Emmanuel Rollinde
email: emmanuel.rollinde@u-cergy.fr

Enacting Planets to Understand Occultation Phenomena

Keywords: Solar System, Science education, Mathematics education, Enation, Newtonian dynamics

The Solar System motivates students to interest themselves in sciences, as a large number of concepts may be easily introduced through the observation and understanding of the planet's motion. Using a large representation of the Solar System at a human scale ("a human Orrery"), we have conducted different activities with 10 to 16 years old children. In this contribution, we discuss the different scientific concepts covered by the Human Orrery, allowing the connection of both science and mathematics subjects in schools. We then detail how this pedagogical tool may serve to introduce abstract concepts required to understand occultation phenomena through a modelling activity.

Introduction

In 2013, a program called "kinaesthetic activities in teaching science and humanities" was granted by Sorbonne University, France, connecting UPMC (departments of physics and biology), and Paris Sorbonne (departments of sports, Italian, and ancient Greek). A "human Orrery" was thus created by [START_REF] Rollinde | Conf. VIIIe colloque Question de Pédagogie dans l'Enseignement Supérieur[END_REF], allowing the learners to enact the planets' movement with correct relative speed (Figure 1). An Orrery is a mechanical or digital device designed to illustrate the motion of the planets around the Sun and their changing positions in the sky. On a human Orrery, the orbits of planets and comets are drawn at a human scale allowing movements in the Solar System to be enacted by the learners. Astronomy, in general, provides a highly motivating context for learners to develop observational skills, discover methods of scientific inquiry, and explore some of the fundamental laws of physics and concepts of mathematics in both an attractive and meaningful way. The implementation of Human Orrery, as a learning experience, involves topics from mathematics and science and illustrates an example of a STEM approach of learning about concepts perceived as abstracts by students and teachers. Through our initiative, seven Human Orreries were built in France (one for a science center, one in a public place in Paris -Le Jardin des Plantes, five in primary or secondary schools), one was drawn in a Lebanon school and one map was purchased by a science center in Vietnam. Human Orreries are used thereafter by different teachers (physics, mathematics, technology) in those places. The topics of different sequences tested since 2015 are described briefly. Details may be found in the references or through direct requirement to the author. (i) Construction of a human Orrery. The description of the Solar System involves different length scales from the diameter of the planets to the distance between planets and the length of one orbit. It also involves different duration scales from the rotation to the orbital period. The construction of a Solar System implies a choice of length scales and orientation and the drawing of ellipses. The links with mathematics are obvious, including placement on a 2D plan, Euclidian division and simple geometry. (ii) Enacting planets to learn physics. In the last 3 years, planets have been enacted by undergraduate students in Paris as well as pupils of primary and secondary schools in Paris, Nantes, and Beirut on their own Orrery or on Sorbonne University one. Through the eccentric orbit of a comet, students have observed its varying velocity. They then realized why force and speed are not directly related, and how the work-energy theorem is connected with real movements… 16-year-old students have observed, measured and plot Kepler's laws and experiment different referential frames in astrophysics context [START_REF] Rollinde | Learning Science Through Enacted Astronomy[END_REF][START_REF] Rollinde | Enacting planets to learn physics[END_REF] with significant improvement in their understanding and use of change of frames. 12-year-old pupils have experienced the complex relationship between duration, distance, and velocity with a specific insight that a larger distance may be travelled in a larger amount of time if the velocity is smaller [START_REF] Abboud | Enhancing mathematics and science learning through the use of a Human Orrery[END_REF]. 10year-old pupils have made a movie about shooting stars (encounter of Earth and a comet during the night, on planetaire.overblog.com) and pre-service teachers were introduced into the use of the Human Orrery and astronomy, in general, to set up different sequences on geometry, velocity, localization of planets and orbital periods. In this contribution, the design and operation of the Human Orrery will be shortly described, together with a short theoretical background on enaction theory. Then, a specific proposition that concerns specifically occultation phenomena will be presented.

The Human Orrery

A Short Manual

The design of a Human Orrery is made such that users may walk along orbits of different bodies around the Sun that is located at the center of the design. The Human Orrery that we use is printed in a large map of 12m by 12m (Figure 1). It allows one to follow the orbits of the inner planets (Mercury, Venus, Earth, and Mars) and Jupiter; the inner planets are located inside the asteroid belt that is materialized with a grey colour together with the orbit of the largest asteroid known, Cérès. The highly eccentric orbits of two comets are also used: Encke and Chury. This choice of objects illustrates different types of movements while keeping the size of the Orrery reasonable. Earth is at one meter from the Sun (spatial scale), while Jupiter's orbit has a diameter of 10,5m.

The orbits of all bodies are materialized by dots at constant intervals of times, with accurate elliptical shapes. Note that orbits are near-circular for the five planets and Ceres. The interval of time may be different for each orbit, but is always a multiple of 16 terrestrial days: For Earth, there are 23 dots separated by 16 days, which would make a period of 368 days. For Jupiter, there are 54 dots separated by 80 days, which makes a period of 4320 days instead of the real period of 4332,59 days. Note that this difference creates a difference between the walk on the Orrery and the walk of the real planet at each round-about 3 days too late for Earth which combines into 15 days after 5 terrestrial years. To account for this difference, one has to jump one point after 5 round on Earth. This may be used as an introduction to the Euclidian division in mathematics -how to divide 365,25 days into intervals of 16 days (365,25 = 23*16 -2,75). A sound (either a clock or hand claps) is heard regularly. Every user makes one step during this time interval. The person that enacts Earth walks from one point to the next (distance) in one step (duration). Hence, the interval of time between two sounds corresponds to 16 terrestrial days (temporal scale). "Jupiter" has to do five steps (five times 16 days) to reach the next point. All rules are described in [START_REF] Rollinde | Learning Science Through Enacted Astronomy[END_REF] and [START_REF] Rollinde | Enacting planets to learn physics[END_REF]. By acting according to those rules, the movements on the Human Orrery illustrate the correct relative velocities of all Solar System bodies.

Scientific Concepts in Use

We review now the concepts associated with the solar system in general and the Human Orrery specifically, and the conceptions that may hinder this knowledge. The order chosen follows the chronology of learning in the French school context. Firstly, the factual knowledge of what makes up the solar system is part of the astronomy literacy that must be taught at school, including the great diversity of bodies in the solar system, from small rockets (asteroid and comets), small planet with or without atmosphere, telluric planets, gaseous planets... The words "meteor, meteorites, shooting stars" will come into the discussion and allow the teacher to clarify all definitions. Phenomena such as seasons, eclipses, phases of the Moon, etc. may be introduced too. Note that since the printed Human Orrery represent the correct elliptical orbits, it may be observed that Earth is closer to the Sun during winter in the northern hemisphere. Human Orrery is best suited for the study of movements, including reference frames, notions of distance and duration related by speed, inertia, forces, and acceleration. There is no doubt that the difficulties associated with these notions are present at almost all levels in schools and for the general public too. Overall, cognitive schemata in the description of movements are very often adapted to the tasks of everyday life and thus lead to erroneous conceptions of more complex situations [START_REF] Viennot | Raisonner en physique: La part du sens commun[END_REF]. In the first place, the transition from one frame to another is difficult because it is not done in everyday life [START_REF] Saltiel | Spontaneous ways of reasoning in elementary kinematics[END_REF]. In particular, the difference between heliocentric and geocentric landmarks has historically led to a paradigm shift. However, it is often misinterpreted as characterizing a fair model and a false model [START_REF] Shen | Justifying Alternative Models in Learning Astronomy: A study of K-8 science teachers' understanding of frames of reference[END_REF]. Secondly, the perception of speed difference is often understood assuming constant distances or constant durations. Taking simultaneously into account speed, distance and duration is problematic. This difficulty is clearly not solved by the usual teaching based on formulas or graphs [START_REF] Thompson | Investigation of student understanding of the concept of velocity in one dimension[END_REF][START_REF] Thompson | Investigation of student understanding of the concept of velocity in one dimension[END_REF][START_REF] Trudel | Favoriser la compréhension des concepts du mouvement rectiligne à vitesse constante à l'aide d'une investigation scientifique assistée par ordinateur[END_REF]). Finally, the concept of time is fundamental in an enacted framework as described here. The word "time" hides very different notions, whether biological, psychological, physical or measurement. We use the terms of [START_REF] Coquide | Espace et temps dans l'enseignement des sciences et des technologies[END_REF] which describe time as "the transformation of space", and, referring to Bergson's work, indicate that "duration is considered as characteristic of a vital momentum, partially biological and evolutionary, and in a philosophy of continuity ". We understand then the essential difficulty of an adequate understanding of time in the study of a movement at all levels of study, including university. The next step is to investigate the causes of speed variation, or acceleration. The movements of bodies in the Solar System and in our Galaxy are governed by one and the same law (for large bodies with a diameter greater than 30-40 km): that of universal gravitation combined with the principle of inertia, in the framework of Newtonian mechanics. The accounting of Einsteinian mechanics is beyond our approach here. In this area, the prevalence of intuitive or incorrect notions of inertia into adulthood has often been studied [START_REF] Viennot | Raisonner en physique: La part du sens commun[END_REF][START_REF] Trudel | Favoriser la compréhension des concepts du mouvement rectiligne à vitesse constante à l'aide d'une investigation scientifique assistée par ordinateur[END_REF]. It has been noticed that difficulties arise when attention is directed to the trajectory of the moving object (circular in particular). In these cases, the false link between the force on the object and its speed is difficult to challenge in the mind of the learner.

Enation Theory as a Theoretical Background

The use of a Human Orrery in education is based on the assumption that bodily perceptions help the learning of abstract concepts. The conversion of learning space into performance space means that embodiment becomes a vehicle for interpretation [START_REF] Lindgren | [END_REF] and movement a medium for choreographing thought (Lapaire, 2017). New connections are set up between mental activity and kinetic activity; sense-making and sensory-motor experience. Such an assumption is based on the cognition theory of enaction [START_REF] Varela | The Embodied Mind: Cognitive Science and Human Experience MIT Press[END_REF] 2014) "a sensuous cognition". Abstract concepts must then be embodied and experienced in sensory experiences to be fully integrated (Glenberg, 2015); the environment becomes then an actor of reflection under the condition that it favours a fine perception [START_REF] Varela | The Embodied Mind: Cognitive Science and Human Experience MIT Press[END_REF]. This distinction between fine perception and coarse perception proposed by Varela brings us closer to the conclusions of didactics.

To be in line with the enaction theory, as shortly described above, the use of the Human Orrery has to provide students with an adapted or congruent sensory pathway to the abstract notions to be learned [START_REF] Segal | Do Gestural Interfaces Promote Thinking? Embodied Interaction: Congruent Gestures and Direct-Touch Promote Performance in Math[END_REF]. We refer the reader to [START_REF] Rollinde | Enacting planets to learn physics[END_REF], which provides a "dictionary of the Human Orrery" that describes in detail how gestures and movements on the Orrery are associated with abstract concepts of kinematics.

Medialisation of Occultation

We propose in this section an illustration of the Human Orrery adapted to the topic of occultation. The first stages of this sequence, up to the observation of one planet that pass another one, has been tested in September 2019 with French 12-year-old pupils of the junior high school Pailleron located in Paris.

Figures in the text are extracted from those sequences. The computational part of the sequence, namely the prediction of periodicity of occultation was tested a few years ago with 1st-year University students in France.

The sequence is based on the modelling principles (Halloun & Hestenes, 1985;[START_REF] Sensevy | An epistemological approach to modeling: Cases studies and implications for science teaching[END_REF] that require regular connections by the learners between the experimental world and the world of theories and models. A model allows to represent salient characteristics of experimental objects; communicate about specificities of the objects; unify different classes of objects. The connection between the two worlds is made through different activities:

(1) Within the experimental world, learner must use welldefined words, make connections, compare and classify objects and actions. (2) Through such activities, a model may emerge with specificities and general rules. (3) Those rules are used to make predictions of specific observations that will be tested in the experimental world. Failure of those predictions will force to come back to actions (1) and ( 2). ( 4) Manipulations within the model such as enrichment of the model, to better define concepts, to infer new predictions, to propose an application of the model for new experimental objects. The Human Orrery is a model of the Solar System. Learners recognize the Solar System primarily by the presence of the Sun at the centre and occasionally through the identification of orbits. The model (representation of real objects) is thus already present with some specific features only (e.g. distances and not sizes). The experimental world cannot be accessed directly, neither duration of movements nor distance between objects. This ability to enact objects (here, planets) with our body and at our human scale is natural to human beings [START_REF] Lapaire | Mental Action as Visible Bodily Performance: An Educational Perspective[END_REF][START_REF] Mcneill | Hand and Mind: What Gestures Reveal About Thought[END_REF]. Modelling process must then be understood from an enacted experimental world to abstract models.

Dance of the Planets

The activity starts with a review of the planets, inquiring about "missing" ones on the Human Orrery. In the middle school where the sequence was tested, the Human Orrery includes only inner planets and one comet. Students easily spot the Sun. The names of planets emerge collectively. Then, some individuals in the group usually search for the moon (whatever the age of the group). They start by choosing one dot that corresponds to the comet Ceres and that is near one of the dot on Earth orbit. Some individuals may also choose Ceres or even Mercury whose image (when printed on the map) looks a bit like the moon. Discussing about the distance between Earth and the moon discards those possibilities: the moon would be under the dot of Earth and is thus not represented on the Human Orrery. This also allows the teacher to remark that the sizes of each object are not at scale. Going further, and following the position of moon orbiting around the Sun would prove that the moon's trajectory is at the first order of approximation a circle around the Sun. The positions of the planets at the date of the day are then provided to the learners or obtained through a specific software (e.g. Stellarium©, Figure 2 The choreography may then be started. The teacher claps regularly, and each planet makes one step between two claps. All actor-planets reach the next point along its orbit at each single step, except Mercury that jump every second point. At least one individual (usually Earth) tried to orbit around itself. This is obviously too difficult to be done for a long time. It must be explicated by the teacher that this specific feature is not conserved in the Human Orrery model. Without explicit instruction by the teacher, each actor may start in a different direction. There is indeed no specific a-priori reason for all planets to orbit in the same direction. This may be an opportunity to discuss the global dynamic of a gravitational system (if appropriate given the age of the learners, and the duration of the activity) that ends up with all planets orbiting in the same direction. Once all those aspects are fixed, all actor-planet walk with correct relative velocity and one may observe the evolution of their respective position with time. One classical prediction, not detailed here, is the evolution of the planets that may be observed from Earth in the nocturnal sky. In many successive attempts, five learners play the role of the inner planet and the Sun while others observe. Different actors are selected each time so that all may experiment with the dance of the planets. A single instruction is given to both observers and actors: "raise your hand as soon as one planet pass by another one. Then, the actors must stop, and we will observe the respective position of the planets at that instant." Figure 3 provides the reader with two examples of such events, involving Mercury and Earth (right), and Venus and Earth (left). The definition of overtaking was quite obvious for all. Yet, we may notice that some learners raised their hand as soon as they observed that one planet was going faster than another one. The gamification of the activity was obvious. Consequently, almost all individuals tried to spot overtaking, even those who were not interested in the activity at first. For a few attempts, the teacher lets the learners spot any planets. Then, the teacher asked to raise their hand only if Mars was involved in an overtaking. It then appears that Mars is never the one that pass by. The connection with the different speed was immediately perceived by most learners who almost shouted "but of course, Mars is travelling so slowly!". Next, the teacher asked to raise their hands only if Mercury was involved. Again, the connection with the speed of Mercury appears to be obvious for learners.

When Planets Cross Each Other

At each stop of the choreography, the teacher asked if observers have noticed something special in the relative position of planets. The answer was not obvious here. In one of the group, the Actor-Sun played a very important role here, revealing to others that s(he) was always in line with the two planets. In other groups, the teacher had to highlight the relative position. In all cases, the concept of alignment emerged at some point from the learners themselves.

The activity for middle school students stop here, with the written conclusion that (1) planets have different speed, (2) when one planet pass by another one, they are aligned with the Sun, (3) Mercury is the fastest planet and overtakes all planets while Mars is the slowest of all inner planet and never overtakes any inner planets. The activity was then illustrated by a photo of the transit of Venus.

Prediction

Playing the choreography during several terrestrial years, learners may directly observe and measure the periodicity of, e.g., the transit of Venus. This period is found to be 575 days… Note that many successive transits should be observed to check the actual periodicity of the transit. It should also become clear that the transit does not occur at the same position every 575 days. Students at University level were asked to create a simple iterative code that computes the position of Earth and Mars every 16 terrestrial days. They also compute the position that Venus should have along its orbit to create an occultation for every single position of Earth along its own orbit (the "occultation position", intersection of Venus orbit with the segment that connects the position of Earth to the Sun). They could then make a graph (Figure 4) showing the distance between the actual position of Venus to the occultation position related to the actual position of Earth every 16 days. The occultation is predicted every time this distance equals zero. Thus, the periodicity of the occultation is made obvious. Doing either one of the methods, one obtains a period of 575 days ±3 (this uncertainty is obtained with the numerical method). As explained in Section 2.1, orbital periods in the Human Orrery have different offset compared to the real ones for each planet. Then, after many turns, the actors are in advance or late along the orbits as compared to the real trajectory. This may be taken into account in both the choreography or the numerical code, and yields to a periodicity of 586 days ±3. Those predictions may be compared to the theoretical values obtained by comparison of orbital periods. One may first wonder when Earth and Mars get back to the same positions simultaneously. This corresponds to 8 terrestrial years, 5 times the period we obtained. Yet, transit occur at different positions. The correct theoretical period is obtained using: 1/T = (1/Tvenus -1/TEarth), which corresponds to a periodicity of 584 days, in agreement with our prediction using the Human Orrery; However, observations of the real phenomena reveal that the transit of Venus occurs very rarely, with "pairs" of transit separated by 8 years with a periodicity of about 120 years.

This huge discrepancy forces the learners to question the validity of the model used for this activity. It appears that the Human Orrery is in 2D, while Venus orbital plane is not aligned with Earth orbital plane. Thus, the transit may occur only in two specific positions where the two planes intersect. This explains why the prediction of the model was wrong.

Conclusion and Perspectives

The perspective that we adopt in this work is an interdisciplinary approach that incorporates an embodied dimension and draws on real-world modelling. The use of the Human Orrery enables students to enact or "experience" scientific concepts and the dynamics of their properties by evolving in an adapted environment. In this contribution, we have detailed the use of a Human Orrery to help understand the occultation phenomena.

The current dynamic of the project is to motivate more classes from different levels and teachers from subjects other than mathematics and physics such as art, technology, music, languages (native or foreign), physical education and sport. In 2018-2019, more schools have decided to join the project in France: two primary schools in the context of interdisciplinary projects and one secondary school through a project involving science and technology. At the European level, a consortium is emerging involving researchers in mathematics and science didactics. Our next objective is then to design sequences that fit into the STEAM approach and that will be used as experimental situations for the theoretical framework of enaction, complemented with the framework of Activity Theory in the context of science education [START_REF] Abboud | Annales de didactique et de sciences cognitives SI[END_REF]. Our focus will be as much on the didactical learning as on the motivation and awareness of the learners during the activity.

Figure 1 .

 1 Figure 1. The "Human Orrery": Pupils enact planets (Mercury to Jupiter), asteroids and comets (in black) along their orbits around the Sun

  , left). Learners will then have to reproduce those positions on the Human Orrery (Figure2, right). To do so, they need to search for specific geometrical configurations. Here, they reproduce the observed alignment of Earth (rightmost in the figure, blue dot), Sun and Mars (leftmost on the Figure, red dots). Some learners explicitly explained to the group that "Earth and Mars are opposite" (omitting the word Sun, that is implicit).

Figure 2 .

 2 Figure 2. Initial positions of the inner planets at the date of the activity as seen on ©Stellarium (left panel; in French) and reproduced by students (middle panel, from left to right: Mars, Venus, Mercury, Sun and Earth).

Figure 3 .

 3 Figure 3.Left: Venus (yellow dots) overtakes Earth (blue dots). Right: Mercury (brown dots, uneasy to distinguish here) overtakes Earth. The black line highlights the alignment of the two planets with the Sun.

Figure 4 .

 4 Figure 4. Evolution of the distance to the occultation point for Venus and Earth with time (see text for details).

  that is already well known and widely used in Science and Mathematics Education, e.g. Johnson-Glenberg et al. (2016). In recent years, research on learning and education has been increasingly influenced by theories of embodied cognition. Several embodiment-based interventions have been empirically investigated, including gesturing, interactive digital media, and bodily activity in general. The setting-up of our designs bring us to the highest level of embodiment as defined by Johnson-Glenberg et al. (2014) as it is using integrated forms of embodied learning and a high level of bodily engagement. Enaction theory considers the interaction between the body and the outside world as the foundation of cognition, what Lapaire (2017) calls "Kineflexion" and Radford (

Acknowledgments

The authors express their special thanks to the teachers of the junior high school Pailleron, who have always supported the implementation of the Human Orrery within their project « Des Pommiers sur Mars » (apple trees on Mars): M. Hauss, Mme Bessonies, Mme Bouchez Pommier, Mme Nardou, Mme Samali, M Saussez, and Arlette.


Claudio Cometto

Lingjing Chen

Elodie Anxolabehere-Mallart

Claire Fave

Tai-Chu Lau

Marc Robert

Molecular Electrochemical Catalysis of the CO 2 -to-CO Conversion with a Co Complex: A Cyclic Voltammetry Mechanistic Investigation

The electrochemical catalytic reduction of CO 2 into CO could be achieved with excellent selectivity and rate in acetonitrile in the presence of phenol with cobalt 2,2′:6′,2″:6″,2‴-quaterpyridine complex [Co II (qpy)(H 2 O) 2 ] 2+ (Co) acting as a molecular catalyst. Upon using cyclic voltammetry at low and high scan rate (up to 500 V/s) two catalytic pathways have been identified. At a low concentration of phenol (<1 M), catalysis mainly occurs after the reduction of Co with three electrons. In that case, the selectivity for CO production is ca. 80% with 20% of H 2 as by product, along with a turnover frequency of 1.2 × 10 4 s -1 for CO production at an overpotential η of ca. 0.6 V. The triply reduced active species binds to CO 2 and the C-O bond is cleaved thanks to the acid. At very large concentration of phenol (3 M), another pathway becomes predominant: the doubly reduced species binds to CO 2 , while its reductive protonation leads to CO formation. As already shown, this later process is endowed with fast rate at low overpotential (turnover frequency of 3 × 10 4 s -1 at η = 0.3 V) and 95% selectivity for CO production. By varying the phenol concentration and the scan rate in voltammetry experiments, it was thus possible to identify, activate, and characterize several pathways for the CO 2 -to-CO conversion and to decipher Co electrochemical reactivity toward CO 2 .

INTRODUCTION

The electrochemical reduction of CO 2 has attracted in the recent years a lot of attention, due to the possibility of producing value added chemical as well as fuels, such as methanol or light hydrocarbons. [START_REF] Jhong | Electrochemical conversion of CO 2 to useful chemicals: current status, remaining challenges, and future opportunities[END_REF][START_REF] Aresta | Catalysis for the valorization of exhaust carbon: from CO 2 to chemicals, materials, and fuels[END_REF] It may have an enormous impact on our industry (electrification of the industry) as well as on our societies. [START_REF] Tatin | A case for electrofuels[END_REF][START_REF] Seh | Combining theory and experiment in electrocatalysis: Insights into materials design[END_REF] In this perspective, CO 2 would be used as a renewable feedstock for storing unlimited, free solar energy (initially transformed into electricity) before reutilizing it for various applications (heating, transportation, etc.). Molecular electrocatalysts are mainly capable of transforming CO 2 with two electrons and two protons into either CO or formic acid, and they have been recently reviewed with great care. [START_REF] Costentin | Catalysis of the electrochemical reduction of carbon dioxide[END_REF][START_REF] Qiao | A review of catalysts for the electroreduction of carbon dioxide to produce low-carbon fuels[END_REF][START_REF] Loewen | Renewable formate from C-H bond formation with CO 2 : using iron carbonyl clusters as electrocatalysts[END_REF][START_REF] Elgrishi | Molecular polypyridine-based metal complexes as catalysts for the reduction of CO 2[END_REF][START_REF] Grice | Carbon dioxide reduction with homogenous early transition metal complexes: Opportunities and challenges for developing CO 2 catalysis[END_REF][START_REF] Grills | Mechanistic aspects of CO 2 reduction catalysis with manganese-based molecular catalysts[END_REF][START_REF] Kuramochi | Reaction mechanisms of catalytic photochemical CO2 reduction using Re(I) and Ru(II) complexes[END_REF][START_REF] Francke | Homogeneously catalyzed electroreduction of carbon dioxide-Methods, mechanisms, and catalysts[END_REF] Catalysts employing noble metals (e.g., Ru, Re, Pd) as well as earth-abundant metals (e.g., Co, Mn, Fe) have been used, although the later ones are less numerous. Among those recently proposed for CO 2 -to-CO catalytic conversion, the cobalt 2,2′:6′,2″:6″,2‴-quaterpyridine [Co II (qpy)(H 2 O) 2 ] 2+ (Scheme 1, noted Co) is one of the most promising examples in terms of selectivity, durability, and catalytic rate, being active in acetonitrile (ACN) as solvent both in electrochemical [START_REF] Cometto | Highly selective molecular catalysts for the CO 2 -to-CO electrochemical conversion at very low overpotential. Contrasting Fe vs. Co quaterpyridine complexes upon mechanistic studies[END_REF] and in photochemical conditions under visible light irradiation. [START_REF] Guo | Highly efficient and selective photocatalytic CO 2 reduction by iron and cobalt quaterpyridine complexes[END_REF] This complex (having 2 Cl atoms as ligand in axial position instead of the water molecules) was recently incorporated in a catalytic ink comprising a mixture of multiwalled carbon nanotubes plus a minimal amount of Nafion and was deposited onto porous carbon paper. [START_REF] Wang | Hybrid Co quaterpyridine complex /carbon nanotube catalytic material for CO 2 reduction in water[END_REF] It led to exceptional catalytic activity for CO production in pure water at close to neutral pH (7.3) with 99% selectivity and current densities up to ca. 20 mA/cm 2 (geometric area) at a low overpotential of 425 mV. It is currently the most active and selective molecular supported catalyst for the electrochemical reduction of CO 2 in water. We have reported a preliminary study on the catalytic mechanism in organic solvent in the presence of a highly concentrated proton source (3 M phenol). [START_REF] Cometto | Highly selective molecular catalysts for the CO 2 -to-CO electrochemical conversion at very low overpotential. Contrasting Fe vs. Co quaterpyridine complexes upon mechanistic studies[END_REF] The role of the acid is to boost the C-O bond cleavage. At such high phenol concentration, a first reduction wave produces the Co I one-electron-reducedspecies (OERS) that reversibly binds two molecules of phenol in axial position. Upon a second reduction process at potential close to ca. -1.25 V vs SCE, the catalyst was shown to activate CO 2 and lead to a maximum turnover frequency of 3.3 × 10 4 s -1 at an overpotential as low as 300 mV. However, a more detailed mechanistic study probing the effect and the importance of the applied electric potential and acid concentration on the catalytic pathway is missing. V vs SCE in the absence of any added acid at a potential corresponding to a reduction wave that was assigned to ligand reduction, suggesting that the CO 2 may be activated at various potentials by different species. [START_REF] Lam | -quaterpyridine as catalysts for electrochemical reduction of carbon dioxide[END_REF] Different catalytic pathways may thus operate depending on the acid content of the solution and the applied potential.

In this study, we have dissected in detail the reaction pathways for the electroreduction of CO 2 into CO as a function of phenol addition to the solution and of the potential applied to the electrode upon using cyclic voltammetry from 0.1 to 500 V s -1 along with complementary controlledpotential electrolysis. It appeared that CO 2 could be activated at various potentials depending on the phenol concentration from a doubly or a triply reduced catalyst, respectively. Mechanisms and catalytic rate constants for both processes have been obtained.

RESULTS AND DISCUSSION

The cyclic voltammetry of [Co(qpy)(H 2 O) 2 ] 2+ under Ar atmosphere (Figure 1) shows a first reversible one-electron reduction wave centered on the metal with E 0 1 (Co II /Co I ) = -0.565 V vs SCE, followed by a second one with a standard redox potential E 0 2 = -0.795 V vs SCE (see below). Upon increasing the H 2 O concentration or the scan rate (Figure 2), the second reduction wave evolves to a small plateau-shape current, indicating that it is preceded by the loss of water molecules from the one-electron-reduced Co I species (C + E mechanism), [START_REF] Saveánt | Elements of Molecular and Biomolecular Electrochemistry: An Electrochemical Approach to Electron Transfer Chemistry[END_REF] as already hypothesized by Che et al. [START_REF] Lam | -quaterpyridine as catalysts for electrochemical reduction of carbon dioxide[END_REF] The following reactions summarize the two redox processes:

[ ] + [ ] + - + Co (qpy)(H O) e Co (qpy)(H O) II 2 2 2 I 2 2 V (1)
[ ] [ ] + + + Co (qpy)(H O) Co (qpy) 2H O I 2 2 I 2 V (2)
[ ] + [ ] + - • - Co (qpy) e Co (qpy)
I I V (3)
The second reduction appears localized on the ligand, [START_REF] Cometto | Highly selective molecular catalysts for the CO 2 -to-CO electrochemical conversion at very low overpotential. Contrasting Fe vs. Co quaterpyridine complexes upon mechanistic studies[END_REF] as suggested by DFT calculation (spin-density analysis) so that the neutral[Co(qpy)] (noted [Co I (qpy)

•-] in reaction (3)) can be viewed as a CO I species coordinated by a ligand radical anion. This doubly reduced species undergoes adsorption on the electrode surface, as inferred from the sharp desorption peak observed upon reoxidation scan (oxidation peak at ca.

-0.78 V vs SCE, Figure 1) and from a second desorption peak (E p = -1.51 V vs SCE, Figure 1) that corresponds to the third reduction wave of the complex. The adsorption/desorption phenomena made it difficult to obtain qualitative and quantitative data for Co reduction from the analysis of cyclic voltammetry (CV) experiments at low scan rate (i.e., typically 0.1 V/s). However, when raising the scan rate, the charge passing through the electrode surface diminishes, and the amount of adsorbed species could thus be minimized. This is what can be seen from Figure 3, where a scan rate of up to 70 V/s was applied, leading to a purely diffusion-controlled third reduction peak (see Figure S1). A standard redox potential E 0 = -1.51 V vs SCE was obtained from this reduction wave that corresponds to the generation of the anionic [Co(qpy)] - species.

As already described, [START_REF] Cometto | Highly selective molecular catalysts for the CO 2 -to-CO electrochemical conversion at very low overpotential. Contrasting Fe vs. Co quaterpyridine complexes upon mechanistic studies[END_REF] upon addition of phenol (PhOH) to a solution of Co, [Co I (qpy)] + binds two PhOH molecules: When a solution of Co is purged with CO 2 in absence of added acid, no catalysis occurs upon the first three cathodic peaks. Conversely a catalytic wave is produced at more negative potential (potentials negative to ca. -1.62 V vs SCE, Figure S3), as already reported by Che et al. [START_REF] Lam | -quaterpyridine as catalysts for electrochemical reduction of carbon dioxide[END_REF] We did not investigate this wave further. Upon gradually increasing the phenol concentration, a current enhancement is observed upon the third reduction peak (Figure 5).

Concomitantly, adsorption phenomena decrease as attested by the progressive disappearance of the reoxidation peak at ca. -0.8 V vs SCE, showing that phenol limits interactions between the electro-generated intermediates and the electrode surface. Indeed, formation of [Co I (qpy)(PhOH) 2 ] + instead of [Co I (qpy)] + upon the first reduction wave avoids [Co I (qpy) •-] production upon the second reduction wave, which is the species prone to adsorption. The complex shape and continuous positive shift of the catalytic wave upon increasing [PhOH] suggests the occurrence of several catalytic processes.

By increasing the scan rate between 50 and 500 V/s for very high phenol concentrations (1-3 M), the catalytic peak was distinctively split into two different waves (Figure 6a). The first one is centered at ca. -1.25 V vs SCE (indicated by an arrow in Figure 6a), while the second one is observed at ca. -1.55 V. Both catalytic waves show plateau current at 500 V/s (Figure 6a). For a fixed scan rate of 250 V/s, the first catalytic wave becomes the only process to be observed for 3 M phenol (Figure 6b). Concomitantly, the reductive wave at -0.95 V vs SCE corresponding to the formation of [Co I (qpy) •-] almost disappears. These data suggest that two catalytic pathways could be activated by adding phenol as proton source, at overpotentials close to ca. 0.3 and 0.6 V respectively (E 0 CO 2 /CO = -0.96 V vs SCE in this solvent conditions). [START_REF] Azcarate | Throughspace charge interaction substituent effects in molecular catalysis leading to the design of the most efficient catalyst of CO 2 -to-CO electrochemical conversion[END_REF] The first, most positive one is triggered by the reduction of [Co I (qpy)-(PhOH) 2 ] + , while the second one requires the reduction of [Co(qpy)] to [Co(qpy)] -. In our previous contribution, we have demonstrated that the first process is highly selective for CO production (catalytic selectivity > 95%). [START_REF] Cometto | Highly selective molecular catalysts for the CO 2 -to-CO electrochemical conversion at very low overpotential. Contrasting Fe vs. Co quaterpyridine complexes upon mechanistic studies[END_REF] Upon performing a controlled potential electrolysis at -1.6 V vs SCE in the presence of 0.1 M PhOH (Figure S4), conditions into which the second catalytic process is dominant, CO is the main product (72% Faradaic efficiency FE, 77% selectivity) with however a significant amount of H 2 (20% FE).

According to these data, two different catalytic mechanisms (Scheme 2) could be proposed for the At 1 M PhOH, a catalytic plateau independent of the scan rate is observed at large scan rate for pathway b, as shown in Figure 7. It is indicative of pure kinetic conditions resulting from mutual compensation between the catalyst diffusion and fast catalytic rate. [START_REF] Azcarate | Throughspace charge interaction substituent effects in molecular catalysis leading to the design of the most efficient catalyst of CO 2 -to-CO electrochemical conversion[END_REF] Catalysis is purely homogeneous as confirmed by the freely diffusive nature of the third reduction wave (see Figure S1). The catalytic current could be obtained from the CVs: Of the plateau current, 72% (Figures 7 andS5) is due to CO 2 reduction (the remaining charge is essentially directed to H 2 evolution, assuming similar selectivity as found in potential controlled electrolysis). This current is related to the rate constant k cat of the catalysis through the following eq 5 corresponding to the mechanism sketched in Scheme 2 (pathway b): 19

= × × × i FS C D k 2 plateau cat 0 cat cat ( 5
)
where S is the electrode surface, C 0 cat is the catalyst concentration, D cat its diffusion coefficient, and v is the scan rate.

After normalization by the one electron diffusion current of the catalyst (obtained from the first wave reoxidation peak, Figure 7):

= × × × × i FS C D Fv RT 0.446 peak 0 cat 0 cat (6)
It led to

= × × × k i i Fv RT 1 (2 2.24) cat plateau peak 0 2 2 i k j j j j j j y { z z z z z z (7)
and thus k cat = 1.2 × 10 4 s -1 , showing the remarkable efficiency of the CO 2 reduction process (see the Supporting Information). The rate for catalysis is close to the rate of pathway a (3.3 × 10 4 s -1 at PhOH 3 M) previously obtained.

In conclusion, we have shown that the electrocatalytic reduction of CO 2 into CO by Co could occur through two different mechanisms, depending on the phenol content in the solution. Both pathways necessitate the assistance of the weak proton source to boost the C-O bond cleavage. The most positive pathway is activated with an overpotential η of ca. 300 mV and is predominant at very large concentration of acid (3 M). The second one occurs with η = 600 mV and is activated at smaller acid concentrations. Both processes are endowed with rate constant in the range of 10 4 s -1 , illustrating the high efficiency of the catalyst. By adjusting the acid concentration and the scan rate, both mechanisms could be characterized independently from analysis of the CVs, showing the power of cyclic voltammetry to access mechanistic and kinetic data for catalytic processes.

■ EXPERIMENTAL SECTION Chemicals. Acetonitrile (Acros, > 99.9%, extra dry over molecular sieves) and supporting electrolyte NBu 4 PF 6 (Fluka, purriss.) were used as received. Co was synthesized as previously reported. [START_REF] Guo | Highly efficient and selective photocatalytic CO 2 reduction by iron and cobalt quaterpyridine complexes[END_REF] Phenol was purchased from Sigma-Aldrich and used without further purification.

Electrochemistry and Spectroscopic Analysis. All the experiments have been performed by using dry acetonitrile as solvent and NBu 4 PF 6 as supporting electrolyte. Before each experiment, the solution was purged with Ar or CO 2 for 20 min.

Cyclic Voltammetry. The working electrode was a 3 mm diameter glassy carbon (Tokai) disk carefully polished using diamond paste of various size (from 15 to 1 μm) and ultrasonically rinsed in absolute ethanol and dried before use. The counter electrode was a platinum wire, and the reference electrode was an aqueous SCE electrode. All experiments were carried out under argon or carbon dioxide atmosphere at 20 °C, the double-walled jacketed cell being temperature-controlled by circulation of water. CV data were obtained by use of a Metrohm AUTOLAB instrument. Ohmic drop was compensated using the positive feedback compensation implemented in the instrument. For high speed cyclic voltammetry, a 1 mm diameter glassy carbon was used as working electrode, and a homemade potentiostat was used for ohmic drop compensation.

Controlled Potential Preparative Scale Electrolysis. Electrolysis reactions were performed using a Princeton Applied Research (PARSTAT 2273) potentiostat. Experiments were carried out in a two-compartment cell with a glassy carbon plate as working electrode. The volume of the solution was 3 mL, and active surface area was 1.8 cm 2 . The reference electrode was an aqueous SCE electrode, and the counter electrode was a platinum wire positioned in a bridge separated from the cathodic compartment by a ceramic frit, containing a 2% H 2 O + 0.1 M NBu 4 PF 6 ACN solution. The electrolysis solution was purged with CO 2 during 20 min prior to electrolysis. The ohmic drop between working electrode and reference electrode was minimized by dipping the former one directly in the solution and positioning it close to the working electrode. All CPE experiments have been performed under stirring.

Gas Chromatography. Analyses from the gas evolved in the headspace during electrolysis were carried out with an Agilent Technologies 7820A GC system equipped with a thermal conductivity detector. CO and H 2 production were quantitatively assessed using a CP-CarboPlot P7 capillary column (27.46 m in length and 25 μm internal diameter). Temperature was held at 150 °C for the detector and 34 °C for the oven. The carrier gas was argon flowing at 9.5 mL/min at constant pressure of 0.5 bar. Injection was performed via a 250 μL gas-tight container (Hamilton). These conditions allowed for separation of both H 2 , O 2 , N 2 , CO, and CO 2 . Calibration curves for H 2 and CO were determined separately by injecting known quantities of pure gas.

For the formic acid analysis, a 500 μL sample of the solution was diluted into 9 mL of H 2 O, filtered and analyzed with a Dionex ICS-1100 Ionic Chromatography System equipped with a IonPac AS15 column (KOH 20 mM as eluent).

UV-Visible Spectroscopy. UV/vis Varian Cary 60 Spectrophotometer has been employed for recording the UV-visible spectra. A 0.05 mM solution of Co was used, and the temperature was kept constant at 25 °C by a thermostat. Each spectrum was recorded under Ar atmosphere in a 1 cm path length quartz cell.

Scheme 1 .

 1 Scheme 1. Structure of the Catalyst (Co) a

  an equilibrium constant K of 0.18 M -2 . The intensity of the second reduction wave decreases with increasing concentration of phenol (Figure 4, inset), due to limited formation of uncoordinated [Co I (qpy)] + . The new species [Co I (qpy)(PhOH) 2 ] + is in turn reduced at ca. -1.25 V, corresponding to a new reduction wave that grows with [PhOH] (Figure 4). At the largest concentration of phenol

Figure 1 .

 1 Figure 1. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) under Ar, scan rate 0.1 V/s. A glassy carbon electrode (3 mm diameter) was used as working electrode. The gray trace is restricted to the first redox process (eq 1).

Figure 2 .

 2 Figure 2. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) under Ar: (a) without added H 2 O, (black); + 1% H 2 O, (red); + 10% H 2 O (blue) (scan rate 2.5 V/s), (b) with 10% added H 2 O: 0.1 V/s (black); 0.5 V/s (red); 2.5 V/s (blue), 5 V/s (pink). A glassy carbon electrode (3 mm diameter) was used as working electrode.

Figure 3 .

 3 Figure 3. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) under Ar: scan rate 5 V/s (black), 20 V/s (blue), 40 V/s (green), 70 V/s (purple). A glassy carbon electrode (1 mm diameter) was used as working electrode.

Figure 4 .

 4 Figure 4. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) under Ar, scan rate 0.1 V/s, in the absence (black) and presence of 1 M (red), and 1.5 M (blue) and 2 M (magenta) PhOH. Each CV curve is normalized to the first reduction peak current in order to take into account viscosity change and dilution effect. Inset: zoom of the first and second reduction waves. A glassy carbon electrode (3 mm diameter) was used as working electrode.

Figure 5 .

 5 Figure 5. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) under CO 2 atmosphere, scan rate 0.1 V/s, in the absence (black) and presence of 10 mM (red), 30 mM (blue), 130 mM (magenta), 210 mM (green), and 800 mM (orange) PhOH. A glassy carbon electrode (3 mm diameter) was used as working electrode.

Figure 6 .

 6 Figure 6. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) solution under CO 2 atmosphere: (a) +1.5 M PhOH at 50 V/s (black), 150 V/s (blue), 250 V/s (green), 350 V/s (purple), 450 V/s (wine), and 500 V/s (red). (b) +1 M (black), 1.5 M (red), 2 M (blue), and 3 M (magenta) PhOH at ν = 250 V/s. A glassy carbon (1 mm diameter) was used as working electrode.

  CO 2 reduction catalyzed by Co, depending on the applied potential and the amount of PhOH added to the solution. For a very large concentration of phenol (3 M), [Co I (qpy)(PhOH) 2 ] + is produced upon the one-electron reduction of Co (pathway a). Although this species is formally considered as the precursor of the active species [Co I (qpy) •-(PhOH) 2 ], deprotonation of one or two phenolic groups should not be excluded. Pathway b is dominant for a smaller amount of added acid (<1.5 M). In this case, [Co I (qpy)] + is reduced to [Co I (qpy) •-] (E 0 = -0.795 V vs SCE). Reduction of this later species (E 0 = -1.51 V vs SCE) to [Co(qpy)] -triggers the catalysis. The partial loss of selectivity detected during electrolysis at -1.6 V vs SCE with 0.1 M phenol can be attributed to the formation of the highly basic species [Co(qpy)] -that may be protonated to an hydride species and further evolve hydrogen, this reaction thus being in competition with CO 2 reduction.

Scheme 2 .

 2 Scheme 2. Mechanism for the Electrochemical Reduction of CO 2 into CO with Co a

Figure 7 .

 7 Figure 7. CVs of Co (0.5 mM) in an ACN solution + NBu 4 PF 6 (0.1 M) and PhOH (1 M) under CO 2 atmosphere, at v = 50 V/s(black), 100 V/s (red), 150 V/s (blue), 200 V/s (magenta), 250 V/s (olive), 300 V/s (navy), 350 V/s (violet), 400 V/s (purple), 450 V/s (wine), and 500 V/s (dark yellow). Inset: variation of the catalytic current observed at potentials negative to -1.5 V (obtained after subtraction of the background current value) as a function of the scan rate; see FigureS5for further details. A glassy carbon (1 mm diameter) was used as working electrode.

  In this connection, in a preliminary study on [Co II (qpy)(H 2 O) 2 ] 2+ , Che et al. have reported a catalytic process starting at ca. -1.7

■ AUTHOR INFORMATION

Corresponding Author *E-mail: robert@univ-paris-diderot.fr.

ORCID

Elodie Anxolabehere-Mallart: 0000-0002-8708-802X Claire Fave: 0000-0001-8146-8702 Tai-Chu Lau: 0000-0002-0867-9746 Marc Robert: 0000-0001-7042-4106

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The work described in this paper was supported by Hong Kong University Grants Committee Area of Excellence Scheme (AoE/P-03-08), the National Science Foundation of China (No. 21703034) and the French National Agency for Research (ANR-16-CE05-0010-01). A Ph.D. fellowship to C.C. from UniversitéSorbonne Paris Cité(USPC) is gratefully acknowledged. Partial financial support to M.R. from the Institut Universitaire de France (IUF) is gratefully acknowledged. G. Thoraval (UniversitéParis Diderot) is warmly thanked for the design and production of the glassy carbon electrodes.


Author. Denis Claude
email: denis.claude@u-bourgogne.fr

Keywords: Energy efficiency, strategic delegation, behavioral bias, energy paradox

Recent attempts at explaining the energy-efficiency gap rely on considerations related to organizational and behavioral/cognitive failures. In this paper, we build on the strategic delegation literature to advance a complementary explanation. It is shown that strategic market interaction may encourage business owners to instill a bias against energy efficiency in managerial compensation contracts. Since managers respond to financial incentives, their decisions will reflect this bias, resulting in lack of investment.

Introduction

The optimization of production processes in combination with the acquisition of innovative energy efficiency solutions enable better energy management and performance. These energy efficiency improvements are beneficial in terms of reducing production costs, strengthening the competitive position of the firm and permiting the achievement of its social and environmental responsibility objectives. In light of this, firms should seize every opportunity to improve their energy efficiency. Yet, ample evidence show that even investments with low initial expenditures and quick returns do not materialize. This paradox -commonly referred to as "the energy efficiency gap" -prompted a large literature on the barriers that prevent firms from improving their energy efficiency. 1The energy-efficiency gap refers to the difference between the energy efficiency levels observed and those deemed privately optimal according to engineering studies [START_REF] Jaffe | The energy-efficiency gap: What does it mean?[END_REF]. The actual extent of this gap is disputed [START_REF] Sutherland | Market Barriers to Energy-Efficiency Investments[END_REF]. Some authors argue that engineering studies may overestimate it by ignoring hidden costs [START_REF] Anderson | Information programs for technology adoption: the case of energy-efficiency audits[END_REF], including those arising from the imperfectly substitutability of technologies, transactions associated with the acquisition of new equipment and services, and firm heterogeneity. Indeed, energy efficiency solutions may have less satisfactory characteristics than standard equipment (e.g., color rendition under cool white Led illumination). Besides, their implementation may involve search, assessment and procurement costs. Lastly, adoption costs vary widely across firms depending on firm specific characteristics [START_REF] Verhoef | The adoption of energy-efficiency enhancing technologies: Market performance and policy strategies in case of heterogeneous firms[END_REF]. Once hidden costs are accounted for, these skeptic authors argue, the remaining energy-efficiency gap, if any, is explained by market failures [START_REF] Koomey | Technical evidence for assessing the performance of markets affecting energy efficiency[END_REF].

Sources of market failures are well documented [START_REF] Fisher | Market failure and energy policy: A rationale for selective conservation[END_REF]. Among these, information deficiencies play an important role in impeding investments [START_REF] Howarth | Market barriers to energy efficiency[END_REF]. Firms may simply lack accurate and dependable information about equipment attributes (e.g., energy-efficiency, running costs and reliability). Alternatively, information may be distributed asymmetrically across market participants allowing opportunistic behavior such as adverse selection, moral hazard and split incentives [START_REF] Gillingham | Energy efficiency economics and policy[END_REF][START_REF] Gerarden | Assessing the Energy-Efficiency Gap[END_REF]. For example, a company that is unable to confirm the energy-efficiency characteristics of a cost-effective equipment will balk at its price. Anticipating such reluctance from prospective customers, suppliers of energy saving technologies will market only cheap low-performance equipment.

Additionally, market failure may arise from externalities [START_REF] Gerarden | Assessing the Energy-Efficiency Gap[END_REF]. The limited appropriability of returns on innovation activities discourages investment in energy-efficiency R&D. Likewise, learning-by-doing spillovers incite firms to delay investment to benefit from the continuous flow of knowledge generated by early-adopters. Finally, the market for energy saving equipment may fail because of capital market failures, incomplete markets and irreversibilities. 2 Market failure arguments may not be sufficient to close the energy-efficiency gap, however. Recent contributions suggest that organizational issues [START_REF] Decanio | Barriers within firms to energy-efficient investments[END_REF] combine with behavioral biases [START_REF] Tientenberg | Energy Efficiency Policy: Pipe Dream or Pipeline to the Future?[END_REF][START_REF] Allcott | Energy policy with externalities and internalities[END_REF][START_REF] Gillingham | Bridging the Energy Efficiency Gap: Policy Insights from Economic Theory and Empirical Evidence[END_REF][START_REF] Frederiks | Household energy use: Applying behavioural economics to understand consumer decision-making and behaviour[END_REF] to impede investments. Top management is made up of executives who differ in their training, skills, experience and goals. Firm behavior will reflect this complex set of idiosyncrasies [START_REF] Decanio | Barriers within firms to energy-efficient investments[END_REF][START_REF] Cagno | A novel approach for barriers to industrial energy efficiency[END_REF]. Admittedly, short-term compensation contracts, rapid job rotation, and individual reputation building behaviors aiming at self-promotion convey incentives that favor projects that offer a quick return on investment, thereby excluding investment in energy saving equipment [START_REF] Decanio | Barriers within firms to energy-efficient investments[END_REF]Sorrell et al., 2004).

Likewise, the scarce and selective managerial attention precludes the balancing of costs against benefits of investment. DeCanio (1993) notes that "[M]anagement attention and resources are scarce and must be concentrated on those areas deemed crucial to the survival of the firm (p. 910)". To economize on managerial attention, top management relies on simplified decision procedures aiming at satisfactory rather than optimal outcomes. For instance, as a way to reduce managerial discretion while saving on control costs, it may demand an unusually fast return on investment on smaller cost reduction projects, particularly those related to energy savings [START_REF] Antle | Capital Rationing and Organizational Slack in Capital Budgeting[END_REF][START_REF] Decanio | Barriers within firms to energy-efficient investments[END_REF]. The adoption of such simple heuristics might be perfectly rational in the face of uncertainty.

Selectivity suggests that managers focus their attention only on the most salient issues facing the firm [START_REF] Frederiks | Household energy use: Applying behavioural economics to understand consumer decision-making and behaviour[END_REF][START_REF] Gerarden | Assessing the Energy-Efficiency Gap[END_REF][START_REF] Cattaneo | Internal and External barriers to energy efficiency: which role for policy interventions[END_REF]. It is reflected in the fact that decisions related to product management or strategic planning clearly enjoy greater standing than is accorded to energy cost savings. It further transpires from the better pay and greater prestige associated with corresponding executive positions. This difference in treatment may, in turn, discourage energy efficiency managers to further invest in human capital [START_REF] Decanio | Barriers within firms to energy-efficient investments[END_REF].

Inattention may be perfectly rational. Indeed, faced with prohibitive search and transaction costs and given limited managerial resources, firms may engage in 'rational inattention', a type of information processing characterized by self-imposed rationality constraints [START_REF] Sallee | Rational Inattention and Energy Efficiency[END_REF]. This could explain why, in actual practice, top managers seem to ignore (or heavily discount) energy cost savings and to focus exclusively on initial investment outlays. However, the differential treatment of benefits and costs may also result from the behavioral biases that top executives hold. Loss aversion --the tendency to avoid losses over acquiring equivalent gains --may provide part of the explanation. As noted by [START_REF] Cattaneo | Internal and External barriers to energy efficiency: which role for policy interventions[END_REF], "it could be that investment costs are evaluated as a loss and are weighted more than gains". Other behavioral biases may explain why firms exhibit an inertia in adopting new technologies [START_REF] Frederiks | Household energy use: Applying behavioural economics to understand consumer decision-making and behaviour[END_REF]. Sunk costs effects may induce firms to retain obsolete equipment in an effort to recoup initial investment outlays. Besides, endowment effects may encourage firms to replace faulty (or tired) equipment by similar (or same) apparatus even though innovative cost-effective energy-efficient alternatives are available.

In this paper, we bridge the above three -economic, organizational and behavioral/cognitiveperspectives. Our aim is to show that strategic market interaction may contribute to and even exacerbate behavioral and cognitive biases that are at the source of the energy-efficiency gap. The intuition for this result is based on the separation of ownership and control in modern corporations. We consider a strategic delegation game in which business owners decide whether or not to delegate executive decisions to a professional managers. As shown by [START_REF] Vickers | Delegation and the Theory of the Firm[END_REF], [START_REF] Fershtman | Equilibrium incentives in oligopoly[END_REF] and [START_REF] Sklivas | The strategic choice of managerial incentives[END_REF], delegation in Cournot markets has strategic value. By hiring a manager, the owner of a firm can credibly commit to pursue a goal that differs from maximizing profit. Indeed, the managerial incentives embedded in managerial compensation contracts make the deviation from profit maximization credible since managers respond to financial incentives. Delegation decisions and deviations from profit maximization may thus arise as an equilibrium behavior. 3In this game, we assume that production implies energy expenditures. Accordingly, when delegation actually occurs, managers are responsible for both production and energy conservation decisions. The previous literature focused on deviations from profit maximization involving ingredients of sales revenue, market share or relative profit maximization. By contrast, here, we allow business owners to design compensation contracts that encourage managers to ignore part (or all) of energy costs. Such compensation contracts would result in an unequal weighting of energy costs savings and investment costs. It is shown that they are selected by business owners at the equilibrium of the strategic delegation game. Consequently, what is usually regarded as a cognitive bias -the unequal weighting of energy cost savings and investment costs-is sustained as an equilibrium behavior.

Our model is related to two previous contributions on managerial delegation decisions for polluting firms. In a Cournot duopoly with homogeneous goods and pollution emissions, [START_REF] Barcena-Ruiz | Environmental taxes and strategic delegation[END_REF] find that business owners favor revenue maximization at the expense of profits. Consequently, strategic delegation results in higher output levels and pollution emissions. The optimal environmental tax is then higher than that required to regulate a ownermanaged oligopoly. [START_REF] Pal | Delegation and Emission tax in a Differentiated Oligopoly[END_REF] generalizes this result to industries supplying differentiated products. In both models, stronger managerial incentives (i.e, a lower profit weight) lead managers to pay less attention to production costs with respect to sales revenues.

Our paper differs from these contributions along several important lines. We consider a different kind of deviation from profit maximization that has nothing to do with sales revenues, market share or relative profit objectives. Furthermore, we resort to strategic delegation arguments to explain why modern corporations may weight differently different type of costs -upfront investment outlays aimed at increasing energy conservation and recurring energy expenditures. In so doing, we show that strategic market interaction may contribute to and even exacerbate observed behavioral biases.

The paper is organized as follows. Section 2 describes the model. Section 3 solves the managerial subgame. Section 4 characterizes Subgame-Perfect Nash Equilibrium (SPNE) delegation decisions. The last section concludes.

The Model

We consider a duopoly market with two identical firms indexed by i = 1, 2. The two firms produce a homogeneous good q and compete in quantities. Let q i denote firm i's output and Q = (q 1 + q 2 ) denote industry output. The inverse demand function is linear and given by p(Q) = a -bQ with a, b > 0. The production of the final good requires energy inputs. We assume that both firms buy energy inputs from the same energy retailer at a rate (i.e., fixed price), r > 0. For sake of simplicity, we assume that firm i's demand for energy inputs is e i (q i , γ i ) = q i -γ i where γ i ≥ 0 denotes firm i's effort at energy conservation. Under this specification, absent of investment in energy efficiency, firms' production technologies are characterized by an unitary energy-output ratio; i.e., e i (q i , 0) = q i . Also, energy conservation may be regarded as a substitute for energy consumption.

Obviously, a reduction in energy input use -or an increase in energy conservation-is highly desirable from the firm's perspective. However, further increases in energy conservation require increasingly experienced staff and increasingly specialized capital. Hence, increases in energy conservation come at an increasing cost. To reduce its overall energy consumption by γ i units, we suppose that firm i incurs a cost equal to K i (γ i ) = k 2 γ 2 i . Firm i's energy bill writes as B(q i , γ i , r) = r (q i -γ i ). We restrict our attention to investments in energy efficiency. In particular, we assume away the possibility for firms to sell excess energy back to the energy retailer. This implies that γ i ≤ q i so that Firm i's energy bill is always non- negative. Also, we assume that consumers (maximal) willingness to pay for the good exceeds the price of energy; i.e., a > r. Firm owner-managers seek to maximize profits, which are given by:4

π i = (a -bQ) q i -r (q i -γ i ) - k 2 γ 2 i , i = 1, 2. (1)
However, if profitable, owner-managers may choose to divorce ownership from control . In other words, each owner i may delegate day-to-day production decisions to a manager. We assume that managers respond to financial incentives and, accordingly, seek to maximize their compensation. By appropriately designing the managerial compensation contract, each owner is able to direct managers to pursue an objective that differ from profit maximization. Since managerial compensation in publicly traded companies is common knowledge, this strategic move will convey to rival firms the credible commitment to a deviation from (pure) profit-maximization. Managerial compensation is assumed to depend on managerial performance. Specifically, we assume a two-part compensation scheme w i = w F i + w V i M i consisting of a base salary w F i ≥ 0 and a commission on performance w V i M i , where w V i > 0 is the rate at which the performance measured by M i is rewarded. Given this scheme, each manager i will maximize M i in order to maximize its compensation w i .

To answer the question whether discounting energy costs could be rational, we consider a class of performance measures defined with respect to δ i by:

M i (δ i ) = (a -bQ) q i -δ i r (q i -γ i ) - k 2 γ 2 i , for all δ i ∈ [0, 1] and i = 1, 2. (2)
By adjusting the value of δ i , the owner of Firm i generates a continuum of managerial incentives that range from pure-profit maximization (δ i = 1) to a complete disregard for energy costs (δ i = 0). One important remark must be made here. Unless δ i is set to one, the above-defined class of performance measure does not weight equally the different types of costs involved in production. While expenditures aimed at improving energy efficiency are fully accounted for, energy input costs are at least partially ignored. This is consistent with empirical evidence which highlight that economic agents (consumers or firms) display a tendency to weight differently savings and costs. Our purpose here is to show that such a bias may sometimes be rational rather than linked to defeating cognitive illusions. To the best of our knowledge, no other paper in the strategic delegation literature includes such a differential treatment of the different types of costs that the firm faces. The timing of the game is as follows (See, Figure 1). At the first stage, both firm owners independently and simultaneously decide whether to hire a manager -and if so, design the managerial compensation contract. To state it differently, both owners set the rate at which input energy costs will be discounted so as to maximize profits. At the second stage, firm managers respond to incentives by independently and simultaneously setting their production and energy conservation levels so as to maximize managerial compensation. The game is solved by backward induction for a Subgame-Perfect Nash-Equilibrium (SPNE). Hence the next section is devoted to the analysis of the managerial subgame.

Managerial production and investment decisions

At stage two, the managerial incentives embedded in compensation contracts (δ 1 , δ 2 ) have be- come public knowledge. Given those incentives, each manager i simultaneously and indepen-

Owner sets δ i

Manager sets q i and γ i (in order to maximize π i ) (in order to maximize M i ) t

Fig. 1: Timing of the game dently sets its output q i and energy conservation effort γ i to maximize own compensation, taking as given the choices of the manager of the rival firm. A Nash equilibrium of the managerial subgame is a profile of strategies

(q ⋆ 1 , γ ⋆ 1 , q ⋆ 2 , γ ⋆ 2 ) such that (q ⋆ i , γ ⋆ i )
solves the problem of managerial compensation maximization faced by Manager i for all i = 1, 2:5 max

q i ,γ i M i = (a -bQ) q i -δ i r (q i -γ i ) - k 2 γ 2 i , (Problem 1) s.t.: a b -q i ≥ 0, γ i ≥ 0, q i -γ i ≥ 0.
The Lagrangean associated with Problem 1 is:

L i = (a -bQ) q i -δ i r (q i -γ i ) - k 2 γ 2 i + λ i ( a b -q i ) + µ i (q i -γ i ) + ν i (γ i ) . ( 3
)
The Karush-Kuhn-Tucker (KKT) conditions yield:

∂ L i ∂ q i = a -2bq i -bq j -δ i r -λ i + µ i = 0, (4)
γ i ≥ 0, ∂ L i ∂ γ i = δ i r -kγ i -µ i ≤ 0, -γ i (δ i r -kγ i -µ i ) = 0, (5)
λ i ≥ 0, ∂ L i ∂ λ i = a b -q i ≥ 0, λ i ∂ L i ∂ λ i = λ i ( a b -q i ) = 0, (6)
µ i ≥ 0, ∂ L i ∂ µ i = q i -γ i ≥ 0, µ i ∂ L i ∂ µ i = µ i (q i -γ i ) = 0. ( 7
)
Due to the concavity of M i and the linearity of the constraints the above necessary conditions are also sufficient for optimality.

The KKT conditions delineate up to four regions of the (δ 1 , δ 2 ) plane, as illustrated in Figure 2a.

For the time being, let us assume that these four regions actually exist. Let the lines

ℓ j (δ i ) = k (a + rδ i ) r (3b + 2k) , ℓ i (δ i ) = ( 2 + 3b k ) δ i - a r , ( 8
)
denote the boundaries that delimit the region in which an interior Nash equilibrium obtains. This region, which is labeled I in Figure 2a, may be formally defined as:

Region I = { (δ i , δ j ) 0 ≤ δ i < ℓ -1 i (δ j ) , 0 ≤ δ j < ℓ j (δ i ) } . ( 9
)
The lines l j (δ i ) and l i (δ i ) intersect at the point This point generates two other boundaries of interest: δ i = δ , i = 1, 2. Consider the two trapezoid areas labeled II (k) , k = 1, 2, in Figure 2a. Region II (i) is delimited by ℓ i (δ i ) and δ j = δ . It describes the set of all pairs of managerial incentives (δ 1 , δ 2 ) that would induce Manager i to stop buying energy; i.e., to set γ i = q i . Conversely, region II ( j) , which is delimited by ℓ j (δ i ) and δ i = δ , describes all (δ 1 , δ 2 )-pairs that would induce Manager j to set γ j = q j . These two regions may be respectively defined as:

Region II(i) = { (δ i , δ j ) ℓ -1 i (δ j ) ≤ δ i ≤ 1, 0 ≤ δ j < δ } , ( 11
) Region II(j) = { (δ i , δ j ) ℓ j (δ i ) ≤ δ j ≤ 1, 0 ≤ δ i < δ } . ( 12
)
A last region, which may be defined as

Region III = { (δ i , δ j ) |δ i ≥ δ , δ j ≥ δ } , ( 13
)
describes the set of all (δ 1 , δ 2 )-pairs that would induce both managers to stop buying energy (γ i = q i , i = 1, 2). Depending on parameter values, all four regions may not exist. We proceed by characterizing conditions under which only interior Nash equilibria obtains.

Let δ = ak r (3b + 2k) . ( 14
)
From Figure 2a, observe that Region I fully encompasses the subset [0, 1] 2 of the coordinate plane whenever δ > 1. This will be the case if the price of energy is sufficiently low; Namely, if

r < r 1 := ak (3b + 2k) . ( 15
)
Now, if δ ≤ 1 but δ > 1, or equivalently, if

r 1 := ak (3b + 2k) ≤ r < ak (3b + k) := r 2 , ( 16
)
then Region III does not exist. Finally, if the price of energy is sufficiently high (r ≥ r 2 ), the KKT conditions partition the [0, 1] 2 into four regions. 6Equilibrium configurations associated with these four regions are given in Table 1.

Regions I II (i) II ( j) III Equilibrium q I i = a-r(2δ i -δ j ) 3b q II (i) i = a+rδ j 3b+2k q II ( j) i = a(b+k)-r(2b+k)δ i b(3b+2k) q III i = γ III i = a (3b+k) q I j = a-r(2δ j -δ i ) 3b q II (i) j = a(b+k)-r(2b+k)δ j b(3b+2k) q II ( j) j = a+rδ i 3b+2k i = 1, 2. γ I i = rδ i k γ II (i) i = q II (i) i γ II ( j) i = rδ i k γ I j = rδ j k γ II (i) j = rδ j k γ II ( j) j = q II ( j) j p I = (a+r(δ i +δ j )) 3 p II (i) = (b+k)(a+rδ j ) (3b+2k) p II ( j) = (b+k)(a+rδ i ) (3b+2k) p III = a(b+k) 3b+k
Tab. 1: The four equilibrium configurations

Having completed the analysis of the managerial subgame, we can now turn to the analysis of strategic delegation decisions.

Delegation and design of managerial compensation contracts

At stage 1, each firm owner decides simultaneously and independently whether to hire a manager, and -if so, sets managerial incentives so as to maximize its profit. As noted above, the KKT conditions delineate up to four different regions, each associated with a different equilibrium configuration. Given this partitioning, the profit function of each owner is defined by a piecewise function. Profit maximization then yields piecewise reaction correspondences. Before solving the whole game, we shall consider a reduced model obtained by assuming that the price of energy is low enough to ensure that Region I encompasses the square [0, 1] 2 .

Low energy price:

Let us recall from section 3 that r < r 1 implies [0, 1] 2 ⊆ Region I. In other words, for any pair of managerial incentives (δ i , δ j ) set by firm owners, the equilibrium of the managerial subgame is unique and given by (q I i , γ I i , q I j , γ I i ). The profit of Owner i evaluated at the equilibrium is defined by:

π I i (δ i , δ j ) := π i ( q I i (δ i , δ j ) , γ I i (δ i , δ j ) , q I j (δ i , δ j ) , γ I i (δ i , δ j ) ) , i (̸ = j) = 1, 2. ( 17
)
A pair of managerial incentives (δ I i , δ I j ) is a Subgame-Perfect Nash-Equilibrium of the game among owners if, and only if,

δ I i ∈ arg max δ i ∈[0,1] π I i ( δ i , δ I j ) for i, j (i ̸ = j) = 1, 2.
The corresponding first-order conditions are:

∂ π I i (δ i , δ j ) ∂ δ i = 0, akr + r 2 (9b (δ i -1) + (4δ i + δ j -6) k) = 0, (18)
for i, j (i ̸ = j) = 1, 2. Then, Manager i's reaction function is given by :

δ i ≡ ϕ i (δ j ) = 3r (3b + 2k) -ak (9b + 4k)r - k (9b + 4k) δ j , i (̸ = j) = 1, 2. ( 19
)
Since the Hessian matrix is negative definite, π I i (δ i , δ j ) is strictly concave, and the above neces- sary conditions are also sufficient for optimality. Solving the system (18) yields

δ ⋆ i = 1 - k (a -r) (9b + 5k) r , i = 1, 2. ( 20
)
Recall our assumption that δ i ∈ [0, 1]. Observe that δ ⋆ i < 1. However, note that δ ⋆ i < 0 for

0 < r < r 0 := ak 3 (3b + 2k) . ( 21
)
In this case, given the positivity constraint on δ i , the best Owner i can do is set δ I i = 0. We have the following proposition.

Proposition 1. If the energy rate r is relatively low, the two-stage game admits a unique Subgame-Perfect Nash Equilibrium. At this equilibrium, business owners provide managers with incentives to discount energy costs. Equilibrium discount factors are given by

δ I i = { 0 if 0 < r ≤ r 0 , δ ⋆ i , if r 0 < r ≤ r 1 , ( 22
)
for i = 1, 2.

From the above proposition it follows that business owners may require management to wholly ignore energy expenditures if the energy rate r is low enough.

Moderate-to-high energy price

When energy becomes expensive, the question arises as to whether the pair of managerial incentives described by Equation ( 20) remains a Subgame-Perfect Nash Equilibrium of the game. This will be the case if the strategic pair (δ ⋆ i , δ ⋆ j ) satisfies two conditions: (i) it is an element of Region I for r > r 1 and (ii) it is immune from unilateral deviations. To begin with, let us check the former condition. Observe that the point (δ ⋆ i , δ ⋆ j ) is located along the 45°line for all r > r 1 . Hence, it is an element of Region I provided that δ ⋆ i = δ ⋆ j < δ or, equivalently, that

r < r 3 := 2ak(2b + k) (3b + k)(3b + 2k) . ( 23
)
We proceed by checking that the strategic pair (δ ⋆ i , δ ⋆ j ) is immune from unilateral deviations. Assume that Owner j contemplates deviating from δ ⋆ j to any δ d j ∈ [ℓ j (δ ⋆ i ), 1]. The resulting change in Owner j profit is given by : π

II ( j) j ( δ ⋆ i , δ d j ) -π I j ( δ ⋆ i , δ ⋆ j ) = π II ( j) j (δ ⋆ i ) -π I j ( δ ⋆ i , δ ⋆ j ) , (24)
= -

(9b + 4k)(r(3b + k)(3b + 2k) -2ak(2b + k)) 2 2bk(3b + 2k) 2 (9b + 5k) 2 < 0. (25)
Therefore, Owner j has no profitable deviation, and neither does Owner i by symmetry. We can state the following proposition.

Proposition 2. Unless the price of energy becomes prohibitive (r > r 3 ), the combination of managerial incentives

(δ ⋆ i , δ ⋆ j ) is a Subgame-Perfect Nash-Equilibrium outcome.
The question now is whether the game has equilibria in other parts of the coordinate space. The following lemma narrows down the search.

Lemma 1. Let (δ c k i , δ c k j ) be any pair of managerial incentives from Region II (k) , k = 1, 2. Then, (δ c k i , δ c k j ) cannot be a Subgame-Perfect Nash-Equilibrium.

Proof. See Appendix A.

As a consequence of Lemma 1, we need only consider Region III. To simplify exposition, in the remainder of this paper, any equilibrium solution in Region I (resp., III) is referred to as an interior (resp., a corner) equilibrium. Note that Region III exists if, and only if, r > r 2 . Furthermore, note that there is an overlap between the interval of energy prices which ensures the existence of an interior solution (0 < r < r 3 ), and that which guarantees that Region III is not empty (r 2 ≤ r < a). For energy prices in the interval [r 2 , r 3 ), interior and corner equilibria may coexist leading to potential coordination problems. The existence of a corner equilibrium is assured for an energy price greater or equal to

r 4 = ak(2b + k) 2 (b + k)(3b + k) 2 , ( 26
)
as detailed in the following proposition.

Proposition 3. If r ≥ r 4 , then any pair (δ i , δ j ) ∈ Region III is a Subgame-Perfect Nash Equilibrium.

Proof. See Appendix B.

The next section discusses our findings.

Interpretation of the results

Let us combine the results from the previous two sections to describe how the equilibrium behavior of business owners is affected by the price of energy. Reference to Figure 3 will be of assistance for the reader here. When the price of energy is very low (r ∈ (0, r 0 ]), competition concerns trump energy saving concerns. Indeed, the increase in sales revenues more than compensates for the increased expenditures in energy inputs. Thus, business owners strategically take advantage of delegation to commit their managers to be more aggressive on the product market. This is done by designing compensation contracts so as to encourage managers to ignore energy costs in their everyday production and investment decisions. Compensation contracts then require that managerial performance be evaluated and rewarded solely in terms of sales revenue net of capital expenditures -thus explicitly excluding energy cost savings. In that event, top managers regard investment as both costly and irrelevant because it does not contribute to their compensation. Investment in energy efficiency is then effectively deterred.

Apart from this specific case, business owners have to strike a balance between competitive and energy saving concerns. Accordingly, for r > r 0 , they set non-zero discount factors. From Equation 20, we have the following comparative statics result:

∂ δ ⋆ i ∂ r = ak (9b + 5k) r 2 > 0, for r 0 < r < r 3 . ( 27
)
Namely, the equilibrium weight δ ⋆ i is monotonically increasing on (r 0 , r 3 ) implying that an increase in the price of energy justifies greater attention to energy costs. However, there are two caveats here. First, note that δ ⋆ i is bounded from above by:

δ ⋆ max = lim r→r 3 δ ⋆ i (r) = 1 - b 2 (2b + k) . ( 28
)
This upper-bound originates from the fact that values of δ i larger than δ are associated with a different equilibrium configuration in the managerial subgame (i.e., that which corresponds to Region III). Accordingly, small changes in the energy rate around r = r 3 may lead to jumps in equilibrium discount rate for there are multiple Nash equilibria on which business owners may coordinate.

When the price of energy becomes cost-prohibitive (r > r 3 ), energy savings become the dominant concern for business owners. Indeed, the increase in sales revenues no longer compensates for increased expenditures in energy inputs. Consequently, at the equilibrium, the two business owners choose relatively high discount factor

( δ i > δ )
. Thereby, they encourage managers to close the energy-efficiency gap by setting q i = γ i . In that case, the energy costs borne by the two firms are zero and, thus, changes in the value of δ restricted to the interval [ δ , 1] alter neither managerial production and investment decisions nor business owners profits. Correspondingly, the set of Subgame Perfect Nash Equilibrium discount rates is given by [ δ , 1] 2 . Besides, profit maximization is a natural focal point because business owners are indifferent between all their equilibrium strategies including δ i = 1. Under the circumstances, business owners may success- fully coordinate their hiring and compensation decisions so that delegation does not occur and the industry remains owner-managed. The differents components of cost will then be weighted the same at the managerial level.

Second, observe however that this set of corner equilibria emerges for relatively high -but not necessarily cost-prohibitive -energy prices (r ≥ r 4 ). Hence, the two equilibrium configurations ('interior' and 'corner') coexist provided that r ∈ [r 4 , r 3 ). Furthermore, there is a threshold

0 ak 3(3b+2k) ak 3b+2k ak 3b+k 2ak(2b+k) (3b+k)(3b+2k) r δ I i = 0 0 < δ I i < 1 [ δ, 1] 2 There is a unique SPNE Infinite number of SPNE r 0 r 1 r 2 r 3 r 4 δ I i ∈]0, 1[∪[ δ, 1] 2 [ δ, 1] 2 ak(2b+k) 2 (b+k)(3b+k) 2
Fig. 3: Existence and unicity of SPNE value7 for the energy rate below (resp., above) which the 'interior' equilibrium Pareto dominates (resp., is Pareto dominated by) 'corner' equilibria. In that case, business owners face a coordination game. Here again, profit maximization provides a focal point for business owners. However, efficiency arguments may support delegation decisions.

Conclusion

Empirical research highlight that firms exhibit behavioral biases in their energy efficiency investment decisions. It appears that managers attribute less weight in their decision-making to potentially achievable savings than to the initial investment cost. Uncertainty and irreversibility of investment fail to explain this difference in treatment between the various components of the firm's profit function. Recent explanations have suggested the possibility that behavioral biases such as loss aversion or sunk cost effects may explain why profit maximizing investments in energy efficiency are sometimes not undertaken.

In this paper, we proposed a complementary explanation based on strategic delegation theory. It was shown that imperfect competition may encourage business owners to manipulate the terms of managerial compensation contracts to reward investment behaviors exhibiting a bias against energy efficiency. In this case, the performance measure used to assess managerial effectiveness and compensation attributes more weight to investment costs than to energy related cost savings.

The rate at which energy costs are discounted is shown to depend on the price of energy. When the price of energy is low, competitive concerns dwarf energy cost concerns. Consequently, business owners encourage managers to ignore energy expenditures. However, as the price of energy increases, energy costs weight more in decision-making. Accordingly, owners encourage managers to be increasingly attentive not to waste energy. Finally, in scenarios where the price of energy is prohibitive, there are multiple rational behaviors and coordination issues arise.

The explanation we propose is not intended to replace existing organizational and behavioral explanations. Rather, it aims to emphasize that these accounts can be reinforced by strategic considerations. Contrary to a widely held belief, market competition may play a role -albeit limited -in explaining the maintenance of the energy-efficiency gap.

An important managerial insight from our analysis is that managerial compensation contracts should reflect energy market conditions. When energy prices are high, managers should be encouraged to monitor carefully energy costs and invest in energy efficiency. Conversely, when energy prices are low, they should be encouraged to focus on competitive issues. Modeling uncertainty regarding energy prices is a possible avenue for extension of the results of the present contribution.

B Proof of Proposition 3

The idea of the proof is illustrated in Figure 4. Consider any δ j ∈

[ δ , 1 ] . Then, δ i is a best- π III i δ δ i π III i , π IIj i (δ i ) π IIj i (δ i ) π III i δ i Fig. 4: The idea behind Proposition 3 reply to δ j if π III i ≥ π II ( j) i (δ i ).
Observe that Firm i's profit function is continuous in δ . Indeed,

CEE-M Working Papers

  Fig. 2: Partitioning of the square [0, 1] 2

For literature surveys, see Sorrell et al.

2004; Tientenberg 2009; Gillingham & Palmer 2014; Gerarden et al. 2017.2 Or, to be more specific, liquidity constraints, risk and losses of future options.

For a recent survey of the strategic delegation literature, see[START_REF] Kopel | Strategic Delegation in Oligopoly[END_REF]

Since managerial compensation is intended to secure the participation of managers, it will be equal to their respective (exogeneouly defined) reservation incomes. Consequently, from an owner's perspective, maximizing profit net of compensation is the same as maximizing profit. For more on this point, see[START_REF] Kopel | Strategic Delegation in Oligopoly[END_REF].

The assumption that a bq i ≥ 0 ensures that firm i's strategy set (i = 1, 2) is compact. This condition has obvious economic meaning. It says that manager i would not select a quantity leading to a negative price when firm i is the only active firm in the market.

If r = r 2 then Region III reduces to the point (1, 1).

We omit the expression of this threshold because it is very lengthy

The authors acknowledge the financial support of the French Agence Nationale de la Recherche (ANR). Mabel Tidball received funding from the project GREEN-Econ (ANR-16-CE03-0005) and the LabEx Entreprendre (ANR-10-LABX-11-01). Denis Claude received funding from the project RILLM. The authors would like to thank the participants of the 18th International Symposium on Dynamic Games and Applications in July 2018 and the seminar participants at Center for Environmental Economics -Montpellier and the University of Valladolid for helpfull discussions. The usual disclaimer applies.

Appendices

A Proof of Lemma 1

Let BR i (x) denote the set of decisions that are best-responses to a decision x for player i

0 since these two sets lie on opposite sides of the 45°line. We conclude that (δ c k i , δ c k j ) are not mutual best-responses. To complete the proof, observe that the same holds for any point (δ

we have: lim

. Now, observe that Owner i's profit function is strictly concave since ∂ 2 π

and reaches a maximum for

Hence, π III i ≥ π

The proof is completed by noting that Condition (31) holds provided that r ≥ r 4 .