Rémy Beugnon

Stephan Hättenschwiler

Nolwenn Le Guyader

Jonathan Lenoir

Alexandru Milcu

Xavier Morin

Jérémy Puissant

Keywords: climate change, microclimate, biodiversity-ecosystem functioning, ecosystem services, thermal tolerance, plant species richness

Changes in climate and biodiversity are widely recognized as primary global change drivers of ecosystem structure and functioning, also affecting ecosystem services provided to human populations. Increasing plant diversity typically enhances ecosystem functioning and stability. Higher plant diversity can also mitigate climate change effects and buffer extreme weather conditions, but the underlying mechanisms remain largely unclear. Recent studies have shown that plant diversity can mitigate climate change (e.g. reduce temperature fluctuations or drought through microclimatic effects) in different compartments of the focal ecosystem, which as such may contribute to the effect of plant diversity on ecosystem properties and functioning. However, these potential plant diversity-induced microclimate effects are not sufficiently understood. Here, we explored the consequences of climate modulation through microclimate modification by plant diversity for ecosystem functioning as a potential mechanism contributing to the widely documented biodiversity-ecosystem functioning (BEF) relationships, using a combination of theoretical and simulation approaches. We focused on a diverse set of response variables at various levels of integration ranging from ecosystem-level carbon exchange to soil enzyme activity, including population dynamics and activity of specific organisms. Here, we demonstrated that a vegetation layer composed of many plant species has the potential to influence ecosystem functioning and stability through the modification of microclimatic conditions, thus mitigating the negative impacts of climate extremes on ecosystem functioning. Integrating microclimatic processes (e.g. temperature, humidity, and light modulation) as a mechanism contributing to the BEF relationships is a promising avenue to improve our understanding of the effects of climate change on ecosystem functioning and to better predict future ecosystem structure, functioning, and services. In addition, microclimate management and monitoring should be seen as a potential tool by practitioners to adapt ecosystems to climate change.

Introduction

The profound impacts of changing climate and biodiversity on ecosystem functioning and stability are widely acknowledged (IPBES, 2019;[START_REF] Pörtner | Scientific outcome of the IPBES-IPCC co-sponsored workshop on biodiversity and climate change[END_REF]. These threats extend to the key ecosystem services that sustain human populations (IPCC, 2021;[START_REF] Pörtner | Scientific outcome of the IPBES-IPCC co-sponsored workshop on biodiversity and climate change[END_REF], particularly as climatic extremes like heatwaves, droughts, and floods become more frequent, pushing ecosystems toward critical thresholds and potential collapse [START_REF] Duffy | How close are we to the temperature tipping point of the terrestrial biosphere?[END_REF][START_REF] Macdougall | Diversity loss with persistent human disturbance increases vulnerability to ecosystem collapse[END_REF].

The intricate relationship between ecosystem functioning and climatic conditions holds on the climate-dependency of the sum of biological processes that contribute to ecosystem functioning. Net Ecosystem carbon Exchange (NEE), for instance, depends on air temperature, radiation, and soil water availability [START_REF] Niu | Thermal optimality of net ecosystem exchange of carbon dioxide and underlying mechanisms[END_REF]. Similarly, organic matter recycling responds to changes in temperature and water availability [START_REF] Joly | Resolving the intricate role of climate in litter decomposition[END_REF][START_REF] Xiao | Rates of Litter Decomposition and Soil Respiration in Relation to Soil Temperature and Water in Different-Aged Pinus massoniana Forests in the Three Gorges Reservoir Area, China[END_REF]. With changing climatic conditions, these processes move along a climate response curve that is rarely linear, and the net outcome on ecosystem properties and services becomes a focal point of concern.

The climatic conditions experienced (i.e. temperature, light availability, concentrations of greenhouse gasses, humidity) by organisms may deviate substantially from those measured by using conventional meteorological standards characterizing macroclimate under highly standardized conditions [START_REF] Haesen | Microclimate reveals the true thermal niche of forest plant species[END_REF][START_REF] Lembrechts | Microclimate alters the picture[END_REF][START_REF] Lembrechts | SoilTemp: A global database of near-surface temperature[END_REF].

The climatic niche experienced by an organism is therefore defined by microclimatic conditions that can differ substantially from macroclimatic conditions, potentially altering the real shape of the species response curve against a given climatic parameter like temperature [START_REF] Haesen | Microclimate reveals the true thermal niche of forest plant species[END_REF]. Unsurprisingly, incorporating microclimate data, rather than macroclimate data, into species distribution models (SDMs) has proven highly valuable for advancing our understanding of ecosystem properties [START_REF] Kemppinen | Microclimate, an inseparable part of ecology and biogeography[END_REF][START_REF] Lembrechts | Microclimate alters the picture[END_REF][START_REF] Stark | Microclimate-based species distribution models in complex forested terrain indicate widespread cryptic refugia under climate change[END_REF], as well as our ability to better predict the activity and survival of species [START_REF] Colinet | Insects in Fluctuating Thermal Environments[END_REF][START_REF] Dingman | Cross-scale modeling of surface temperature and tree seedling establishment in mountain landscapes[END_REF].

The physical structure of ecosystems and their species composition, above all that of the vegetation compartment contributing to most of the biomass and controlling the key ecosystem processes like the fluxes of energy and matter, play a critical role in influencing differences between macro-and microclimate conditions [START_REF] De Frenne | Global buffering of temperatures under forest canopies[END_REF], 2021;[START_REF] Gril | Using airborne LiDAR to map forest microclimate temperature buffering or amplification[END_REF][START_REF] Milling | Habitat structure modifies microclimate: An approach for mapping fine-scale thermal refuge[END_REF]. Consequently, manipulating the species composition and canopy structure of a given ecosystem offers a promising strategy to modulate microclimate, and thus, the overall functioning of the focal ecosystem. Indeed, recent research highlighted the potential of plant diversity, and thus ecosystem structure, to influence microclimatic fluctuations [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF][START_REF] Ray | Tree diversity increases productivity through enhancing structural complexity across mycorrhizal types[END_REF][START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF][START_REF] Zhang | Tree species diversity effects on microclimate buffering, throughfall regulation and litter decomposition in young forest plantations[END_REF]. This included the further reduction of temperature extremes experienced underneath a vegetation canopy that is composed of more species [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF][START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF], and changes in sensible heat flux and evapotranspiration resulting in higher ecosystem CO2 uptake [START_REF] Milcu | Plant functional diversity increases grassland productivity-related water vapor fluxes: An Ecotron and modeling approach[END_REF]. Higher plant diversity also promotes ecosystem functioning and stability by increasing and securing essential functions such as productivity and carbon storage [START_REF] Beugnon | Abiotic and biotic drivers of tree trait effects on soil microbial biomass and soil carbon concentration[END_REF][START_REF] Huang | Impacts of species richness on productivity in a large-scale subtropical forest experiment[END_REF][START_REF] Liang | Positive biodiversity-productivity relationship predominant in global forests[END_REF][START_REF] Liang | Consistent stabilizing effects of plant diversity across spatial scales and climatic gradients[END_REF][START_REF] Schnabel | Drivers of productivity and its temporal stability in a tropical tree diversity experiment[END_REF]. However, the nexus between biodiversity, microclimate, and ecosystem functioning remains relatively unexplored, also lacking a unified framework.

The aim of this study is to better understand the biodiversity -microclimate -ecosystem functioning relationships by exploring the cascading effects of biodiversity-induced changes on microclimatic processes and ecosystem functioning. While climate encompasses various environmental factors, such as temperature, air pressure, humidity, precipitation, solar radiation, cloudiness, and wind, in this study, we specifically concentrate on the effects of temperature. To address this, we take a four-fold approach (Fig. 1). First, we emphasize the climate-dependency of ecosystem functions across organizational levels. Second, we examine the evidence supporting the effect of plant diversity on microclimate. Third, through the implementation of a common simulation framework, we quantify the potential consequences of increased plant diversity for temperature-dependent ecosystem functions. Finally, we discuss the implications of our findings for the general understanding of the BEF relationship and their practical applications. [START_REF] Bertrand | Changes in plant community composition lag behind climate warming in lowland forests[END_REF][START_REF] Lembrechts | Microclimate alters the picture[END_REF], (2) Pörtner et al., 2021, (3) Beugnon et al., 2021;Messier et al., 2021, (4) [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF][START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF]Zhang, 2022, (5-6) [START_REF] Gottschall | Tree species identity determines wood decomposition via microclimatic effects[END_REF][START_REF] Huang | Air temperature optima of vegetation productivity across global biomes[END_REF][START_REF] Joly | Tree species diversity affects decomposition through modified microenvironmental conditions across European forests[END_REF].

Climate context dependencies of process rates across organizational levels

Formulated by Brown and colleagues (2004), "the metabolic theory of ecology" proposes a temperature dependency of biological processes across organizational levels, from specific processes at the molecular scale to integrated processes at the ecosystem scale. Beyond its primary emphasis on temperature, this theory proposes that similar causal relationships exist for other climatic parameters, such as humidity and light, all influencing the performance of organisms and ultimately their contribution to ecosystem functioning [START_REF] Brown | Toward a Metabolic Theory of Ecology[END_REF].

At the molecular level, temperature acts as a strong driver of enzymatic activity due to the strong dependence of the reaction rate on temperature. The temperature dependence of enzyme catalytic reactions shows a reaction-specific optimum and an upper-temperature limit that is ultimately reached when thermal inactivation of enzymes occurs [START_REF] Alster | Temperature Sensitivity as a Microbial Trait Using Parameters from Macromolecular Rate Theory[END_REF][START_REF] Peterson | The dependence of enzyme activity on temperature: Determination and validation of parameters[END_REF]. This temperature-enzyme activity relationship has historically been modeled with the Arrhenius law, although recently proposed models better represent the experimentally observed bell-shaped response curve [START_REF] Alster | Embracing a new paradigm for temperature sensitivity of soil microbes[END_REF]. Concurrently, other climatic variables also impact enzyme functioning, for instance, changes in soil moisture alter substrate diffusion and the likelihood of substrate-enzyme encounters and the production of microbial extracellular enzymes, ultimately leading to changes in enzyme activity, as observed for the extracellular beta-glucosidase enzyme, a key player in soil organic carbon decomposition [START_REF] Puissant | Seasonal influence of climate manipulation on microbial community structure and function in mountain soils[END_REF][START_REF] Puissant | Seasonality alters drivers of soil enzyme activity in subalpine grassland soil undergoing climate change[END_REF][START_REF] Wallenstein | Emerging tools for measuring and modeling the in situ activity of soil extracellular enzymes[END_REF][START_REF] Zhang | Kinetic parameters of soil β-glucosidase response to environmental temperature and moisture regimes[END_REF]. Changes in microclimate can also affect substrate concentration, such as an increase in CO2 concentrations that enhance the activity of the photosynthetic enzyme Rubisco [START_REF] Galmés | Potential improvement of photosynthetic CO2 assimilation in crops by exploiting the natural variation in the temperature response of Rubisco catalytic traits[END_REF].

The climate, or more generally the environmental control of enzymatic kinetics, affects individual performance and the functions the organisms have in the ecological context via the organisms' metabolic activities. This holds particular significance for species unable to regulate their body temperature, such as ectotherms. For instance, insects, being ectotherms, demonstrate an increase in activity with rising temperatures, reaching an optimal point before experiencing a drastic decline [START_REF] Colinet | Insects in Fluctuating Thermal Environments[END_REF][START_REF] Sinclair | Can we predict ectotherm responses to climate change using thermal performance curves and body temperatures?[END_REF][START_REF] Terlau | Microhabitat conditions remedy heat stress effects on insect activity[END_REF]. Although primarily assessed at the microbial community scale, the temperature response of microbes plays a critical role in nutrient recycling and ecosystem functioning [START_REF] Bradford | Cross-biome patterns in soil microbial respiration predictable from evolutionary theory on thermal adaptation[END_REF][START_REF] Dacal | Soil microbial respiration adapts to ambient temperature in global drylands[END_REF]. The temperature dependency of microbes is a subject of intense debate, representing a major uncertainty in the projections of carbon cycle models in response to climate warming [START_REF] García-Palacios | Evidence for large microbial-mediated losses of soil carbon under anthropogenic warming[END_REF][START_REF] Duffy | How close are we to the temperature tipping point of the terrestrial biosphere?[END_REF]. Consequently, microclimate buffering could emerge as a critical factor in regulating the rate of soil carbon decomposition. Such control over specific activities of organisms is also exerted by other environmental factors such as light availability. When light penetrates a plant canopy, it is increasingly absorbed following the Beer-Lambert law [START_REF] Larcher | Physiological plant ecology: Ecophysiology and stress physiology of functional groups[END_REF] affecting, for example, leaf functional traits and microclimatic conditions along the vertical profile [START_REF] Govaert | Trait-micro-environment relationships of forest herb communities across Europe[END_REF][START_REF] Messier | Trait variation and integration across scales: Is the leaf economic spectrum present at local scales?[END_REF][START_REF] Niinemets | Photosynthetic Acclimation to Simultaneous and Interacting Environmental Stresses Along Natural Light Gradients: Optimality and Constraints[END_REF], growth of understorey plants, as well as their composition and cover in the understorey [START_REF] Messier | Effects of overstory and understory vegetation on the understory light environment in mixed boreal forests[END_REF][START_REF] Tinya | Congruence of the spatial pattern of light and understory vegetation in an old-growth, temperate mixed forest[END_REF].

Climate responses of individual organisms ultimately transition to the community level by impacting their survival [START_REF] Allen | A global overview of drought and heat-induced tree mortality reveals emerging climate change risks for forests[END_REF][START_REF] Dingman | Cross-scale modeling of surface temperature and tree seedling establishment in mountain landscapes[END_REF][START_REF] Mcdowell | Mechanisms of woody-plant mortality under rising drought, CO2 and vapour pressure deficit[END_REF], phenology [START_REF] Cleland | Shifting plant phenology in response to global change[END_REF][START_REF] Møller | Forest management drives evolution of understorey herbs[END_REF], and productivity [START_REF] Huang | Air temperature optima of vegetation productivity across global biomes[END_REF], which all determine competitive interactions and community composition. Through its influence on individuals, biotic intercations and communities, climate effects also transcend trophic levels and food-web structure and dynamics [START_REF] Sentis | Towards a mechanistic understanding of temperature and enrichment effects on species interaction strength, omnivory and food-web structure[END_REF].

Collectively, these responses to climate variables across various organizational levels from single biochemical reactions to organisms, communities, and food web structure may have cascading effects on ecosystem-level processes such as photosynthesis [START_REF] Crous | Temperature responses of photosynthesis and respiration in evergreen trees from boreal to tropical latitudes[END_REF][START_REF] Kirschbaum | Direct and Indirect Climate Change Effects on Photosynthesis and Transpiration[END_REF], evapotranspiration [START_REF] Kirschbaum | Direct and Indirect Climate Change Effects on Photosynthesis and Transpiration[END_REF], organic matter recycling (Bradford et al., 2017;[START_REF] Xiao | Rates of Litter Decomposition and Soil Respiration in Relation to Soil Temperature and Water in Different-Aged Pinus massoniana Forests in the Three Gorges Reservoir Area, China[END_REF], and soil respiration [START_REF] Jones | Carbon sequestration in a temperate grassland; management and climatic controls[END_REF]. A notable example is the temperature dependence of the Net Ecosystem CO2 Exchange (NEE) that can vary between net C-influx at average temperatures to net C-efflux at extreme temperatures (Fig. 2, [START_REF] Niu | Thermal optimality of net ecosystem exchange of carbon dioxide and underlying mechanisms[END_REF][START_REF] Yuan | Thermal adaptation of net ecosystem exchange[END_REF]. Other climate-related variables such as atmospheric CO2 concentrations and water availability additionally or interactively affect NEE [START_REF] Milcu | Functional diversity of leaf nitrogen concentrations drives grassland carbon fluxes[END_REF]. Similarly, net ecosystem nitrous oxide (N2O) emissions are contingent upon soil moisture and temperature [START_REF] Schindlbacher | Effects of soil moisture and temperature on NO, NO 2 , and N 2 O emissions from European forest soils[END_REF]. [START_REF] Dingman | Cross-scale modeling of surface temperature and tree seedling establishment in mountain landscapes[END_REF], (4) insect activity [START_REF] Colinet | Insects in Fluctuating Thermal Environments[END_REF], and (5) Rubisco activity [START_REF] Galmés | Potential improvement of photosynthetic CO2 assimilation in crops by exploiting the natural variation in the temperature response of Rubisco catalytic traits[END_REF].

The climate-modulating impact of plant diversity

The dominating contribution of primary producers in terms of biomass in terrestrial ecosystems confers to plants a particularly important role in structuring and engineering their proximal environment. The soil development during successional dynamics with a continuous modification of soil properties resulting from plant colonization in early successional stages and the accumulation of organic matter and nitrogen (through symbiotic N-fixation) with ongoing succession is a well-documented example of how plants reshape environmental properties [START_REF] Northup | Intraspecific variation of conifer phenolic concentration on a marine terrace soil acidity gradient; a new interpretation[END_REF][START_REF] Rahmonov | Changes in Soil Features and Phytomass during Vegetation Succession in Sandy Areas[END_REF][START_REF] Wardle | A critique of the microbial metabolic quotient (qCO2) as a bioindicator of disturbance and ecosystem development[END_REF]. Specific plant species can alter environmental conditions for associated species through Facilitation by "nursing" can indirectly modify biodiversity, then making it less exposed to climate extremes [START_REF] Bulleri | Facilitation and the niche: Implications for coexistence, range shifts and ecosystem functioning[END_REF] and sometimes improving access to limited resources such as water (hydraulic lift, [START_REF] Caldwell | Hydraulic lift: Consequences of water efflux from the roots of plants[END_REF] or nitrogen for N-fixing plants.

Due to their large size, trees often have a particularly strong influence on both their proximal and distal environments by modifying biogeochemical cycles and providing food and shelter for many other species in their environment. Coniferous trees, for example, typically reduce litter and soil pH in forests where they dominate with far-ranging consequences for soil microbial and fauna communities, nutrient availability, and associated plant species in the understorey [START_REF] Burgess-Conforti | Are soils beneath coniferous tree stands more acidic than soils beneath deciduous tree stands?[END_REF][START_REF] Phillips | The mycorrhizal-associated nutrient economy: A new framework for predicting carbon-nutrient couplings in temperate forests[END_REF][START_REF] Wardle | A critique of the microbial metabolic quotient (qCO2) as a bioindicator of disturbance and ecosystem development[END_REF]. The tree canopies of forests as a whole have been shown to substantially alter microclimatic conditions underneath [START_REF] De Frenne | Global buffering of temperatures under forest canopies[END_REF], generally providing climatically more favorable conditions to understorey plants by alleviating climate extremes [START_REF] Bertrand | Changes in plant community composition lag behind climate warming in lowland forests[END_REF][START_REF] De Frenne | Forest microclimates and climate change: Importance, drivers and future research agenda[END_REF]. Forest tree canopies affect microclimate via different mechanisms, such as shading, reducing air movement that diminishes mixing between air layers, and active cooling by alive foliage through transpiration [START_REF] Geiger | The Climate Near the Ground[END_REF][START_REF] Zellweger | Forest microclimate dynamics drive plant responses to warming[END_REF]. Increasing canopy thickness and structural complexity consequently enhance the strength of the buffering capacity of the upper canopy layer [START_REF] Ehbrecht | Effects of structural heterogeneity on the diurnal temperature range in temperate forest ecosystems[END_REF][START_REF] Gril | Using airborne LiDAR to map forest microclimate temperature buffering or amplification[END_REF].

Plant diversity was found to foster canopy thickness and complexity [START_REF] Peng | Leaf area increases with species richness in young experimental stands of subtropical trees[END_REF][START_REF] Perles-Garcia | Tree species richness promotes an early increase of stand structural complexity in young subtropical plantations[END_REF][START_REF] Williams | Spatial complementarity in tree crowns explains overyielding in species mixtures[END_REF], potentially also enhancing the buffering capacity of vegetation cover on microclimatic processes in the understorey. Results from experiments manipulating plant diversity in the field (BEF experiments) indicate that microclimatic conditions are even more stable with increasing plant species richness [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF][START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF][START_REF] Zhang | Tree species diversity effects on microclimate buffering, throughfall regulation and litter decomposition in young forest plantations[END_REF] and that microclimate interacts with stand complexity in promoting forest productivity [START_REF] Ray | Tree diversity increases productivity through enhancing structural complexity across mycorrhizal types[END_REF]. Similar effects by more diverse plant communities on microclimatic conditions via thicker canopies were also observed in grasslands [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF]), yet empirical evidence in grassland ecosystems remains limited, particularly regarding comprehensive global quantification across various ecoregions. Further, in-situ measurements are necessary to fully understand the scope of this relationship on a global scale, especially, long-term measurements are essential to quantify the potential mitigation of climatic extremes within the context of climate change.

Such diversity-driven canopy feedbacks may be particularly important in mitigating effects on soil temperature that have recently been shown to be more frequently affected by temperature extremes than air temperature (García-García et al., 2023) at a global extent.

The present, still limited, evidence suggests that higher plant diversity potentially modifies ecosystem functioning through climate-dependent ecosystem properties. Among the possible (and non-exclusive) mechanistic hypotheses proposed to explain the effect of plant diversity on ecosystem properties (e.g. hydraulic lift, architecture complementarity leading to canopy packing), microclimatic processes appear as a promising explanation allowing a direct link with physiology-based mechanisms. However, the effects of plant diversity on microclimate dynamics depend on seasons [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF][START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF][START_REF] Zhang | Tree species diversity effects on microclimate buffering, throughfall regulation and litter decomposition in young forest plantations[END_REF], owing to structural mediation in winter (e.g. branch structure in forests) and leaf mediation in summer [START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF]. Consequently, the impact of plant diversity on climate-dependent ecosystem functions is anticipated to be dynamic over time. It still could accumulate over larger time scales, because increase in productivity with species-rich communities is expected to increase with time [START_REF] Dietrich | Tree diversity effects on productivity depend on mycorrhizae and life strategies in a temperate forest experiment[END_REF][START_REF] Huang | Impacts of species richness on productivity in a large-scale subtropical forest experiment[END_REF][START_REF] Perles-Garcia | Tree species richness promotes an early increase of stand structural complexity in young subtropical plantations[END_REF][START_REF] Weisser | Biodiversity effects on ecosystem functioning in a 15-year grassland experiment: Patterns, mechanisms, and open questions[END_REF].

Consequences for ecosystem functioning and services

Plant diversity effects on microclimate have the potential to influence ecosystem functioning and services through climate-dependent processes across various levels of organization (Fig. 2). However, the extent of these effects remains largely unexplored, partly because they are not easy to experimentally tease apart microclimatic processes from other mechanisms of diversity effects. Therefore, it is unclear how diversity-driven climate modulation cascades through different levels of organization and how large the resulting consequences on ecosystem functioning may be. For instance, some studies on litter decomposition indicate a positive influence of tree species richness on leaf litter decomposition through microclimate modulation [START_REF] Gottschall | Tree species identity determines wood decomposition via microclimatic effects[END_REF][START_REF] Joly | Tree species diversity affects decomposition through modified microenvironmental conditions across European forests[END_REF][START_REF] Makkonen | Do physical plant litter traits explain non-additivity in litter mixtures? A test of the improved microenvironmental conditions theory[END_REF]. However, the quantification of these causal relationships remains limited and the resulting consequences on ecosystem carbon and nutrient cycling are untested.

Here, we chose a different approach to experimental testing in an attempt to move forward with this critically important question. We employed a simple simulation approach to assess the effects of species diversity on ecosystem functioning via microclimate, using temperature as an example for a microclimate variable and soil respiration as a temperature-dependent ecosystem process (Fig. 3.A, for details on the workflow see Suppl. S1). Initially, we simulated soil temperature for grasslands with various plant species richness scenarios (bare soil, monoculture, sixty-species mixtures), using air temperature records from central Germany (2017-2019, ERA5 hourly data, Copernicus Climate Change Service, 2019) and the soil temperature-grassland species richness relationship from [START_REF] Huang | Plant diversity stabilizes soil temperature[END_REF].

Subsequently, based on the soil respiration-temperature relationship from [START_REF] Jones | Carbon sequestration in a temperate grassland; management and climatic controls[END_REF], we predicted soil respiration for each scenario and calculated cumulative soil respiration (Fig. 3.B). Our simulation illustrated the potential of plant species diversity to modulate soil respiration by modifying microclimate. We demonstrated that diversity primarily affects soil respiration during warm hours, days, and months, rapidly increasing the divergence between scenarios during these periods. Despite the fact that the magnitude of this climate-mediated effect of diversity on soil respiration varied among different time periods, it accumulated over time with an increasing difference in total respiration between the low and high plant diversity scenarios (Fig. 3.B). The simulations that illustrate how soil respiration depends on microclimate highlight the possible effects of climate modulation on ecosystem functioning. However, these simulations oversimplify the relationship by considering only temperature as the factor affecting soil respiration's dependence on microclimate. In reality, the relationship between soil respiration and microclimate is multifaceted and is modulated by additional factors such as soil humidity [START_REF] Wood | Sensitivity of Soil Respiration to Variability in Soil Moisture and Temperature in a Humid Tropical Forest[END_REF], nutrient availability and CO2 diffusivity [START_REF] Davidson | The D ual A rrhenius and M ichaelis-M enten kinetics model for decomposition of soil organic matter at hourly to seasonal time scales[END_REF][START_REF] Skopp | Steady-State Aerobic Microbial Activity as a Function of Soil Water Content[END_REF]. To accurately predict long-term trends in soil respiration, it is imperative to consider these various aspects, their interactions, and the direct effects of plant diversity on soil quality [START_REF] Weisser | Biodiversity effects on ecosystem functioning in a 15-year grassland experiment: Patterns, mechanisms, and open questions[END_REF].

These simulations underscore the potential of plant diversity to influence ecosystem processes through microclimate modulation, particularly when considering cumulative processes such as carbon fluxes that occur not only during soil respiration but also during organic matter decomposition or through biomass production, which can greatly accentuate overall effects beyond the slight differences measured at specific points in time. Given that numerous ecosystem functions across various organizational levels are contingent on climatic conditions, the vegetation-microclimate-ecosystem functioning nexus may play a pivotal role in overall ecosystem functioning. This simulation exercise emphasizes the necessity for empirically measuring the vegetation-microclimate-ecosystem functioning nexus to better understand the consequences at the ecosystem level. A way forward to tackle the issue of confounding mechanisms influencing ecosystem functions at a yearly extent could then be to increase the temporal resolution of sampling allowing to identify potential mechanisms and their temporality.

Perspective for research and management

Microclimate is an ecologically important environmental parameter that regulates ecosystem functioning to an important degree (e.g. [START_REF] Gottschall | Tree species identity determines wood decomposition via microclimatic effects[END_REF][START_REF] Joly | Tree species diversity affects decomposition through modified microenvironmental conditions across European forests[END_REF][START_REF] Kemppinen | Microclimate, an inseparable part of ecology and biogeography[END_REF], which can be co-determined by plant diversity and canopy structure (Huang et al., 2023;[START_REF] Schnabel | Tree diversity increases forest temperature buffering[END_REF][START_REF] Zhang | Tree species diversity effects on microclimate buffering, throughfall regulation and litter decomposition in young forest plantations[END_REF]. Thus, integrating microclimate into the BEF relationship seems critically important and promises an enhanced understanding of the mechanisms underlying BEF relationships, particularly given the multitude of climatedependent processes at different organizational levels ultimately determining ecosystem functions. The mediation of microclimate in the relationship between biodiversity and ecosystem functioning, coupled with its direct correlation to macroclimate fluctuations, could offer an explanation for the observed climate and context dependency in the strength of the biodiversity-ecosystem functioning relationship [START_REF] Liu | Drought resistance enhanced by tree species diversity in global forests[END_REF][START_REF] Spohn | The positive effect of plant diversity on soil carbon depends on climate[END_REF][START_REF] Cesarz | Tree diversity effects on soil microbial biomass and respiration are context dependent across forest diversity experiments[END_REF]. Incorporating microclimate mediation into the biodiversity-ecosystem functioning framework may advance our understanding of ecosystem functioning by providing mechanistic insights.

Microclimate spans a continuum from the top of the canopy to the soil layers, with specific microclimates affecting distinct processes. For example, the soil microbial community will respond to variations in the topsoil microclimate rather than to variations in the microclimate within the plant canopy [START_REF] Kemppinen | Microclimate, an inseparable part of ecology and biogeography[END_REF]. Consequently, microclimate variations will have to be quantified in space and time simultaneously with the assessment of how plant diversity affects microclimate dynamics in space and time. Additionally, there is a growing need to quantify the climate responses of ecosystem functions over time, particularly when considering functions that integrate temporal dynamics in microclimate effects.

The impact of plant diversity on microclimate acts primarily through modifications of canopy parameters such as thickness and structural complexity [START_REF] Ehbrecht | Effects of structural heterogeneity on the diurnal temperature range in temperate forest ecosystems[END_REF][START_REF] Gril | Using airborne LiDAR to map forest microclimate temperature buffering or amplification[END_REF]. It is therefore critical to include stand age and structure beyond the diversity of species composing the canopy for a detailed understanding of plant diversity control over microclimate [START_REF] Au | Younger trees in the upper canopy are more sensitive but also more resilient to drought[END_REF].

Recognizing the importance of plant diversity in promoting ecosystem functioning and services, as well as their stability, has been emphasized to ecosystem managers [START_REF] Messier | For the sake of resilience and multifunctionality, let's diversify planted forests! Conservation Letters[END_REF], particularly in the context of climate change [START_REF] Beugnon | Diverse forests are cool: Promoting diverse forests to mitigate carbon emissions and climate change[END_REF][START_REF] Pires | Is Biodiversity Able to Buffer Ecosystems from Climate Change? What We Know and What We Don[END_REF]. Understanding the biodiversity-microclimate-ecosystem functioning nexus offers new perspectives. Microclimate monitoring becomes a valuable non-invasive tool for assessing climate change mitigation, allowing stakeholders to quantify the effects of plant diversity, both taxonomic and structural diversity, on ecosystem functioning (Fig. 4). Microclimate monitoring, being low-cost, effortless, and non-invasive [START_REF] Kemppinen | Microclimate, an inseparable part of ecology and biogeography[END_REF], emerges as an important approach for tracking ecosystem functioning, given its influence on numerous functions across all organizational levels.

Conclusions

We showed that the influence of plant diversity on microclimate is promising for advancing our understanding of the relationship between biodiversity and ecosystem functioning, especially considering the climate-dependency of ecosystem functions and services.

Therefore, future research should strengthen our understanding of biodiversity-microclimateecosystem functioning by developing and testing these connections theoretically as much as experimentally. Consequently, the manipulation of microclimate, through plant diversity, both in terms of taxonomic and structural diversity, presents an opportunity to effectively manage ecosystems functioning in the face of climate change.

Figure 1 :

 1 Figure 1: Conceptual framework linking the nexus between plant diversity, microclimate, and ecosystem functions. Examples of pairwise relationships from the

  It is particularly important to better understand all these ecosystem-level responses to climate variables with ongoing climate change (IPBES, 2019; IPCC, 2021; Pörtner et al., 2021), especially since the responses across organizational levels span different time scales, ranging from immediate reactions to long-lasting effects.

Fig. 2 :

 2 Fig. 2: Climate dependencies of various processes across different levels of

Figure 3 :

 3 Figure 3: Climate-mediated biodiversity-ecosystem functioning (BEF) relationships:

Figure 4 :

 4 Figure 4: Perspectives and consequences for Biodiversity-Ecosystem Functioning (BEF) research and ecosystem management.

Acknowledgments

Funded by the Saxon State Ministry for Science, Culture and Tourism (SMWK) -[3-7304/35/6-2021/48880]. We gratefully acknowledge the support by the German Centre for Integrative Biodiversity Research (iDiv) funded by the German Research Foundation (DFG-FZT 118, 202548816). JL acknowledges funding from the Agence Nationale de la Recherche (ANR), under the framework of the collaborative research program funding scheme (Grant N°ANR-21-CE32-0012-03: MaCCMic project) and the young researcher funding scheme (ANR-19-CE32-0005-01: IMPRINT project). JL also acknowledges the Région Hauts-de-France, the Ministère de l'Enseignement Supérieur et de la Recherche and the European Fund for Regional Economic Development for their financial support to the CPER ECRIN program.

Data and code availability

All datasets and codes used for the simulations are available in the supplementary material S1.

Conflict of Interest

The authors declare no conflict of interest.


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Charles Labrousse
email: charles.labrousse@psemail.eu

Yann Perdereau
email: yann.perdereau@psemail.eu

Geography versus income: the heterogeneous effects of carbon taxation

Keywords: C61, E37, E62, H23, H30, Q43, Q48, Q58, R11, R13 Carbon taxes, Energy, Fiscal policy, Emissions, Macroeconomic effects, Inequalities, Geography JEL classification -C61, E37, E62, H23, H30, Q43, Q48, Q58, R11, R13 Keywords -Carbon taxes, energy, fiscal policy, emissions, macroeconomic effects, inequalities, geography

Redistributive effects of carbon taxation are key for its political acceptability. We build a dynamic equilibrium model incorporating heterogeneity in wealth, income and living areas. Energy serves as both a household consumption good and a firm intermediate input. We position our model as a comprehensive policy toolkit, extendable to diverse countries and several environmental policies. Using calibrated French micro data, we evaluate the redistributive effects of carbon taxation and obtain three key results. First, geography is more important than income to assess the distributive effects of carbon taxation, as rural households bear more substantial losses. Second, the carbon tax on households is regressive, while the carbon tax on firms may be progressive. Third, it is possible to achieve welfare increase and emission reduction through appropriate recycling of carbon policy, but a trade-off between equity and efficiency emerges if we want to reduce the rural-urban gap.

Introduction

Carbon taxes reduce emissions, but they induce strong distributional costs, as energy represents a larger share of consumption for poor and rural households. These regressive effects are likely to reduce the political acceptability of carbon taxation, as shown in [START_REF] Douenne | French attitudes on climate change, carbon taxation and other climate policies[END_REF], and illustrated in France with the Yellow Vests protest and the subsequent carbon tax freezing.

Therefore, a socially acceptable carbon taxation design should account for its distributive effects. A trade-off emerges between equity and (climate) efficiency, as redistributing carbon tax revenue towards poor and rural households will likely increase emissions. In this paper, we use a dynamic general equilibrium model with rich household heterogeneity to assess the distributive effects of carbon taxation, and we consider different revenue-recycling scenarios. The main features of our model, and the associated findings, can be put into three categories.

First, geography is more important than income to assess the distributive effects of carbon taxation. [START_REF] Cronin | Vertical and Horizontal Redistributions from a Carbon Tax and Rebate[END_REF] for the U.S. and [START_REF] Douenne | The Vertical and Horizontal Distributive Effects of Energy Taxes: a Case Study of a French Policy[END_REF] in the French context, find that most of energy consumption heterogeneity, and therefore winners and losers of carbon taxation, lie within income classes. As shown in Figure 1 using French micro data, the energy share in total consumption expenditures varies more across living areas than across income levels. Rural areas inhabitants tend to have higher incompressible energy needs related to transport and heating systems, while urban households benefit from public transportation and live in smaller housing. The vertical heterogeneity is less relevant, as rich households tend to have bigger houses and travel more, mitigating the decline of their energy share compared to poor households. We use the [START_REF] Aiyagari | Uninsured Idiosyncratic Risk and Aggregate Saving[END_REF] framework to introduce vertical heterogeneity related to income and wealth, and we add several household types with different incompressible energy consumption to take into account the horizontal heterogeneity related to living area. We find that in the absence of targeted transfers, carbon taxation hits disproportionately more rural households (with a consumption equivalent equal to -5.8% after the carbon tax increase, against -3.2% for urban households), and is mildly flat across income groups (-4.8% for low income and -4.3% for top income).

Second, taxing households is much more regressive than taxing firms. In our model, energy is both a consumption good for households, and an intermediate input in production for firms, hence we have two carbon taxes. Our mechanisms are closely related to the classical [START_REF] Diamond | Optimal Taxation and Public Production I: Production Efficiency[END_REF] optimal result. Taxing households' energy consumption is regressive, because of the nonhomotheticity of energy consumption. Therefore, poor and rural households are disproportionately hit by the policy. However, taxing firms' energy consumption distorts their optimal input allocation, reducing activity incomes. Those who rely more on labor incomes are more affected than poor people living mostly from public transfers. Those with a higher share of financial income are less affected because taxing firms' emissions reduces the income risk and reallocates towards a more capital intensive sector.

Third, it is possible to reduce emissions and make the policy progressive with respect to income. Uniform lump-sum rebate of carbon tax revenue allows to reduce CO 2 emissions by 18%, while increasing general welfare and reducing income inequality. However, this uniform transfer widens the rural-urban gap: compensating the loss for rural households implies a trade-off between equity and climate efficiency, as they have a high marginal propensity to consume energy. We quantify this tradeoff, and propose different redistribution scenarios with targeted transfers. Moreover, we show that the optimal utilitarian scenario is to make a transfer contingent to both income and location (with a consumption equivalent of +4.5%, against +2.7% for uniform transfers), but this creates much more losers (45% against 10%).

Our paper contributes to the literature assessing the distributive effects of carbon taxation. A first strand of the literature is based on micro-simulations, as [START_REF] Cronin | Vertical and Horizontal Redistributions from a Carbon Tax and Rebate[END_REF] for the U.S. and [START_REF] Douenne | The Vertical and Horizontal Distributive Effects of Energy Taxes: a Case Study of a French Policy[END_REF] in the French context, and generally concludes that carbon taxes, when put on final consumption, are regressive. Our findings on the households carbon tax are similar, but we use a general equilibrium model to take into account the firm intermediate input part of the carbon tax.

As highlighted in [START_REF] Metcalf | The distributional impacts of U.S. energy policy[END_REF], carbon taxes also yield "source side" effects. Therefore, a second part of the literature relies on Computable General Equilibrium (CGE) models. [START_REF] Rausch | Distributional impacts of carbon pricing: A general equilibrium approach with micro-data for households[END_REF] and [START_REF] Goulder | Impacts of a carbon tax across US household income groups: What are the equity-efficiency trade-offs?[END_REF] conclude that the progressivity of source-side effects tends to offset the regressive use-side effects in the U.S., while [START_REF] Ravigné | Is a fair energy transition possible? Evidence from the French Low-Carbon Strategy[END_REF] estimates that the overall effect is still regressive. We also compare the "use-side" and "source-side" effects of the tax, but our income and wealth distribution is endogenous, based on idiosyncratic income risks.

In this regards, we are closer to a third literature, based on modern macroeconomic heterogeneous-agent models. [START_REF] Fried | Understanding the Inequality and Welfare Impacts of Carbon Tax Policies[END_REF] focuses on the optimal recycling of carbon tax revenue, allowing the government to use the carbon tax revenue to lower any existing tax; we focus on targeted transfers, as this option is often favored by government and clearer for citizens. [START_REF] Benmir | The Distributional Costs of Net-Zero: A Heterogeneous Agent Perspective[END_REF] investigates the distributional effects of the U.S. net-zero emissions target; [START_REF] Känzig | The unequal economic consequences of carbon pricing[END_REF] focuses on the EU-ETS market; [START_REF] Langot | The Macroeconomic and Redistributive Effects of Shielding Consumers from Rising Energy Prices: the French Experiment[END_REF] studies the tax-shield implemented in France in 2021. We build on these works by adding living areas heterogeneity through different incompressible energy consumption levels, producing a rich and realistic heterogeneity across households. This framework allows us to consider targeted transfers contingent on location or income, creating a variety of "equity-efficiency" trade-offs for the government. Compared to some of these papers, we also focus on a permanent increase in carbon tax rather than a temporary one, reflecting the fact that the carbon transition will shift the economy towards a new steady state.

Finally, we contribute to the literature comparing the taxation of firms (intermediate goods taxation) and consumers (final commodity taxation) going back to [START_REF] Diamond | Optimal Taxation and Public Production I: Production Efficiency[END_REF], in the the specific context of carbon taxation. In our heterogeneousagent model framework, we obtain that taxing households is more regressive than taxing firms. Traditional mechanisms exposed in the literature to explain the distributive effects of carbon taxation: (i) income effects in [START_REF] Rausch | Distributional impacts of carbon pricing: A general equilibrium approach with micro-data for households[END_REF], (ii) indexation of social income on prices in [START_REF] Metcalf | The distributional impacts of U.S. energy policy[END_REF], (iii) use of multisector macroeconomic modelling in [START_REF] Ohlendorf | Capital is Back: Wealth-Income Ratios in Rich Countries 1700-2010[END_REF], (iv) the crossed effect between subsidies for low-carbon technologies and a carbon tax in [START_REF] Lamb | What are the social outcomes of climate policies? A systematic map and review of the ex-post literature[END_REF] or (v) indirect income effects in [START_REF] Känzig | The unequal economic consequences of carbon pricing[END_REF].

The remainder of the paper is organized as follows. Section 1 presents our quantitative model. Section 2 discusses our calibration choices using French data. Section 3 presents the quantitative results. Finally, Section 4 concludes.

A quantitative Heterogeneous-Agent model

Our main focus is the distributive effect of carbon taxation, so we introduce a rich heterogeneity on the households side, with idiosyncratic productivity shocks leading to income and wealth heterogeneity, and different incompressible energy consumption levels by living areas. Our productive sector is composed of a electricity firm, a final good firm using fossil fuel and electricity as intermediate inputs, and an imported fossil energy. Finally, the fiscal authority has a complete set of instruments, and uses the carbon tax revenue either to increase public spending or to implement targeted transfers.

Households

The economy is populated by an infinite amount of households indexed by i that are heterogeneous in two dimensions. The "vertical" heterogeneity is related to the idiosyncratic productivity process z, creating a distribution for wealth and income. The "horizontal" heterogeneity is related to the living area, with 5 household types k ranking households from "rural" to "urban", depending on the size of the city they live in. The living area determines the level of incompressible energy consumption ē(k), the energy mix parameter γ h (k), and the mean and variance of the idiosyncratic productivity shock, so that the individual productivity is denoted z i (k).

Households maximize intertemporal utility, choosing consumption c, asset a, energy bundle e h (composed of electricity N h and fossil fuel F h with the carbon tax τ h ), subject to their budget constraint, their idiosyncratic productivity process and a borrowing constraint. Each household i of type k solves the following problem:

max {c i,t ,a i,t ,e h i,t ,l i,t ,F h i,t ,N h i,t } +∞ t=0 E 0 ∞ t=0 β t u 1-θ i,t -1 1 -θ -ϕ l 1+ψ i,t 1 + ψ subject to: Λ 1 σ c c i,t u ϵc i,t σ-1 σ + Λ 1 σ e e h i,t -ē(k) u ϵe i,t σ-1 σ = 1 (1) e h = (1 -γ h (k)) 1 ϵ h (N h ) ϵ h -1 ϵ h + γ h (k) 1 ϵ h (F h ) ϵ h -1 ϵ h ϵ h ϵ h -1 (2) (1 + τ VAT ) c i,t + p N t N h i,t + (p F t + τ h t )F h i,t Total consumption expenditures + a i,t+1 -a i,t Savings = Γ(z i,t (k)w t l t ) Net labor income + (1 -τ k )r t a i,t Net capital income + T i,t (k) Transfers (3) z i,t (k) = e x i,t (k) , x i,t (k) = (1 -ρ z )µ z (k) + ρ z x i,t-1 (k) + ϵ i,t , ϵ i,t ∼ N (0, σ z (k)) (4) a i,t ≥ a (5)
Equation 1 implicitly defines utility following [START_REF] Comin | Structural Change With Long-Run Income and Price Effects[END_REF], which is appealing for three reasons. First, it introduces a non-homotheticity for the energy consumption that does not vanish with income: energy represents a higher share of total consumption expenditure for poor households, and stays a non-homothetic good even for high income. Second, this utility function introduces an imperfect substitution between energy and other goods, with a constant elasticity of substitution. Third, we introduce an incompressible consumption level ē(k) that differs across living areas, accounting for higher energy requirement in rural areas compared to urban areas.

Equation 2 is the energy bundle of the household. The elasticity of substitution between fossil fuel and electricity is determined by the parameter ϵ h , and the energy mix depends on the living area with the parameter γ h (k).

Equation 3 is the budget constraint of the household, subject to 4 taxes. Good and energy consumptions are subject to a VAT tax at a rate τ VAT . Fossil fuel with relative price p F t is subject to an excise tax τ h . Labor income is taxed according to a progressive tax rule Γ defined later. Capital income is subject to a flat tax at rate τ k . Finally, households receive transfer from the fiscal authority, that may be contingent to their productivity or their living area.

Equation 4 is the idiosyncratic productivity process. Productivity follows an AR(1) process with normally distributed shocks. We allow the mean µ z and the variance σ z to depend on the type k, which allows us to match the cross-distribution across income and living areas.

Finally, Equation 5 is the borrowing constraint leading to imperfect capital markets. Households cannot borrow more than -a, so that some agents will be constrained and "hand-to-mouths", resulting in high marginal propensity to consume.

Three-sector model 1.2.1 Goods & Services sector

Consumption good y is consumed by households (c), government (G) or foreigners (X t ), or invested by the energy firm (I e ) or the final good firm (I y ). The consumption good is produced competitively using labor l y , capital k y and energy bundle e y (composed of electricity N y and fossil fuel F y with the carbon tax τ f ), according to the following program: max {ly,ky,e y ,F y ,N y ,y}

Π y = y -(r + δ)k y -wl y -(p F + τ f )F y -p N N y such that y = (1 -ω y ) 1 σy k α y l 1-α y σy -1 σy + ω 1 σy y (e y )
σy -1 σy σy σy -1

e y = (1 -γ y ) 1 ϵy (N y ) ϵy -1 ϵy + γ 1 ϵy y (F y )
ϵy -1 ϵy ϵy ϵy -1 [START_REF] Hassler | Directed technical change as a response to natural-resource scarcity[END_REF] points toward a very low short-run substitutability between energy and other inputs once the technology factors have been chosen. This motivates our choice for a CES production function. Moreover, we assume constant return to scale since Lafrogne-Joussier et al. ( 2023) finds a full pass-through of positive energy price shocks on French firm microdata. Finally, the energy used by the firm is a bundle of electricity and fossil fuel, with an elasticity of subtitution governed by the parameter ϵ y .

Electricity sector

Electricity N in our model is a consumption good for households (N h ) and an intermediary input for firms (N y ). We assume electricity is produced competitively using labor l N , capital k N and fossil fuel F N , according to the following program:

max {l N ,k N ,F N ,N } Π N = p N N -(r + δ)k N -wl N -(p F + τ f )F N such that N = l η N k ζ N F 1-η-ζ N
As explained in Section 2, we calibrate l N and F N to be very small, reflecting the fact that energy production is capital-intensive and is a small share of the total labor force and total fossil consumption.

Fossil fuel sector and the rest of the world

Fossil fuel F is imported from the rest of the world, at a fixed price p F . The rest of the world uses this revenue to import goods and services X from the domestic economic.

The budget constraint of the rest of the world is then:

X = p F (F Y + F N + F h )

State

The fiscal authority gets revenue from taxes on labor income, capital income, consumption, and carbon taxation. It uses its revenue to fund lump-sum transfers (T ), public spending (G) and public debt repayment (r t d). Denoting the aggregation x t = 1 0 x i,t di for x ∈ {a, c, e h }, the government has the following budget constraint:

T t + G t + r t d = 1 0 [z i,t w t l t -Γ(z i,t w t l)] di + τ k r t a t + τ VAT c t + p N t N h t + p F t F h t + τ h t (1 + τ VAT )F h t + τ f t (F y t + F N t )
Carbon tax revenue

Following [START_REF] Heathcote | Optimal tax progressivity: An analytical framework[END_REF], we assume a progressive labor tax of the form:

Γ(zwl) = λ(zwl) 1-τ
Apart for the carbon tax revenue, the budget constraint clears with G t . However, the carbon tax revenue can be separately allocated either to finance an increase in public spending, or to fund lump-sum transfers towards households, possibly contingent on income and location. We explore these different scenarios in Section 3.

Market clearing conditions and equilibrium

Finally, to close the model, we have the following market clearing conditions: quantities, such that, for every period t, (i) households and firms maximize their objective functions taking as given equilibrium prices and taxes, (ii) the government budget constraint holds, and (iii) all markets clear.

               1 0 a i,t di = k y,t + k N,t + d (Savings) i z i,t ldi = l y + l N (Labor) y t = 1 0 c i,t di + I e,t + I y,t + G t + X t (G&S) N t = N y t + 1 0 N h i,

Calibration on French macro and micro data

As this paper assesses the distributive effects of carbon taxation, the main point of the calibration is to reproduce the energy mix used by households and firms in France, with a special focus on the consumption heterogeneity related to living area and income. The complete calibration is summarized in Appendix A.

Households

Energy consumption: first, we need to fit the energy consumption heterogeneity both between and within income quantiles. We use ē(k) to match the average energy share in each city types, and γ(k) to have the right energy mix, as shown in Figure 2.a. We use Λ e to match the average energy share in Paris, and ϵ e is used to fit the nonhomotheticity in energy consumption (Figure 2 We estimate σ, the elasticity of substitution between energy and G&S consumption, using National Accounts longitudinal data from Insee (Insee 2022 -NA), ranging from 1959 to 2021. Our regressions are described in Appendix A.3. We get σ = 0.26, significant at the 1% threshold. Thus, we consider energy consumption as a substitute rather than as a complement to other consumption goods. Finally, we set the elasticity of substitution between fossil fuel and electricity to ϵ h = 0.2 because we focus on the upper bound effect of carbon taxation.

Income process: as changes in transfer, labor and capital incomes account for a large part of the distributive effects of carbon taxation, we calibrate carefully the distribution of each type of income. We fit the disposable income distribution1 (Figure 3.a), using the AR(1) persistence parameter ρ that we set equal for all types. We use the mean of the idiosyncratic productivity process for each type µ z (k) to match the ratio of total consumption between types (Figure 3.c), and the variance σ z (k) to match the proportion of each geographical location type within each disposable income quintile (Figure 3.b). Our model recovers that high-and low-income households are concentrated in largest cities. We do not target the MPC, but as shown in Figure 3.d, we obtain an average MPC of 18%, which is in the range of empirical estimates, and is closed to [START_REF] Kaplan | Monetary Policy According to HANK[END_REF]. Other parameters: we set the annual discount factor β to match the French capital to income ratio from Piketty and Zucman (2014) when excluding public debt and housing: a GDP = 2. The borrowing constraint is set at ā = 0.

Firms

Goods and services firm: the energy share is set to ω y = 0.43 to account for the fact that the G&S sector represents 60% of total energy. The elasticity of substitution between energy and the capital-labor bundle is set to σ y = 0.4 following [START_REF] Artus | Fonctions de production avec facteur énergie: Estimations pour les grands pays de l'OCDE[END_REF] and [START_REF] Hassler | Directed technical change as a response to natural-resource scarcity[END_REF]. The capital share is set to α = 0.2 to match the share of labor revenue wl GDP = 65% following [START_REF] Cette | Labor Shares in Some Advanced Economies[END_REF]. The share of fossil fuel in the policy mix is set to γ y = 0.22 such that the G&S firm accounts for 59% of the total fossil fuel. We set the elasticity of substitution between fossil fuel and electricity to ϵ y = 0.2, as discussed above. Finally, the depreciation rate is set to δ = 11.28% like in [START_REF] Auray | Markups, Taxes, and Rising Inequality[END_REF].

Electricity firm: the electricity sector is capital intensive, so we set η = 0.12 to have l N l = 2% and ζ = 0.874 to have F N F = 1%. Finally, the exogenous price p F of the imported fossil fuel is set such that fossil fuel imports account for 2% of the GDP.

Fiscal authority

We set lump-sum transfers according to the rule T (z) = T z i

1 z i di -1
to match the share of transfer in each disposable income quintile, as shown in Figure 4, left panel. We set the labor tax progressivity to τ = 0.1 following [START_REF] Auray | Markups, Taxes, and Rising Inequality[END_REF]. The level of the tax λ is set such that public spending Ḡ makes approximately 29% of GDP. Finally, we calibrate τ h and τ f initial levels so that energy taxes account for 3% of total government revenues2 . The resulting amount of tax paid by each households is shown in Figure 4, right panel. In this section, our welfare results are presented in "consumption equivalent" term: we compute the change in steady-state consumption that would make the household indifferent between the steady-state statu-quo forever and the carbon tax increase path. Then we compute for each initial wealth a 0 and productivity z 0 the following equality:

E 0 ∞ t=0 β t u i,t (c SS (1 + CE), e SS h ) 1-θ -1 1 -θ -ϕ (l SS i,t ) 1+ν 1 + ν |a 0 , z 0 = E 0 ∞ t=0 β t u i,t (c carbon , e carbon h ) 1-θ -1 1 -θ -ϕ (l carbon i,t
) 1+ν 1 + ν |a 0 , z 0 with x SS the path of the variable x without carbon tax increase, and x carbon the path with the carbon tax increase and the new steady state. Finally, in subsection 3.1 and 3.2, we assume the governement clears the budget constraint only by adjusting public spending, to isolate the distributive effects associated to carbon taxation. In 3.3, we consider alternative recycling scenarios for carbon tax revenue, such as lump-sum transfer towards households, that may be contingent to income and location.

Geography trumps income

First, rural households lose more than poor households on average. In Figure 6, we find that the carbon tax is slightly regressive with respect to disposable income, but increases widely the rural-urban gap. In other words, geography trumps income to understand the distributive effects of carbon taxation, as the burden of the tax depends more on living area than on income.

On average, poorer households incur a slightly stronger loss (-4.8%) than top incomes (-4.3%). Indeed, the progressive distortion on labor and capital income, coming from the firms' carbon tax, mitigates the regressive effect of households' carbon tax, coming from the downward nonhomotheticity of energy consumption.

Rural households suffered from a 5.8% welfare loss while Parisian households' welfare only dropped by 3.2%. This is because the intermediate input tax is homogeneous across living areas while the final consumption tax affect disproportionately households with higher incompressible energy levels.

Taxing firms or households: a Diamond-Mirrless story

In this section, we focus on the distributive effects of taxing only households' energy consumption (final consumption tax, τ h ) or only firms' energy consumption (intermediate input tax, τ f ). Therefore, our story is closely related to the classical [START_REF] Diamond | Optimal Taxation and Public Production I: Production Efficiency[END_REF] paper. To provide a relevant comparison of the distributive effect of the two taxes, we calibrate the level of each tax such that they yield the same aggregate welfare loss (-2.5% CE). We conclude that taxing households is regressive while taxing firms is rather progressive or flat, as shown in Figure 7. Two main distortions of our model drive the result: the nonhomotheticity of energy consumption and the heterogeneous distribution of labor and capital income. Taxing households' energy consumption is regressive because it affects disproportionately more households with a higher energy share in total consumption, i.e. poor and rural households. The household tax also reallocates resources from a capital intensive sector (energy) towards a more labor intensive sector (G&S). Therefore, capital income decreases and labor income raises after the tax. The borrowing constraint does not play a significant role at this point.

On the other hand, taxing firms' energy consumption is progressive. As in [START_REF] Diamond | Optimal Taxation and Public Production I: Production Efficiency[END_REF], it pulls away the economy from its technological productivity fron-tier and reduces real wage and energy price. This affects disproportionately more rich households who earn capital and labor incomes, than low income households who rely on social transfers. Yet, the reallocation towards a capital intensive sector (electricity sector N ) mitigates the loss at the top of the wealth distribution. Plus, the drop in activity incomes reduces the idiosyncratic risk for the top of the distribution 4 . Therefore, taxing firms hurts more the middle of the distribution (Q3) that holds a bigger labor income share, and is somewhat flat across geographical types.

The equity-efficiency trade-off of recycling policies

We have shown in subsection 3.1 that geography is more important than income to understand the losers of carbon taxation. Therefore, a natural question arises about the possibility to compensate these losses, using the carbon tax revenue, while preserving the positive effect of emissions reduction. In this section, we allow the government to use the carbon tax revenue to either (i) increase public spending (Benchmark G), (ii) increase uniform lump-sum transfers (lump-sum transfer ), or (iii) implement targeted transfers toward low-skilled and rural households (Target poor × rural ). In Figure 8, we compare these three scenarios in the light of three objectives: optimizing households' welfare, reducing CO 2 emissions and minimizing the number of losers. Indeed, the aggregate welfare is not enough to deal with the political acceptability of the tax: if the redistribution increases total welfare by compensating the more affected households, but creates many little losers, the carbon taxation may still be rejected. While our benchmark increase in public spending was bad for welfare (as G does not enter the utility function), rebating the carbon tax revenues to households through lumpsum transfers is welfare enhancing. In our framework, lump-sum transfers mitigates the uninsured idiosyncratic risks. In other words, the borrowing constraint distortion is strong enough to offset the intermediate input tax distortion. Indeed, the share of losers drops from 100% in the Benchmark G scenario to 10% with the uniform transfer. Additionally, total emissions drop less than in Benchmark G. But the gap is not huge: lump-sum transfers, by mitigating idiosyncratic productivity risks, reduce labor supply and savings motives. Therefore, it is possible to reduce both emissions and disposable income inequalities, and to increase aggregate welfare, at the same time.

However, the "uniform transfer" scenario widens the rural-urban gap, and concentrates losers in rural areas. If we try to target this geographical heterogeneity, a trade-off arises. The bigger policy losers are rural households: a social planner may want to compensate these disproportionate losses through targeted transfers. As shown in Figure 8.a, this "poor and rural" transfer simultaneously reduces vertical and horizontal heterogeneity, and is the best scenario to increase aggregate welfare, which increases by 4.5% compared to 2.7% in the uniform scenario. However, emissions drop by less: -18% vs -17.6%. This is because targeted transfers reallocate resources towards households with high incompressible energy consumption levels. Furthermore, the share of losers bounce back with almost 45% of households losing from the policy. In this scenario, we have much more losers but those losers incur very little losses. They are now overrepresented in large cities and uniformly distributed across disposable income groups.

All in all, we show that there exists a dilemma between compensating rural households, reducing the number of losers to favor political acceptability, and reducing emissions. Through uniform transfers, the government can achieve a high emissions reduction and increase welfare, at the cost of widening the urban-rural gap and concentrating the losers in rural areas. Targeted transfers allow to reach a higher aggregate welfare while compensating the rural households, but this also reduces the decline in emissions, and creates more small losers. It is left to the social planner to choose between these different objectives, taking into account that communication is key in the implementation of carbon taxation. heterogeneity, we highlight living areas heterogeneity, as households face higher incompressible energy consumption levels in rural areas. We simulate a linear 15-year increase in carbon tax, and first show that geography is more important than income to assess the distributive effects of carbon taxation. Second, as energy is both a final consumption good for households and an intermediate input for firms, we compare the distributive effects of taxing only consumers or firms. We find that households' carbon tax is regressive as it affects people with a high energy share, while firms' carbon tax is progressive as it reduces labor and capital incomes. Third, we show that there exists different ways to recycle the carbon tax revenue. A uniform transfer allows to reduce emissions and to increase welfare, but it widens the rural-urban gap, while a targeted transfer towards poor and rural households is more welfare-enhancing but reduces emissions by less and yields more losers among large cities.

We leave for future research the optimal carbon tax revenue recycling policy. We studied here polar scenarios for transfers, leaving aside the possibility to use the revenue to lower existing taxes or invest in the reduction of incompressible energy consumption. We yet believe that transfers are of primary importance for communication and political acceptability, as it explicitly separates the carbon tax revenue from the state budget, making clear that this tax aims at distorting behavior and not at financing public deficit. Finally, a possible research avenue would be to improve our geographical economy, allowing households to move across areas and to work on specific segmented local labor markets.

A Complete calibration A.1 Parameter values

p F F h (k) p F F h (k)+p N N h (k) ϵ h ES between F h and

A.2 Effective carbon taxes

In France, CO 2 emissions by firms and households are not taxed at the same level.

There are three reasons for that. First, energy taxes vary across sectors, energy products and geographical location. Since firms and households do not have the same location, consumption basket or energy mix (households use more oil, especially gasoline, firms consume more electricity and gas), this leads to different effective tax rates. Fuels represent 49% of households' energy consumption when oil products only make 27% of firms' energy consumption. Second, some sectors are part of the European Union Emissions Trading System (EU ETS). This explains why CGDD computations of effective carbon tax rates vary a lot for firms. In 2020, effective carbon tax rate reached 60.8€/tCO 2 against 83.2€/tCO 2 in 2022. Plus, there exists multiple reduced rates and exemptions for firms. To calibrate the initial carbon tax rates for households and firms, we take 2018 estimates, implying an effective rate for households three times larger than for firms. We finally get σ = 0.26192, significant at the 1% level. Restricting our estimation to the 1960-1990 period, we get σ = 0.27242, and taking only the 1990-2021 period we get σ = 0.2477. Therefore, the households' elasticity of substitution between energy and other goods may be a structural parameter of the economy. We use households' budget constraint to decompose between "use-side" and "sourceside" effects:

∂c i ∂τ +

Figure 1 :

 1 Figure 1: Energy share in total consumption

  .b).

Figure 2 :

 2 Figure 2: Energy share in total consumption

Figure 3 :

 3 Figure 3: Consumption across areas and MPCs across income quintiles

Figure 4 :

 4 Figure 4: Income composition and taxes by income quintile

Figure 5 :

 5 Figure 5: Quinet (2019) carbon tax shocks

Figure 6 :

 6 Figure 6: Welfare change in transition (in % CE)

Figure 7 :

 7 Figure 7: Welfare effects of carbon taxes by living area and by income

Figure 8 :

 8 Figure 8: Comparing recycling policies

Figure 9 :Figure 10 :

 910 Figure 9: Effective carbon tax for France 2020

Figure 11 :

 11 Figure 11: Effective carbon tax for France, CGDD computations

Figure 12

 12 Figure12shows this decomposition by productivity types for rural households. Top panels show what happens when you increase only τ h while bottom ones deal with an increase of τ f . The change in savings show how the borrowing constraint interact with the two carbon taxes.

  t di (Electricity) Households' savings are invested in capital in both sectors and in public debt, and labor supply is also allocated within both sectors. By no-arbitrage, we only have one wage and one interest rate in the model. The G&S production (y t ) is consumed by households (c t ), government (G t ) or foreigners (X t ), or invested by firms (I e,t , I y,t ). Electricity N t is consumed as intermediate inputs by firms (N f t ), or as a commodity good by households (N h t ). An equilibrium in this economy is defined as paths for households decisions {c t , N h t , F h t , l t , a t } t , G&S firm decisions {y t , l y,t , k y,t , F y t , N y t } t , electricity firm decision {N t , l N,t , k N,t , F N t } t , relative prices {r t , w t , p N t } t , fiscal policies {τ h t , τ f t } t , public expenditures {T t , G t } t , and aggregate

Table 1 :

 1 Parameter values and steady-state targets.

	Parameter Description	Value	Notes
	Households
	β	Discount factor	0.92	a GDP = 2
	θ	Intertemporal ES	1	Authors choice
	1/ψ	Frisch elasticity	3	Lee (2021)
	ϕ	Labor disutility	1	Normalization
	σ	ES between c and e h	0.26	Authors calculation
	Λ e	Energy share	0.155	p h e h p h e h +c
	ϵ e	Non-homotheticity parameter	0.8	Energy share by disposable income
	Λ c , ϵ c	Utility parameters	1	Normalization like in Comin et al. (2021)
	γ h (k)	Fossil share	[0.60, 0.67, 0.685, 0.695, 0.73]

from the

Insee survey "Revenus et patrimoine des ménages"(RPM 2021)

Additionally, we use the effective carbon tax estimates from the French Sustainable Development Agency (CGDD) to account for different energy mix and exemptions across households and firms. See Appendix A.2 for more details.

See Appendix A.2. In our online Appendix, we perform the same increase in carbon taxes, but considering that the household-firm gap in effective carbon taxation rates will maintain over time.

ConclusionIn this paper, we assess the distributive effects of carbon taxation with a quantitative heterogeneous-agent framework. While most of the literature has focused on income