∂y i = 0 ⇒ y i = σa + ȳi + (1 -σ) w i p . This is a valid slope 1 slope σa - slope σa + f i (y -i ) y̅ i (1-σ)w i /p
+ ȳi + (1 -σ) w i p > ȳi .
Otherwise, the maximum lies at the kink y i = ȳi .

We illustrate Proposition 1 in Figure 1. We depict how agent i's consumption of the status good y i depends on the average consumption among her peers, ȳi . Three domains appear. When the social reference level is low, the agent is in a domain of status gains. Her consumption level is linear in ȳi with slope σa + < 1. When the social reference level is high, the agent is in a domain of status losses. Her consumption level is also linear in ȳi with slope σa -< 1. The slope in the loss domain is higher than in the gain domain due to loss aversion, a -> a + . Note, also, that these two straight lines have the same intercept, (1-σ) w i p . Crucially, we see the emergence of an intermediate domain,

when ȳi ∈ [ 1-σ 1-σa + w i p , 1-σ 1-σa - w i p ].
In this domain, the agent behaves as a pure conformist and sets her consumption level equal to the social reference level, y i = ȳi . Intuitively, the agent in this domain can avoid social losses, but cannot afford social gains. This conformism domain only appears under loss aversion when α -> α + and its size increases when the wedge between social gains and social losses expands.

An important implication of Proposition 1 is that the best response of an agent is strictly increasing over her strategy space. This implies that the consumption game is supermodular. 8 A well-known consequence is that there exist a lowest and a highest Nash equilibrium, y min and y max , such that for any Nash equilibrium y, ∀i, y min i ≤ y i ≤ y max i . In addition, an increase in w i leads to a weak increase in the best response of agent i, and hence to a weak increase in the action of every agent in both the lowest and highest Nash equilibrium. We will be using these properties in the proof of our main Theorem below.

Nash equilibria

To provide some intuition for our main result, we show how to determine Nash equilibria graphically with two agents. We depict the best responses of the two agents in the same graph and under three scenarios in Figure 2. A profile is a Nash equilibrium iff it lies at the intersection of the two curves. In the upper panel, the two agents have equal incomes, w 1 = w 2 . We see that there is a continuum of Nash equilibrium, where both agents choose the same level of status good, and this continuum corresponds precisely to the conformist portions of the best-responses. In the middle panel, we assume that agent 2 is now richer than agent 1, w 2 > w 1 , and that the income difference is not too high. Agent 2's best response is now shifted upwards. We see that there is still a continuum of conformist Nash equilibria, corresponding to the portion of the 45 degree line which is common to both best-responses.

In the lower panel, we depict a case where agent 2 is now much richer than agent 1. Agent 2 best response is shifted upwards even further. There is now a unique Nash equilibrium, at the intersection of the domain of status gains for agent 2 and of status losses for agent 1. In this equilibrium, y 2 > y 1 , the richer agent earns strict status gains while the poorer agent earns strict status losses.

We can now state our main Theorem, which shows that the logic of this example extends to any connected network among n agents. Let w min and w max denote the minimal and maximal wealth levels among agents. A richest agent is an agent with wealth w max while a poorest agent has wealth w min . Say that agent i earns strict satus gains in Nash equilibrium y when ȳi < 1-σ 1-σa + w i p , and hence by Proposition 1, y i = σa + ȳi +(1-σ) w i p > ȳi . She earns status gains if ȳi ≤ 1-σ 1-σa + w i p , and hence y i ≥ ȳi . Similarly, agent i earns strict status losses when ȳi > 1-σ 1-σa - w i p and y i = σa -ȳi + (1 -σ) w i p < ȳi . She earns status losses when ȳi ≥ 1-σ 1-σa - w i p , and y i ≤ ȳi .

y i = y = ȳi is a best response when 1-σ 1-σa + w i p ≤ y ≤ 1-σ 1-σa - w i p . These inequalities hold since 1 -σ 1 -σa + w i p ≤ 1 -σ 1 -σa + w max p ≤ y ≤ 1 -σ 1 -σa - w min p ≤ 1 -σ 1 -σa - w i p
This shows that the conformist profiles described in the first part of the Theorem are indeed Nash equilibria.

(1.2) Let us show that these are the only Nash equilibria in this domain. Recall, y min and y max denote the lowest and highest Nash equilibria of the game. By (1.1) we know that ∀i, y min i ≤ 1-σ 1-σa + wmax p . This implies that ȳmin i ≤ 1-σ 1-σa + wmax p . Since by assumption

1-σ 1-σa + wmax p ≤ 1-σ 1-σa - w min p , we have ȳmin i ≤ 1-σ 1-σa - w i p .
By Proposition 1, this implies that no agent is in the domain of strict social losses and hence ∀i, y min i ≥ ȳmin i . We can then invoke the following elementary graph-theoretic property. Consider a directed, connected network such that ∀i, y i ≥ ȳi . Then ∀i, y i = y.

To see why, let i 0 be an agent with lowest value of y i . By assumption, y i 0 ≥ ȳi 0 . However,

ȳi 0 = j∈N i 0 y j |N i 0 |
and since y j ≥ y i 0 , ȳi 0 ≥ y i 0 . Therefore, y j = y i 0 , for every j ∈ N i 0 . Apply the same argument to the neighbors of the neighbors of i 0 . Then, repeat until the whole network is covered, which is possible since the network is connected.

Therefore, everyone must play the same action in the lowest equilibrium. By (1.1), this implies that y min i = 1-σ 1-σa + wmax p . Similarly, since y max is the highest Nash equilibrium, ∀i,

y max i ≥ 1-σ 1-σa - w min p . This implies that ȳmax i ≥ 1-σ 1-σa - w min p , and hence ȳmax i ≥ 1-σ 1-σa + w i p .
By Proposition 1, no agent is in the domain of strict social gains and ∀i, y max i ≤ ȳmax i . Since the network is connected, all agents also play the same action in the largest equilibrium and

y max i = 1-σ 1-σa - w min p .
To conclude, note that any other Nash equilibrium y must satisfy

y min i ≤ y i ≤ y max i . This implies that 1-σ 1-σa + wmax p ≤ y i ≤ 1-σ 1-σa - w min p and hence 1-σ 1-σa + w i p ≤ y i ≤ 1-σ 1-σa - w i p .
By Proposition 1, every agent is then in the conformist range: y i = ȳi , implying that everyone plays the same action.

(2) Assume now that wmax 1-σa + > w min 1-σa -and let S = {y : ∀i,

1-σ 1-σa - w min p ≤ y i ≤ 1-σ 1-σa + wmax p }.
(2.1). Let us first show that all Nash equilibria belong to S and, moreover, that f (S) ⊂ S.

Consider a decrease in incomes w such that w max 1-σa + = w min 1-σa -and w min = w min . From the first part of the Theorem, we know that at incomes w , there is a unique Nash equilibrium where every agent plays y = 1-σ 1-σa - w min p . Since the lowest equilibrium decreases weakly when incomes decrease, this implies that ∀i,

y min i ≥ 1-σ 1-σa - w min p . In particular if i is a poorest agent, ȳmin i ≥ 1-σ 1-σa - w i
p and i earns status losses. Similarly, consider an increase in incomes w such that w min 1-σa -= wmax 1-σa + and w max = w max . At incomes w , there is a unique Nash equilibrium where all agents play y = 1-σ 1-σa + wmax p . Since the highest equilibrium increases weakly following an increase in incomes, this implies that ∀i,

y max i ≤ 1-σ 1-σa + wmax p . If i is a richest agent, ȳmax i ≤ 1-σ 1-σa + w i
p and i earns status gains. Therefore, for any Nash equilibrum y, 1-σ 1-σa -

w min p ≤ y min i ≤ y i ≤ y max i ≤ 1-σ 1-σa + wmax p .
Any Nash equilibrium thus belongs to S.

Next, consider y ∈ S. We have:

1-σ 1-σa - w min p ≤ y i ≤ 1-σ 1-σa + wmax p . Therefore, since i's best response is increasing, f i ( 1-σ 1-σa - w min p ) ≤ f i (y i ) ≤ f i ( 1-σ 1-σa + wmax p ). Since 1-σ 1-σa - w min p ≤ 1-σ 1-σa - w i p , 1-σ 1-σa - w min p
lies in the domain where f i lies weakly above the 45 degree line. Therefore,

f i ( 1-σ 1-σa - w min p ) ≥ 1-σ 1-σa - w min p . Similarly, since 1-σ 1-σa + wmax p ≥ 1-σ 1-σa + w i p , 1-σ 1-σa + wmax p
lies in the domain where f i lies weakly below the 45 degree line and hence

f i ( 1-σ 1-σa + wmax p ) ≤ 1-σ 1-σa + wmax p . This implies that 1-σ 1-σa - w min p ≤ f i (y i ) ≤ 1-σ 1-σa +
wmax p , and hence f i (y) ∈ S. (2.2) We now show that the overall best response f is contracting over S. Let i 0 be a richest agent, w i 0 = w max , and j 0 be a poorest agent, w j 0 = w min . For any y ∈ S,

ȳi 0 ≤ 1-σ 1-σa + wmax p = 1-σ 1-σa + w i 0 p . By Proposition 1, this implies that f i 0 (y) = σa + ȳi 0 +(1-σ) w i 0 p . Similarly, ȳj 0 ≥ 1-σ 1-σa - w min p = 1-σ 1-σa - w j 0
p and hence f j 0 (y) = σa -ȳj 0 + (1 -σ) w j 0 p . Next, observe that for any i, y, y , |f i (y) -f i (y )| ≤ |ȳ i -ȳ i |. This holds by Proposition 1 when ȳi and ȳ i belong to the same domain. In these cases, f i is a linear function of ȳi with slope lower than or equal to 1. This also holds when ȳi and ȳ i belong to different domains.

For instance, if ȳi

≤ 1-σ 1-σa + w i p and ȳ i ≥ 1-σ 1-σa - w i p , then f i (y) ≥ ȳi while f i (y ) ≤ ȳ i . Thus, 0 ≤ f i (y ) -f i (y) ≤ ȳ i -ȳi .
Introduce h the linear function that h i (y) = ȳi if i = i 0 , j 0 , h i (y) = σa + ȳi if i = i 0 and h i (y) = σa -ȳi if i = j 0 . This function is represented by the matrix H built from G by multiplying row i 0 by σa + < 1, row j 0 by σa -< 1 and leaving other rows unchanged. Since G is row-normalized with non-negative entries, the spectral radius of G is 1. From Corollary 2.6 in [START_REF] Azimzadeh | A fast and stable test to check if a weakly diagonally dominant matrix is a nonsingular m-matrix[END_REF], we know that the spectral radius of H is strictly lower than 1 if and only if there is a walk connecting every i = i 0 , j 0 to i 0 or to j 0 . Since the network is connected, this property holds.

Finally, let ||.|| 2 denote the Euclidean norm. Then, for any y, y ∈ S,

||f (y) -f (y )|| 2 2 = (f i 0 (y) -f i 0 (y )) 2 + (f j 0 (y) -f j 0 (y )) 2 + i =i 0 ,j 0 (f i (y) -f i (y )) 2 ||f (y) -f (y )|| 2 2 ≤ (σa + (ȳ i 0 -ȳ i 0 )) 2 + (σa -(ȳ j 0 -ȳ j 0 )) 2 + i =i 0 ,j 0 (ȳ i -ȳ i ) 2 ||f (y) -f (y )|| 2 2 ≤ ||h(y) -h(y )|| 2 2 ≤ ρ(H)||y -y || 2 2
Therefore, the best response f is contracting with respect to the Euclidean norm on S, and hence has a unique fixed point.

(2.3) Finally, let us show that status losses (gains) earned by poorest (richest) agents are strict. Let i be a poorest agent, w i = w min . Suppose that i's status losses are not strict, p , a contradiction. Therefore, poorest agents earn strict status losses and, through similar arguments, richest agents earn strict status gains.

y i = ȳi = 1-σ 1-σa - w min p . Since ȳi = j∈N i y j |N i | and y j ≥ 1-σ 1-σa - w min p , y j = 1-σ 1-σa -
Theorem 1 uncovers the existence of a conformism domain, where all agents consume the same level of status good even when they have different incomes. Interestingly, conformism emerges even though agents do not have a direct preference for conformism. Rather, they display loss aversion with respect to the social reference level. In other words, agents have an extra incentive not to fall below the reference level. Theorem 1 shows that the interplay of loss averse social comparisons over a connected network yields full conformism in status good consumption when income heterogeneity is not too high.

How does consumption varies with incomes across agents? In the absence of social comparison, when α -= α + = 0, consumption of the status good varies linearly with income with slope (1-σ)/p while consumption of the standard good varies linearly with income with slope σ. In the conformism domain, consumption of the status good does not depend on income. By contrast and since x i = w i -py i , consumption of the standard good varies linearly with income across agents and with slope 1. An income difference between two agents is passed on one-to-one into a difference in the consumption of the standard good. Thus, conformism on the consumption of the status good is associated with excess variation with income on the consumption of the standard good.

A continuum of conformist Nash equilibria appears when wmax 1-σa + < w min 1-σa -. This key condition is always satisfied under loss aversion and income homogeneity, when w max = w min . This implies, in particular, that under income homogeneity, equilibrium uniqueness is non-generic in the parameter space. Even an arbitrarily small departure from no loss aversion α -= α + leads to a continuum of equilibria, and this holds even when the interaction parameters α -and α + are arbitrarily small. 9 For a given magnitude of loss aversion, this condition is satisfied when income heterogeneity, as measured by the ratio of the highest to lowest income, is not too high. Formally, wmax w min < 1-σa + 1-σa -. Conversely, for a given level of income heterogeneity, a continuum of conformist equilibria appears when loss aversion is large enough. Formally, a --a + > wmax-w min wmax ( 1 σ -a + ). By contrast when wmax 1-σa + > w min 1-σa -, the consumption game has a unique Nash equilibrium where agents do not all consume the same level of status good. This holds when incomes are heterogeneous, w max > w min , and when loss aversion is absent (as in [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF]) or small enough. In this domain, whether an agent earns status gains or status losses generally depends on incomes and on agents' positions in the comparison network. For a given income profile w, if w min < w i < w max there exist networks where i earns status gains and others where i earns status losses.10 Thus in the uniqueness domain, the relative position with respect to the reference level is independent on the network only for poorest and richest agents. Note that conditional on which agent lies in which domain, consumption levels in equilibrium solve a linear system of equations. While there is no simple explicit formula to determine which agent lies in which domain, we can leverage algorithmic results from the literature on supermodular games to compute the Nash equilibrium. We know, in particular, that a process of synchronous, iterated best responses starting at y = 0 converges quickly, and via an increasing sequence, to the unique Nash equilibrium.

We can also leverage standard results of comparative statics for supermodulare games.

Note that the whole best response of agent i increases weakly following an increase in w i , α -, α + or a decrease in p. Therefore, the actions of all agents in the lowest and in the highest Nash equilibrium also increase weakly following an increase in w i , α -, α + or a decrease in p.

Corollary 1. Let ŵ ≥ w, α-≥ α -, α+ ≥ α + and p ≤ p. Assume ŵmax 1-σâ + > ŵmin 1-σâ -and wmax 1-σa + > w min 1-σa -. The consumption game has a unique Nash equilibrium ŷ for parameters ŵ, α-, α+ , p and y for parameters w, α -, α + , p, and ∀i, ŷi ≥ y i .

In the uniqueness domain, all agents weakly increase their consumption of the status good when agents' incomes increase, interaction parameters increase, or the relative price of the status good decreases.

Extensions

Heterogeneity

In our benchmark analysis, agents may only differ in their income levels. We now show that our main result extends to a setup where agents may also differ in how much they care about status and in their level of loss aversion. Formally, assume that agent i has individual specific interaction parameters α - i , α + i with 0 ≤ α + i ≤ α - i and w i ≥ If ω + max > ω - min , then there is a unique Nash equilibrium y such that ∀i, ω - min ≤ y i ≤ ω + max and not all agents play the same action.

Theorem 2 shows that the emergence of two mutually exclusive domains -one with a continuum of conformist equilibria and another one with a unique equilibrium with different actions -is robust to the introduction of heterogeneity in status concerns and loss aversion.

This heterogeneity affects the equilibria and tends to reduce the emergence of conformism.

For instance under homogeneous incomes, the key condition ω + max ≤ ω - min is equivalent to α + max ≤ α - min with α + max = max i α + i and α - min = min i α - i . A continuum of conformist equilibria then appears when the heterogeneity in interaction parameters is not too high. This setup notably covers the specifications of peer effects in [START_REF] Ghiglino | Keeping up with the neighbors: social interaction in a market economy[END_REF] where the strength of interaction depends on the number of neighbors, through increasing function S(.). In that case, α - i = S(|N i |)α -and α + i = S(|N i |)α + , and a continuum of conformist equilibria appears when the dispersion in degrees and in incomes is not too high.

Utility functions

Our benchmark analysis relies on the assumption that agents have Cobb-Douglas utility.

Providing a full-fledged analysis of the consumption game under status concerns and loss aversion and for arbitrary networks and utility functions is an interesting -and challengingdirection of future research. We present some preliminary results here and conditions under which the emergence of a continuum of conformist equilibria is guaranteed.

Figure 1 :

 1 Figure 1: Individual best response under loss aversion

Figure 2 :

 2 Figure 2: Nash equilibria with two agents

  w min p for every j ∈ N i . Therefore, y j ≤ 1-σ 1-σa - w j p and hence by Proposition 1, y j ≥ ȳj . Thus, ȳj ≤ 1-σ 1-σa -

  every k ∈ N j , y k = 1-σ 1-σa -w min p . Repeating the argument and since the network is connected, ∀k, y k = 1

  extends as follows. 11 Theorem 2. Consider any connected comparison network and heterogeneous interaction parameters. If ω + max ≤ ω - min , then a profile y is a Nash equilibrium if and only if y = (y, y, ..., y) with y ∈ [ω + max , ω - min ].

Figure 3 :

 3 Figure 3: Non linear best response with loss aversion

See, e.g.,[START_REF] Abel | Asset prices under habit formation and catching up with the joneses[END_REF],[START_REF] Clark | Comparison-concave utility and following behaviour in social and economic settings[END_REF],[START_REF] Ljungqvist | Tax policy and aggregate demand management under catching up with the joneses[END_REF].

See e.g.[START_REF] Milgrom | Rationalizability, learning, and equilibrium in games with strategic complementarities[END_REF] and[START_REF] Vives | Nash equilibrium with strategic complementarities[END_REF] for classical references on supermodular games.

The length of the interval of possible equilibrium values converges towards zero when α --α + converges to zero.

Agent i earns status gains when she compares herself to poorest agents and status losses when she compares herself to richest agents.

The proof of Theorem 2 follows the same steps as the proof of Theorem 1, we omit it for brevity.

We thank Ingrid Vargas for the initial collaboration in this research project and participants in conferences and seminars for helpful comments. This work was supported by French National Research Agency Grant ANR-17-EURE-0020 and by the Excellence Initiative of Aix-Marseille University, A * MIDEX.

Consider some utility function u(x, y), increasing and quasi-concave in both arguments.

Denote by x(p, w) and y(p, w) the usual Walrasian demands in the absence of relative comparisons, i.e., the solutions to the consumer problem max x,y≥0 u(x, y) under x+py ≤ w. With status concerns, assume that agent i seeks to maximize u(x i , ϕ(y i , y -i )) under the budget constraint x i + py i ≤ w i . An important first step of the analysis is to determine an agent's best response under status concerns but without loss aversion, i.e., when α -= α + = α. In that case, the best reponse

and hence crucially depends on properties of the Walrasian demand. For instance, the best response is linear in ȳi iff the Walrasian demand is linear in income, which covers both the Cobb-Douglas and CES cases.

In general, the best response without loss aversion may be non-linear, in which case the best response with loss aversion is not piecewise linear. However, the best response under loss aversion can still have an intermediate range with pure conformist behavior. This notably happens when f i is increasing in ȳi and in α and, for a given α, crosses the 45 degree line only once from above. We illustrate a situation where f i is concave in ȳi in Figure 3. As in Proposition 1, the best response under loss aversion is continuous, increasing and formed of three pieces. It is first equal to f i (., α + ) until it crosses the 45 degree line; it is then equal to the 45 degree line until f i (., α -) crosses it, above which it is equal to f i (., α -). Therefore, even with general utility functions the kink induced by loss aversion may lead agents to act as pure conformists.

12 To see why, express the budget constraint as a function of x i and ϕ = (1 + α)y i -αȳ i .