William Bialek

Thomas Gregor

Gašper Tkačik

Action at a distance in transcriptional regulation

There is increasing evidence that protein binding to specific sites along DNA can activate the reading out of genetic information without coming into direct physical contact with the gene. There also is evidence that these distant but interacting sites are embedded in a liquid droplet of proteins which condenses out of the surrounding solution. We argue that droplet-mediated interactions can account for crucial features of gene regulation only if the droplet is poised at a non-generic point in its phase diagram. We explore a minimal model that embodies this idea, show that this model has a natural mechanism for self-tuning, and suggest direct experimental tests.

In multicellular organisms, the transcription of genes into messenger RNA is controlled by the binding of transcription factor proteins to "enhancer" sites that can be separated from the gene by tens of thousands of base pairs along the DNA sequence [START_REF] De | Topology of mammalian developmental enhancers and their regulatory landscapes[END_REF][START_REF] Levine | Looping back to leap forward: Transcription enters a new era[END_REF][START_REF] Furlong | Developmental enhancers and chromosome topology[END_REF][START_REF] Stadhouders | Transcription factors and 3D genome conformation in cell-fate decisions[END_REF][START_REF] Van Steensel | The role of transcription in shaping the spatial organization of the genome[END_REF][START_REF] Schoenfelder | Long-range enhancerpromoter contacts in gene expression control[END_REF]. Close approach of enhancers to their target promoters has been inferred from cross-linking experiments [START_REF] Mifsud | Mapping long-range promoter contacts in human cells with high-resolution capture Hi-C[END_REF], and there is direct evidence that the action of the enhancer requires physical proximity to the promoter site where transcription is initiated [START_REF] Chen | Dynamic interplay between enhancer-promoter topology and gene activity[END_REF]. But proximity is not contact: the most recent measurements indicate that the enhancers and their target promoters remain separated by 150 -350 nm even during active transcription [START_REF] Chen | Dynamic interplay between enhancer-promoter topology and gene activity[END_REF][START_REF] Lj Mateo | Visualizing DNA folding and RNA in embryos at single-cell resolution[END_REF][START_REF] Alexander | Live-cell imaging reveals enhancer-dependent Sox2 transcription in the absence of enhancer proximity[END_REF][START_REF] Heist | Large distances separate coregulated genes in living Drosophila embryos[END_REF][12].

How is the apparent action at a distance possible? Interactions between the enhancer and promoter could be transmitted along the length of the DNA molecule, but it seems more plausible that this interaction is transmitted across the shorter three dimensional distance [START_REF] Benabdallah | Regulatory domains and their mechanisms[END_REF]. Recent observations indicate that there is a medium for this transmission, a condensed droplet of the protein "mediator" which surrounds the promoter [START_REF] Cho | Mediator and RNA polymerase II clusters associate in transcription-dependent condensates[END_REF]; these droplets also contain high concentrations of RNA polymerase [START_REF] Ii Cisse | Real-time dynamics of RNA polymerase II clustering in live human cells[END_REF], are associated with foci of active transcription [START_REF] Cho | RNA Polymerase II cluster dynamics predict mRNA output in living cells[END_REF], can form in vitro [START_REF] Boija | Transcription factors activate genes through the phase-separation capacity of their activation domains[END_REF], and contain other co-activating factors [START_REF] Br Sabari | Coactivator condensation at superenhancers links phase separation and gene control[END_REF]. We propose that the droplet acts as a larger scale version of an allosteric protein [START_REF] Monod | Allosteric proteins and cellular control systems[END_REF][START_REF] Monod | On the nature of allosteric transitions: A plausible model[END_REF][START_REF] Mf Perutz | Mechanisms of Cooperativity and Allosteric Regulation in Proteins[END_REF]: in the same way that the protein structure allows binding of small molecules at one site to influence binding or enzymatic activity at a distant site, the droplet would allow binding of transcription factors (TFs) at an enhancer site to influence activity at the distant promoter site (Fig 1). We will argue that this is possible only if the droplet is at a non-generic point in its phase diagram, and that the collective interactions among the enhancer sites can drive the system toward such points. Two facts will be crucial to our discussion. First, to the extent that the mediator droplets are similar to other examples of intracellular phase separation [START_REF] Banani | Biomolecular condensates: organizers of cellular biochemistry[END_REF], they will be liquid-like [START_REF] Shin | Liquid nuclear condensates mechanically sense and restructure the genome[END_REF], and thus in general will not transmit structural changes across hundreds of nanometers. Second, gene expression can be controlled in a quantitative, graded fashion in response to changing concentrations of transcription factors [START_REF] Colman-Lerner | Regulated cell-to-cell variation in a cell-fate decision system[END_REF][START_REF] Takahashi | Membrane localization of scaffold proteins promotes graded signaling in the yeast MAP kinase cascade[END_REF][START_REF] Giorgetti | Noncooperative interactions between transcription factors and clustered DNA binding sites enable graded transcriptional responses to environmental inputs[END_REF][START_REF] Rogers | Morphogen gradients: from generation to interpretation[END_REF].

If we did not have the constraint of graded responses, we could imagine that binding of TFs to enhancer sites triggers droplet condensation, and that this is the essential mechanism of regulation, as proposed for "superenhancers" [START_REF] Br Sabari | Coactivator condensation at superenhancers links phase separation and gene control[END_REF][START_REF] Hnisz | A phase separation model for transcriptional control[END_REF]. But this is an all-or-none mechanism, and it is difficult to harness the triggering of phase separation to generate a quantitatively graded response to changes in TF concentration. The existence of droplets by itself does not solve the problem.

FIG. 1: The DNA strand (black) surrounds a condensed droplet (orange). Promoter site is marked by an arrow (red), enhancer sites as blocks (green) with transcription factors (magenta and blue) both bound to these sites and freely diffusing.

arXiv:1912.08579v1 [q-bio.SC] 18 Dec 2019

Even if transcription requires droplet condensation, there are pathways for regulation once the droplet has formed. In eukaryotes transcription involves a very large number of different proteins, and it is plausible that many of these components condense into the droplet. With multiple components the phase diagram is more complicated than just two phases [START_REF] Sear | Instabilities in complex mixtures with a large number of components[END_REF][START_REF] Mao | Phase behavior and morphology of multicomponent liquid mixtures[END_REF], so droplets can condense and still have additional degrees of freedom related to the addition or expulsion of different molecular species. Let us summarize these variables by an order parameter φ( r), which can vary with position r inside the droplet. These are the degrees of freedom that can propagate interactions through the droplet.

The simplest model envisions a set of K identical binding sites for a single class of transcription factors (at positions r i ), plus one promoter site (at r a ), all embedded in a droplet. These binding sites typically will be arrayed across multiple enhancers, all of which can contribute to regulating transcription. The relevant variables are σ i = ±1 for empty and occupied binding sites, and A = {0, 1} for the inactive and active states of the promoter. All of these variables couple to the order parameter, and it is important that these couplings are spatially local. The free energy is then

F[φ( r); {σ i }, A] = F 0 [φ( r)] + E 0 A - k B T 2 ln(c/c 0 ) K i=1 σ i + K i=1 g i σ i φ( r i ) + g a Aφ( r a ), (1)
where g i is the interaction between the order parameter and binding of TFs to the enhancers, g a is the interaction between the order parameter and the active vs inactive state of the promoter, E 0 is the free energy difference between the two states of the promoter in the absence of TFs, c is the concentration of these factors, and c 0 is the "bare" binding constant of the TF to the enhancer sites.

Let's assume that, as in conventional models of allostery, the transmission of information can be described as an equilibrium thermodynamic effect [START_REF]Many allosteric proteins are enzymes, and the nonequilibrium nature of an active catalyst could make an equilibrium description impossible[END_REF][START_REF] Marzen | Statistical mechanics of Monod-Wyman-Changeux (MWC) models[END_REF][START_REF] Einav | Statistical mechanics of allosteric enzymes[END_REF]. Hence, we define an effective free energy by integrating out the fluctuations of the order parameter,

e -F eff ({σi}, A)/k B T = Dφ exp - F[φ( x); {σ i }, A] k B T , (2)
where Dφ is the measure for integration over φ( r). F eff is composed of independent and interacting parts, F eff = F 0 + F int , and to leading order in the couplings we have

F 0 = E 0 A - k B T 2 ln(c/c 0 ) K i=1 σ i (3)
F int = - K i, j=1 g i g j 2k B T C(r ij )σ i σ j - K i=1 g a g i k B T C(r ia )σ i A, (4)
where we choose coordinates so that the average order parameter is zero in the Boltzmann distribution defined by F 0 , and C(r) is the correlation function of the order parameter fluctuations in this distribution,

φ( r i )φ( r j ) ≡ C(r ij ), (5)
with

r ij = | r i -r j |.
The question of whether the droplet transmits information from the enhancer to the promoter becomes the question of whether fluctuations in the order parameter are correlated over these long distances [START_REF] Sethna | Statistical Mechanics: Entropy, Order Parameters and Complexity[END_REF].

In liquids at generic parameter values, density fluctuations have a short correlation length ξ, so that C(r) e -r/ξ , with ξ on the nanometer scale, and thus these modes cannot support action at a distance. There can be additional degrees of freedom associated with the orientational ordering of molecules in the droplet, or with conformational changes of these molecules, but again with generic parameters we expect to find small ξ. The alternative is that the parameters describing the droplet are at a non-generic point in the phase diagram, where the correlation length can become long, and this is the "critical droplet" scenario we explore here.

Close to a first-order phase transition the free energy F 0 has two nearly degenerate minima [START_REF] Sethna | Statistical Mechanics: Entropy, Order Parameters and Complexity[END_REF]. In a sufficiently small droplet, fluctuations in the order parameter are dominated by flickering between these minima, and there is an effective surface tension that keeps the entire droplet in one minimum, so that C(r) becomes only weakly dependent on distance [35]. Close to a secondorder phase transition the correlation length diverges, and C(r) decays very slowly, as power of distance. Either of these scenarios seems to require some tuning of the droplet parameters, to which we return below.

If all the transcription factor binding sites couple to the droplet in the same way, then we should have all the g i = g. We can capture the essential predictions of this model if all the distances r ij and r a are roughly equal to a typical R, in which case we can simplify to

F int = - J 2 K i, j=1 σ i σ j -J a K i=1 σ i A, (6)
with two parameters J = (g 2 /k B T )C(R) and J a = 3) and ( 6) with K = 8 sites, J = 0.2kBT , Ja = kBT , and E0 = kBT ln(100), compared with the corresponding MWC model (Ja = 0.2kBT , J = 0). Note that interaction energies are on the order of kBT or less, but the droplet generates a very steep response to changing TF concentrations.

(gg a /k B T )C(R). To gain intuition, we note that if J = 0 then Eq (6) becomes identical to the Monod-Wyman-Changeux (MWC) model for allosteric proteins [START_REF] Monod | On the nature of allosteric transitions: A plausible model[END_REF][START_REF] Marzen | Statistical mechanics of Monod-Wyman-Changeux (MWC) models[END_REF], with the A = 1/0 switch in promoter activation playing the role of the R/T conformational transition [START_REF] Monod | On the nature of allosteric transitions: A plausible model[END_REF][START_REF] Mf Perutz | Mechanisms of Cooperativity and Allosteric Regulation in Proteins[END_REF]. A critical droplet thus generates cooperative interactions among distant sites that are similar, qualitatively, to more familiar examples of allostery (Fig 2).

Transcriptional activators become repressors just by changing the sign of the coupling g i , which describes interaction of the TF with the surrounding droplet, and is determined by the face of the protein opposite from the DNA binding domain. A generalization is to imagine that we have K a binding sites for activators at concentration c a and K r sites for repressors at concentration c r . Then at low repressor concentrations there is cooperative activation, but at higher repressor concentration the system approximates a switch that depends on the ratio of powers of the concentrations, c Ka a /c Kr r . A second generalization is to have multiple nearby binding sites within one enhancer interact more strongly, perhaps through additional degrees of freedom, and then let the emergent states of multiple enhancers couple to the droplet. The system could then approximate logical operations corresponding to combinations of ANDs within each enhancer and ORs among enhancers.

An important implication of this model is that transcription is regulated not by the binding of transcription factors to individual binding sites, but rather by an integrated signal from multiple binding sites that are dis-tributed around the droplet of size R. If we think of the transcriptional output as a "measurement" of the TF concentration, then the accuracy of this measurement is limited by the random arrival of molecules [START_REF] Berg | Physics of chemoreception[END_REF][START_REF] Bialek | Physical limits to biochemical signaling[END_REF][START_REF] Js Van Zon | Diffusion of transcription factors can drastically enhance the noise in gene expression[END_REF]; the smallest concentration differences δc to which a system can respond reliably is given by

δc c 1 √ D cτ . ( 7
)
where c is the background concentration, D is the diffusion constant, τ is the time over which the system can average, and is the linear size of the sensitive element.

If the response is driven by a single binding site, then is the size of that site, but if the system integrates over many binding sites, then can approach the linear dimensions of the entire array of sites, in our case the size of the droplet, which is ∼100× larger than individual binding sites. From Eq [START_REF] Mifsud | Mapping long-range promoter contacts in human cells with high-resolution capture Hi-C[END_REF], responses which would require hours of integration at a single site thus become reliable in minutes. Transcription factor concentrations are so low that this difference can be crucial [START_REF] Bialek | Physical limits to biochemical signaling[END_REF][START_REF] Gregor | Probing the limits to positional information[END_REF].

Poising a condensed droplet near a critical point seems to require fine tuning of its parameters. Cells can exert exquisite control over protein and nucleic acid concentrations [START_REF] Gregor | Probing the limits to positional information[END_REF][START_REF] Md Petkova | Maternal origins of developmental reproducibility[END_REF], but matching the concentrations of crucial molecules to their critical values still seems difficult. In our case, however, there is a thermodynamic driving force that pushes the system toward conditions where correlation lengths are long. To estimate this effect, let's assume that the droplet has a critical point when one of its components is at concentration x 0 . The chemical potential of the surrounding solution holds the concentration close to some mean concentration x, and variations ∆x around this mean cost a free energy F 1 (nk B T /2)(∆x/x) 2 , where n R 3 x is the mean number of molecules in the droplet. But at a concentration x the correlation length will be ξ = a|x 0 /(x -x 0 )| ν [START_REF] Sethna | Statistical Mechanics: Entropy, Order Parameters and Complexity[END_REF], where a is a microscopic length scale. Away from criticality, the gain in free energy from interaction among K binding sites is F 2 -(J 0 /2)K(K -1)e -R/ξ , and in total we have

2F nk B T (∆x/x) 2 -A exp - R a x 0 ∆x 0 + ∆x ν , (8)
with ∆x 0 = x -x 0 and A = (J 0 /k B T )K(K -1)/n. The dominant component of the droplet is present at only n ∼ 100 [START_REF] Cho | Mediator and RNA polymerase II clusters associate in transcription-dependent condensates[END_REF], so with K ∼ 10 binding sites it is easy to have A ∼ 1; to be conservative we consider A = 0.25. We could plausibly have R ∼ 150 nm and the molecular scale a ∼ 5 nm, but again to be conservative we choose R/a = 5. Assuming that the chemical potential alone sets x = 1.5x 0 , we see that the possibility of mediating interaction among binding sites creates a sharp minimum of the free energy at the critical point, sufficient to pull the system very close to criticality (Fig 3).

Taking this thermodynamic driving force seriously, we note that when transcription is active, enhancer bind- ing sites with states that are "aligned" to this activation have a free energy that is lower by J a ∝ C(R), and since C(R) decreases with r this generates a small force pulling the enhancer toward the promoter. In contrast, enhancers in states that are not contributing to activation of transcription have a free energy that is higher by J a and a force pushing enhancer and promoter apart. These small forces are balanced by a stiffness, which also determines the thermal fluctuations in the enhancer-promoter distance. The result is that aligned vs anti-aligned enhancers should be at different mean distances from the promoter site, and this displacement is ∆R/R ∼ (J a /k B T )(δR/R) 2 , where δR is the standard deviation of the distance R, and we assume that d ln C(R)/d ln R ∼ 1. These displacements should be directly observable, for example by measuring the positions of different enhancers for the pair-rule genes in the early fly embryo [START_REF]Pair rule genes are expressed in a seven-striped pattern along the long axis of the Drosophila embryo, and different stripes are activated by different enhancers [JB Jaynes and M Fujioka[END_REF]. More generally this suggests that singlemolecule observations of enhancer motions could be connected, quantitatively, to the energetics of cooperative transcriptional activation.

To summarize, a large number of transcription factor binding sites, embedded in a droplet that surrounds the promoter, will generate cooperative regulation if the droplet is poised near special points or lines in its phase diagram where correlation lengths become long. In this scenario the droplet functions much like an allosteric protein, but this is possible only because of the proximity to criticality. This is similar to the long-ranged interactions between proteins that we expect to see in a membrane [START_REF] Bb Machta | Critical Casimir forces in cellular membranes[END_REF] if lipid compositions are close to a critical point, as observed [START_REF] Sl | Critical fluctuations in plasma membrane vesicles[END_REF][START_REF] Ar Honerkamp-Smith | An introduction to critical points for biophysicists; observations of compositional heterogeneity in lipid membranes[END_REF][START_REF] Sp Rayermann | Hallmarks of reversible separation of living, unperturbed cell membranes into two liquid phases[END_REF]; it has also been suggested that chro-matin itself is close to a sol/gel phase boundary [START_REF] Khanna | Chromosome dynamics near the sol-gel phase transition dictate the timing of remote genomic interactions[END_REF]. There is a much wider range of ideas about how criticality could play a role in biological function [START_REF] Mora | Are biological systems poised at criticality?[END_REF][START_REF] Ma Muñoz | Colloquium: Criticality and dynamical scaling in living systems[END_REF], but what is special in our example is that we have identified, as an intrinsic part of the functional behavior, a mechanism that drives the system toward its critical point, and perhaps this is more general. Consequences of this thermodynamic driving force should be directly observable in the physical positions of enhancer and promoter sites.

FIG. 2 :

 2 FIG.2: Mean promoter activity as function of the TF concentration. Results from Eqs (3) and (6) with K = 8 sites, J = 0.2kBT , Ja = kBT , and E0 = kBT ln(100), compared with the corresponding MWC model (Ja = 0.2kBT , J = 0). Note that interaction energies are on the order of kBT or less, but the droplet generates a very steep response to changing TF concentrations.

FIG. 3 :

 3 FIG.3: Free energy as a function of concentration in the droplet, from Eq[START_REF] Chen | Dynamic interplay between enhancer-promoter topology and gene activity[END_REF], with n = 100, ∆x0 = x0/2, A = 0.25, R/a = 5, and ν = 1/2. Note the weak minimum at the x = x0 + ∆x0, set by the chemical potential, which is dominated by the minimum at criticality, x = x0.

We thank L Barinov, SA Kivelson, and MS Levine for helpful discussions. This work was supported in part by the US National Science Foundation, through the Center for the Physics of Biological Function (PHY-1734030) and Grant PHY-1607612; by National Institutes of Health Grants P50GM071508, R01GM077599, and R01GM097275; and by Austrian Science Fund grant FWF P28844.


Francesca D ' Angelo

MEASURE TO ASSESS BILINGUALS' LEARNING SKILLS IN THIRD OR ADDITIONAL LANGUAGE ACQUISITION: THE CASE OF LLAMA_F

Keywords: Third Language Acquisition, Artificial Language Task, Bilingual Education, Metalinguistic Aw areness, Cross-Linguistic Transfer

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Third Language Acquisition (TLA) is a relatively new field of study, and area of research, developed considerably in the last few years. It refers to the study of a non-native language by learners who have previously acquired or are acquiring two languages. Specifically, it has been defined as 'the acquisition of a language that is different from the first and second and is acquired after them' (Cenoz 2013, 71). The study of TLA brings together two fields that have traditionally worked separately: i.e. second language acquisition (SLA) and the study of the effects of bilingualism on cognition and language learning. Despite the similarities of SLA and TLA, there are several linguistic and cognitive factors to consider TLA as a distinct process and area of research from SLA. Second and third language learners come to the process of language acquisition with a linguistic and cognitive background that differs considerably, both quantitatively and qualitatively [START_REF] Grosjean | The on-line processing of speech: Lexical access in bilinguals[END_REF]. That is, a second language learner is a monolingual at the initial state of SLA, whereas a third language learner is already bilingual (potentially early/late, simultaneous/consecutive, etc.). This entails that having at least two languages in their linguistic repertoires allows third language learners to relate new structures, new vocabulary, or new ways of expressing communicative functions to the two languages they already know, not just one of them, as in the case of monolinguals. Moreover, third language learners show more refined skills and strategies for achieving the language-learning task [START_REF] Garraffa | Il cervello bilingue[END_REF]. Thus, if one takes into account the complexity of the field, due to the aforementioned quantitative and qualitative differences, the traditional assessing tools employed in SLA research may not be adequate. Indeed, not only does TLA differ considerably from SLA but the effect and interaction of the factors involved become difficult to observe and record experimentally. Specifically, to assess bilinguals' skills in TLA several problems arise: i.e. level of competence reached in each language, age and order of acquisition, exposure to the L3, learning context (implicit/explicit), typological proximity etc. The present work aims at discussing the different advantages of using an Artificial Language Task, i.e. LLAMA_F test, in research dealing with multiple language acquisition. In particular, LLAMA_F is propounded as an effective technological tool to assess bilinguals' Metalinguistic Awareness (MLA), developed in previous languages, and to predict successful performance in TLA. It is a test of grammatical inference, which focuses on the participants' ability to learn additional languages in an informal setting. Loosely based on the Modern Language Aptitude Tests by Carrol & Sapon (1959), it is part of the LLAMA battery developed by [START_REF] Meara | LLAMA Language Aptitude Tests: The Manual[END_REF] as shorter, free, language-neutral tests. The first attempt at works in this area appeared as [START_REF] Meara | Swansea language aptitude tests (LAT) v2.0[END_REF] and included a set of five tests assessing different aspects of language learning aptitude, i.e. vocabulary learning, grammatical inference, sound-symbol association, phonetic memory, and a test of memory for unusual sound sequences. The rise of interest generated in the research community, since the first publication, prompted the authors to adapt the tests for people with a different L1 from English. Moreover, another problem was that some of the material languages developed as part of the original test started to be familiar to potential participants. For instance, Polish and Turkish, although not widely recognised in the UK, are more familiar to test takers with Hungarian or Azeri as an L1. Hence, the need for a new set of tests, which was largely independent of the participant's L1, lead the designers to develop the actual version. The main, practical reason making the test particularly suitable for research on bilingualism is that it is entirely based on picture stimuli and has eliminated the need for an L1 database. Second, not only does it allow to control for participants' L1, but it also allows to control for the amount of exposure and level of competence in L3 (i.e. the tested language). In other words, being an artificial language, test takers are not supposed to be familiar with the language of the task presented. Indeed, it would be difficult to find participants with the same level of proficiency in any natural language as an L3, same amount of exposure to the language, same method of acquisition and level of formal instruction received. Third, the task does not explicitly focus on grammar and MLA of the language as no instruction or request about the formal aspects of the grammar was included. Instead, as it will be better described in the following sections, the grammatical rules behind the artificial language are only implicitly inferred to match the sentence description with the picture. Additionally, the restricted time that participants have at their disposal to passively observe the unknown language and figure out the mechanism beyond it, as well as the fact that they are not allowed to take notes, make the task more similar to the implicit learning process of a language. Accordingly, the task does not directly facilitate those bilinguals with higher levels of MLA and language learning experience that are supposed to perform better in TLA.

METHODOLOGY

Participants are required to take the LLAMA_F, test of grammatical inference, as the last task of the session. The box marked  controls the length of time available the grammar of the new language, i.e. five minutes. The instructions provided are the following: "Your task is to use this time to learn as much as you can about a new language. You do this by clicking on the small buttons in the main panel. For each button you click, a picture and a sentence describing it will be displayed, as in the picture below" (see attachment 1). Unak -ek eked-ilad is the sentence that describes the picture. The presentation phase of the program shows the test-taker a series of pictures depicting shapes and objects, and a short sentence in an artificial language that describes each image. Participants are expected to work out how the descriptions related to the pictures. From this, they should be able to intuit some of the grammatical and morphological features of the language such as word order, gender, singular, dual and plural numbers, conjugating prepositions etc. During the second phase, they are presented with a new set of pictures that incorporate new elements as well as some taken from the training phase. Each picture is accompanied by two sentences that might describe it, and test-takers are required to select the correct description. They should be able to do this if they have internalised the grammatical rules evidenced in the presentation phase. Five points are awarded for a correct answer and five points deducted for an incorrect choice. One sentence is grammatically correct, while the other contains a major grammar error. The program gives them feedback in the form of a ding for a correct answer, and a bleep for an incorrect answer. To see the next test item, they need to click the  button. There the score, as they work through the test, and shows how many items are left to complete it. At the end of the test, the score, ranging between 0 and 100, is displayed on the bottom panel.

DISCUSSION

As already mentioned, LLAMA_F, test of grammatical inference, is entirely based on picture stimuli and has eliminated the need for an L1 database. This makes it an effective assessment tool for additional language learning as not only does it allow to control for participants' L1, but it also allows to control for other important variables, i.e. amount of exposure and level of competence in L3. Indeed, being an artificial language, test takers are not supposed to be familiar with the language of the task involved. In addition, the test resolves another important issue when recruiting participants in TLA studies. That is, the difficulty of finding a homogeneous sample of bilinguals with the same level of proficiency in any natural language as an L3, the same amount of exposure to the language, the same method of acquisition and level of formal instruction received. The second reason deals with a factor considered to be crucial in multilingual education, the relationship between Metalinguistic Awareness (MLA) developed in previous languages through formal instruction and level of attainment in TLA (D'Angelo 2020). Indeed, the test does not explicitly focus on grammar and MLA of the language as no instruction or request about the formal aspects of the grammar is included. In other words, the grammatical rules behind the unknown language are only implicitly inferred to match the sentence description with the picture. What is more, the restricted time that participants have at their disposal to passively observe the artificial language and figure out the mechanism behind it, together with the impossibility to take notes, make the task more similar to the implicit learning process of the language. Accordingly, it can be claimed that the task does not directly facilitate bilinguals with higher levels of MLA, supposed to perform better in TLA. Moreover, the test allows assessing another crucial aspect of language learning, i.e. the ability to use and transfer the skills developed through bilinguals' previous experience as language learners, as it reproduces the initial stage of learning additional languages. Interestingly, the type of transfer does not apply to lexical items or grammatical structures from one language to the other(s), as suggested by the Typological Proximity Model [START_REF] Rothman | L3 syntactic transfer selectivity and typological determinacy: The typological Primacy Model[END_REF]. Indeed, the performance in the additional language refers to an artificial grammar that is not typologically related to any of the languages already known by the test takers. Accordingly, the transfer refers to the practice of internalising grammatical patterns and exploiting those strategies when dealing with additional languages. This is confirmed by the fact that, during the performance of the LLAMA_F, participants are required to make grammatical inferences on the basis of a short passive training where they are supposed to figure out how the language works. It is important to highlight that test-takers did not receive any explicit instructions about the grammatical rules and no linguistic terminology was employed. Therefore, what the test assesses in the second phase is precisely their ability to exploit their language learning strategies, grammatical inference, and their ability to generalise principles based on observation of the language by decoding and interpreting it. In other words, when learning another language, participants use their capacity to learn grammar through previous language learning rather than using their knowledge of any specific individual grammatical structures. Finally, as participants were not allowed to take notes during the training phase, the test requires additional effort involving the Working Memory as they had to remember the relevant patterns involved in the training to understand the mechanism of the language they were learning.

The validity of Artificial Language Tasks

Despite the acknowledged importance of ALL experiments as tools exploring the principles of language, as well as language learning ability, a persistent question is whether ALL studies can be considered as ecologically valid assessments of natural language ability. In particular, the aspect of artificial language tasks that have been questioned is that they cannot be compared to natural language learning. Nonetheless, several studies assessed the validity of artificial language tasks by comparing their performance with that recorded in natural language tasks, with a number of internal factors being controlled. [START_REF] Ettlinger | The Relationship Between Artificial and Second Language Learning[END_REF], for instance, bridged the gap between ALL and natural language learning research by comparing the performance of adult learners of Spanish as an L2 and ALL enhancement. The findings suggest that performance in ALL tasks correlates positively with indices of L2 learning even after controlling for IQ, general intelligence and the potential mediation of these internal factors. The authors also considered the effects of specific features of ALL tasks such as including or not a semantic aspect as well as presenting a complex or simple grammar. Interestingly enough, they inferred that ALL studies that incorporate a semantic component and involve more complicated grammatical systems may closely resemble the learning process of natural languages. The specific validity of the LLAMA tests has been confirmed by a study by Rogers, Meara, and colleagues (2017) including 240 participants. In particular, the issues raised by the authors concern the following factors: whether the tests are language-neutral, the effects of bilingualism and age on LLAMA tests scores, and the amount of variance that background factors can account for in the LLAMA test results. Interestingly, the authors did not find any significant difference in terms of language background, suggesting that LLAMA tests are indeed language-neutral as there are no differences between groups once other factors (i.e. L2 instruction) are controlled for. Concerning the effects of bilingualism, specifically, the difference referred to monolingualism, bilingualism, and instruction in L2, instructed L2 learners significantly outperformed the monolingual and bilingual group on two sub-components: LLAMA_B and LLAMA_F. The hypothesis of a training effect is also related to the results found in terms of the effects of age. Younger groups (10-11 years old) were outperformed by older groups (20-21 Years old) suggesting that LLAMA tests are not suitable for children in that older learners may have developed more refined learning strategies over the years. Finally, the effects of six background individual factors were also considered to see how much of the variance in the LLAMA test score could be accounted for by each of them: i.e. L1, L2 status, age, highest formal education, gender, and logic training (puzzles). The multiple regression results from 240 participants show that LLAMA tests can generally be used across different L1s, with male and female participants of different education levels and with different ages as these factors did not consistently affect the variance in scores. The only individual variable to predict most of the variance was prior instruction in a second language in LLAMA_B (6%) and LLAMA_F (2.6%). Therefore, from the analysis provided, it can be argued that LLAMA tests can be considered as robust and reliable measures of language learning ability.

CONCLUSIONS

The main focus of the work was to demonstrate the validity and effectiveness of LLAMA_F, test of grammatical inference, in multilingual research. Specifically, the test was used to overcome several issues due to the complexity and large amount of variation found in TLA research, being not subject to significant external factors or individual variables that could potentially affect the results. More specifically, the quantitative and qualitative complexity of the factors characterising studies on TLA, including number of languages, typological proximity, learning context, age and order of acquisition etc., requires different assessing instruments which allow to deal with the variability of the aforementioned factors as well as to take into account their interaction. In terms of context of acquisition, it has been proved to be a valid tool to address the relationship between bilinguals' language learning experience, their explicit and implicit MLA, developed in other languages, and their ability to learn a third (or additional) language. Indeed, the ability to think about the language as an abstract system, made of different levels interacting among each other, and to focus on the grammatical form, to analyse and manipulate it, independently of the specific language involved, is hypothesised to be the most important predictor of success when learning additional languages. Bilinguals are also expected to enhance their explicit level of MLA in proportion to the amount of formal instruction they received. The number of languages mastered is another factor supposed to enhance MLA on the whole, which in turn assists the process of language learning. As it has been highlighted, the input from the LLAMA_F is unrelated to the linguistic structure of any other potential language known by the test takers. It allows inferring that it is not the specific grammatical knowledge of a particular language that boosts the process of TLA. Instead, it is the abstract knowledge of the language as a system that assists participants to think about the language critically, as an abstract object, and to resort to this awareness and ability when dealing with additional language learning. In other words, when they come to the process of third or additional language acquisition, they do not need to relearn the fundamental principles of language structures. They make use of these explicit and abstract fundamental rules to figure out how the new language works by applying those principles to additional languages. Thus, the implication of this condition of MLA enhancing the process of additional language learning in bilinguals is that the content of metalinguistic knowledge must be broader than any that applies to the knowledge of a specific linguistic structure. In this perspective, MLA refers to the 'explicit representation of abstract aspects of linguistic structure that become accessible through the knowledge of a particular language' (Bialystok 2001, 124). Specifically, in TLA research, the aforementioned knowledge is reinforced through the knowledge of at least two other languages as it deals with bilinguals' additional language acquisition and has proved to be the most important predictor of language attainment. Accordingly, the notion of transfer of the language skills from the first to the second language if there is sufficient exposure to the L2 and motivation to learn the language can be applied by extension to TLA. That is to say, the language skills developed in an L2, in a broader and more abstract knowledge, will be transferred to the L3. In line with this hypothesis, the test confirms that the level of attainment reached in the third language is affected by participants' ability to manipulate, analyse, and think about the L2 as an abstract object. The empirical evidence that the more languages bilinguals have gained literacy and study experience in, the better they are at learning additional languages with an implicit focus on grammatical form confirms the claim supported by many scholars in the field. As Cenoz (2013) points out, MLA is one of the key factors associated with bilinguals' better performance in TLA together with learning strategies and a broader linguistic repertoire. These three elements affect each other and are closely related to the number of languages known by bilingual learners. First, the higher level of MLA can be considered as both a cause and effect of bilinguals' success in additional language learning. On one hand, based on their previous experience of the task of learning a language and their knowledge of two linguistic systems they enhance their level of MLA. On the other, it is precisely the more developed MLA skills and the idea that they manage to think about the language in an abstract way and regard it as an object to assist them in the process of learning additional languages. Indeed, the claimed bilingual advantage in TLA is related to bilinguals' experience as language learners itself as they develop a wider range of learning strategies. Specifically, it has been argued that they look for more sources of input, make an early effort to use the new language, and show selfdirection and a positive attitude towards the task [START_REF] Bowden | Individual Differences: Age, Sex, Working Memory, and Prior Knowledge[END_REF]. This argument, mostly developed on the basis of research using artificial language tasks, has also been confirmed by other studies dealing with natural languages. Also, the broader linguistic repertoire that bilinguals have at their disposal has been associated with a better performance in TLA. The majority of the studies have explained it in terms of language distance, that is, closely related languages would be more useful for bilinguals learning a third (e.g. [START_REF] Angelis | Third or Additional Language Acquisition[END_REF][START_REF] Ringbom | Cross-linguistic Similarity in Foreign Language Learning[END_REF][START_REF] Rothman | L3 syntactic transfer selectivity and typological determinacy: The typological Primacy Model[END_REF]. However, as already mentioned, this is not the case for LLAMA_F, as the task has been proved to be language-neutral and to have no typological relationship with the other languages known by the participants. This suggests that the broader linguistic repertoire on behalf of bilinguals supports them in TLA independently of nature and specific linguistic features of the languages involved. To sum up, for all the reasons discussed in the present paper, it can be claimed that LLAMA_F represents a valid and reliable technological instrument allowing to overcome several specific issues when dealing with TLA research. In particular, what the test assesses is precisely the participants' ability to exploit their language learning strategies, grammatical inference, and their ability to generalise principles based on the observation of the language by decoding and interpreting it. In other words, when learning another language, participants use their capacity to learn grammar through previous language learning rather than using their knowledge of individual grammatical structures. What is more, in the recruitment phase, it eliminates the need for an L1 database, being entirely based on picture stimuli. Also, it permits to control for the amount and type of exposure to the L3, since test takers are not supposed to be familiar with the artificial language presented. For the same reason, it does not affect the typological relation with any previous language known by participants since, as discussed, the test has been proved to be "language neutral". Finally, the test reproduces the implicit language learning process and does not facilitate those participants with a higher level of explicit knowledge of the language, allowing to test the effects of MLA, i.e. a crucial variable affecting the process and outcome of additional language learning.


Damien Bestard

François Thomas Farges

Acoustical reconstruction and thunder energy localisation in lightning flashes measured over Corsica during EXAEDRE field campaign

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Acoustical reconstruction and thunder energy localisation in lightning flashes measured over Corsica during EXAEDRE field campaign

Damien Bestard 1 , Thomas Farges 2 , and François Coulouvrat 1 From September 13th to October 12th 2018, the EXAEDRE field campaign took place in Corsica, dedicated to the characterisation of thunderstorm clouds and electrical activity. Among a wide range of observation instruments, an array of 4 microphones, arranged on a 30-m wide triangle located near the island eastern coast, recorded over the frequency band 1 to 80 Hz the acoustical signal, or thunder, associated to lightning flashes. The search for coherent signals between the four sensors within prescribed frequency bands allows to determine the thunder apparent sound velocity and azimuth (PMCC algorithm [1]). Knowing the flash emission time provided by Meteorage low frequency (1-350 kHz) electromagnetic ground lightning locating system, as well as the local speed of sound, it is possible to reconstruct the various positions of coherent sound sources within a single lightning flash. Co-localisation of acoustic sources with in-cloud detections provided by SAETTA high-frequency (60-66 MHz) electromagnetic lightning locating system, and with ground impacts provided by Meteorage, ensures the efficiency and precision of the method. This one was already used successfully in a previous field campaign (HyMeX-SOP1) in Cévennes in 2012 [2,3,4]. The detection algorithm PMCC also provides the various recorded signal intensities.

Assuming each sound point source radiates a spherical wave, the different propagation distances between the sources and the recording array can be compensated, so that the thunder source energies can also be localised within the flash with their relative levels. For EXAEDRE, two storms have been studied from an acoustical point of view, one with a low electrical activity on October 2nd mainly over the Mediterranean sea, and one with an intense activity on 17th September mainly overland. A significant number of flash events has been analysed, reconstructed and their energy distribution determined. For the 17th of September, acoustical events of large amplitudes are well correlated to (mostly negative) Cloud to Ground flash events. Energy localisation indicates a strong heterogeneity of its distribution within the flash, with intense sound sources concentrated inside the return strokes, mostly within the two first kilometres above the ground. Intracloud parts of the flashes appear much less energetic from an acoustical point of view. For the 2nd October, overversea events turn out quite different.

  [The authors acknowledge the EXAEDRE program, lead by E. Defer, for supplying the data. Present results have been obtained within the frame of the LETMA Contractual Research Laboratory between CEA, CNRS, Ecole Centrale Lyon, C-Innov, and Sorbonne Université.]

Powered by TCPDF (www.tcpdf.org)


Adèle Chabert

Emmanuel Fredon

Romain Rémond

Beech wood improvement by in-situ polyesterification using bio based polycarboxylic acids and polyols

Results: Impregnation -ASE -LR -Swelling

Context

• Important CO 2 stocking properties.

Conclusion

• Considering the results obtained in LR and ASE, especially for the XCiG and AnSuG treatments, it can be safely assumed that a polymerization and/or grafting occurs on the sorption sites of the wood affecting then, the wood ability to bond with moisture.

• Thermal treatment is also induced when curing the samples. This explains the losses of mechanical resistance in both bending and compression. However it is compensated by the chemical treatment effect.

Beech wood improvement by in-situ polyesterification using bio-based polycarboxylic acids and polyols

•

  Hardness values improved compared to the value of native as high as 4.71 HB compared to 2.85 HB. • Bending strength decreased, Young Modulus improved for two of the treatments • Young Modulus E c,90 increased in transversal compression from 7% to 12%, for XCiG and AnSuG compared to native beech wood.

  Chabert Adèle 1 , Fredon Emmanuel 1 , Rémond Romain 1 1: LERMAB, 27 rue Philippe Seguin, 88000 Epinal


Roberto P Ferreira

D A Cunha
email: rcunha@thinkbrg.com

Antoine Missemer
email: missemer@centre-cired.fr

THE HOTELLING RULE IN NON-RENEWABLE RESOURCE ECONOMICS: A REASSESSMENT

Harold Hotelling's 1931 contribution is known for providing a basic principle-the Hotelling rule-to the economics of non-renewable resources. Nearly 90 years later, empirical tests conclude the rule lacks empirical validity, requiring strong amendments to describe the long-term, aggregate behaviour of its target object. On the basis of Hotelling's unpublished archival material, this paper revisits the place given to the Hotelling rule in non-renewable resource economics. Our reconstruction shows that Hotelling's 1931 paper has been misinterpreted: from the outset, the Hotelling rule was not valid for mineral resources. In contrast, the consideration of two inherent geological constraints, alongside exhaustibility, offered the opportunity for an alternative basic framework, capable to generate bell-shaped and U-shaped equilibrium trajectories for supplies and prices, respectively. Inspired by this unknown aspect of Hotelling's work brought to light by our archival investigation, we sketch this alternative basic model, enabling non-renewable resource economics to circumvent the empirical shortfalls of the Hotelling rule.

Introduction

"The Economics of Exhaustible Resources," published in April 1931, is considered a seminal contribution for non-renewable resource economics. Harold Hotelling proposed an extensive analysis of the intertemporal optimization of extraction paths under various competitive conditions. He laid the foundations for many research questions, including the role of taxation and public regulation in extractive sectors. Even if the Hotelling model is no longer taught to students in the original form, the lessons drawn from the 1931 article are still at the ground of contemporary research and teaching, in which the equation p t = p 0 • e r•t plays the role of basic principle.

In 1974, Robert M. Solow paid tribute to Hotelling, labelling this equation as "the Hotelling rule, the fundamental principle of natural-resource economics" (Solow 1974, p. 12). As far as we know, that was the first occurrence of the expression "Hotelling rule," and Solow's impulse, within the favourable context of the oil price shock, reignited research on the subject of non-renewable resources (e.g., Levhari and Liviatan 1977, Dasgupta and Heal 1979, Devarajan and Fisher 1981, Farzin 1984, 1992). Since then, the literature retains finiteness, in the form of Hotelling's hard constraint ∫ 0 ∞ q • dt ≤ a, as the central defining factor of the class of non-renewable natural resources.

Tests on the empirical validity of the Hotelling rule, by itself, have concluded it is unable to stick to reality, both in terms of assumptions and capacity to predict real market prices. Gaudet (2007), analyzing the price evolution of 10 mineral resources throughout the 20th century, concluded the rate of change in prices is approximately centred at zero, and not at the interest rate. Livernois (2009) reviewed 34 empirical studies on the Hotelling rule, concluding data does not provide overwhelming support for it. Slade and Thille (2009), reviewing roughly the same studies, concluded for the rejection of the basic model by data. Further empirical tests, of specific aspects of Hotelling's model, or of elementary extensions of the basic model, reached the same conclusion (e.g., Hart and Spiro 2011, Gaugler 2015, Atewamba and Nkuiya 2017, Karp 2017).

The rejection of the basic model by data has been remediated by the consideration of additional real-world factors. The main ones capable of bringing the basic model closer to reality are technological progress (Slade 1982), degradation costs (Pindyck 1978), durability (Levhari and Pindyck 1981), market structure (Salant 1976) and uncertainty (Kemp 1976). 1 Each extension inaugurated a modern branch in non-renewable resource economics, expanding and detailing the level of sophistication of each factor within the original basic framework. Nonetheless, the implementation of different additional factors leads to questions about the robustness and readability of the basic theory: by varying the choice of factors to implement on the model, one can attain diverging results. By requiring strong amendments to represent its target object, its value as a theoretical startingpoint comes into question.

This paper inquires whether a deeper understanding of Hotelling's 1931 contribution can help us circumvent the empirical invalidity of the Hotelling rule. We hypothesize the prevailing understanding stems from a problematic interpretation of the original contribution, inaugurated through Solow's 1974 labelling of Hotelling's basic result as a rule. To test this hypothesis, we investigated Hotelling's project on non-renewable resources through the history of economic thought methodology, allowing us to leverage unpublished materials towards demonstrating potential theoretical gaps left behind along the evolution of thought, possibly changing the way we understand and use theory going forward. 2 We based our investigation on an unprecedented exploration of Hotelling's archives, stored at Columbia University (hereafter HHP). 3 These materials seem to have 1 Many other extensions of Hotelling's model have been proposed in the literature over the past decades. For the most recent years, we can notably cite Holland (2008), Venables (2014), Okullo et al. (2015) and Anderson et al. (2018). This list is not exhaustive. 2 Previous works in the field of history of economic thought referring to non-renewable resources include Robinson (1980, 1989), Kula (1998) and Missemer (2017, 2018 -Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -3 been consulted only by a few other scholars for previous research (e.g., Crabbé 1986; Darnell 1988, 1990; Hands and Mirowski 1998), but none of these works offer a comprehensive view on Hotelling's project regarding the economics of non-renewable resources. The documents stored at Columbia University consist of several thousand sheets, filed in 58 boxes. Items related to exhaustible resources are scattered in different boxes. Thanks to a systematic review, we found about 20 to 30 significant unpublished drafts, preparatory documents and letters from 1924 to 1925 onwards worthy of interest for our inquiry. 4 Our paper is organized as follows. In section 2, we investigate what sort of research Hotelling was conducting in the 1920s, showing his hesitation between a theoretical abstract ambition and empirical concerns for what he named at the time exhaustible resources. 5 This section allows a clearer picture on the area of validity, from the outset, for his basic resultwhich is different than what has been retained from his article. In section 3, we closely review Hotelling's work on geological constraints for his 1931 article. We reveal a neglected fundamental role for them in his analysis. Hotelling considered cumulative production x(t) and the change in production q'(t) as fundamental variables, and not additional factors, constituting a potential alternative basic model capable of attaining U-shaped trajectory for prices-the now empirically accepted stylized fact for non-renewable resources. In section 4 we provide our concluding remarks, reassessing the Hotelling rule as the starting point of mineral resource analysis. 4 This paper is part of the research project "Bifurcations in Natural Resource Economics" (BNRE), sponsored by the European Society for the History of Economic Thought (ESHET). The objective has been to explore the genesis of modern natural resource economics, notably through the creation of a digital database of Hotelling's economic archives (7000+ images taken of 3000+ documents). The project team mobilized researchers in natural resource economics and history of economic thought over 2.5 years, producing to date several coordinated articles and working papers on complementary aspects related to Hotelling's research (e.g., Franco et al. 2019, Gaspard and Missemer 2019, Missemer et al. 2020, Missemer and Nadaud 2020). 5 What Hotelling reffered to as "exhaustible resources" are now called "non-renewable resources." In the following, we use the modern term, except when it comes to evoking and analyzing Hotelling's project in its historical context.

Hotelling's research on exhaustible resources in the 1920s

From (natural) resources to assets, and vice versa

Until the submission of his paper to the Journal of Political Economy, most likely in the fall of 1930, 6 Hotelling's archives reveal, regarding exhaustible resources, that he actually worked along two approximately parallel lines of research: a generic project on the valuation of exhaustible assets, in the line of his theory of depreciation, and a specific project on exhaustible natural resources (minerals, fossil fuels), more empirically grounded. In his final article, Hotelling uses several terms to describe his research: "exhaustible (natural) resources," "exhaustible assets," "irreplaceable (natural) resources," "irreplaceable assets," "mineral content," "resources of the earth," etc. By definition, the set of "natural resources" here contains both fossil energy sources (coal, oil, gas) and minerals. The set of "assets" is broader, containing "natural resources" as a subset. A mineral stock is an asset. A financial bond is also an asset, but obviously not a natural resource. Focusing on "natural resources" suggests that the precise subject of research is related to concrete forms of natural items. On the opposite side, focusing on "assets" opens the reflection to other case studies, beyond natural items. 7 Hotelling's drafts, correspondence and notes throughout the 1920s confirm the co-existence of two levels of analysis: a generic level, illustrated by the term "exhaustible assets," and a more 6 Two letters from Hotelling to R. A. Fisher and Chamberlin (dated July 23, 1930, and Aug. 8, 1930, respectively) indicate that Hotelling was working on the final version of the paper in the summer of 1930 (RAFP, digitized version online; Guicherd 2017, p. 179). A letter from Mills (dated Jan. 17, 1931) shows that the final version of the paper was already circulating at the very end of 1930 (HHP, Box 1, Mills, Frederick C.). Arrow (1974, p. 1103; 1980; 1987, p. 670) reports that Hotelling (who was his PhD supervisor) submitted his paper to the Economic Journal before submission to the Journal of Political Economy. As shown in Gaspard and Missemer (2019), there is, however, no clear evidence that he did so. 7 This statement is not retrospective. In the early 20th century, Irving Fisher (1906) had already discussed the generic nature of the words "assets" and (not necessarily natural) "resources." specific level, with the term "exhaustible natural resources." We found Hotelling hesitated between which of the two to use in his final paper. Until only a few weeks before submission, the title of the paper was "The Economics of Exhaustible Assets," as indicated in a letter to the statistician Ronald A. Fisher: 8 The book [on statistics] is gradually progressing, though I am leaving it alone for a few weeks at present to finish a paper on "The Economics of Exhaustible assets" which I have been at for five years and for which various people have been calling.

Only in January 1931, four months before publication, had the final title been chosen, as another letter from Fredrick C. Mills (NBER) testifies to:9 Schultz was kind enough to let me read a copy of your paper "The Economics of Exhaustible resources." I returned this to him. It seemed to me to be an extremely suggestive attack upon a problem which economists have largely neglected.

Hotelling therefore finally chose an intermediary title, aiming at covering both his generic, theoretical ambition and his specific, more concrete concerns. The archives further reveal that from 1924/1925 to the final version, Hotelling constantly changed the title of his research. A first draft from 1924 was entitled "Exhaustible Assets under Competition."10 Another draft from 1925 was labelled "The Exploitation of Irreplaceable Resources." 11 In December 1929, a full abstract was named "The Economics of Exhaustible Assets." 12 The contents of these drafts are not exactly the same. When assets are considered, the analysis makes a few mentions of mineral resources or concrete questions, instead focusing on the use of advanced mathematics to deal with abstract principles. 13 When natural resources, or minerals, are scrutinized, references to realworld situations are more present, as shown by the 1925 draft entitled "The Exploitation of Irreplaceable Resources": 14 The complete monopoly and the perfectly fluid competition of traditional economics are extremes between which lie all the cases which actually occur in the world.

The generic subject (exhaustible assets) offered opportunities to draw theoretical principles, such as the equation p t = p 0 • e r•t (i.e., the Hotelling rule). The specific subject (exhaustible natural resources), a particular case of exhaustible assets, required more attention to the real world.

The chronology of the archives helps us to reconstruct Hotelling's early treatment of exhaustible resources. The first draft on the subject dates back to 1924; it was written for a meeting of the American Mathematical Society in Chicago. These notes outline the close connection between Hotelling's research on depreciation and on exhaustible assets, as already mentioned. 15 The work on asset valuation was published in 1925 under the title "A General Mathematical Theory of Depreciation." As noted by Crabbé (1986), assuming Hotelling's asset valuation function from Hotelling (1925) is a negative exponential with a constant parameter-the rate of interestyields the Hotelling rule published in 1931.

We found that it was in the winter of 1925 that Hotelling considered for the first time engaging with dedicated research on exhaustible assets. It is also when he decided to explore in more detail the calculus of variations to "simplify" the issue: 16 -Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -5

The recent work on the economic of irreplaceable assets had during the past month been undergoing revision and extension. The revision is made necessary by a simplification of the Calculus of variations method which is found to be applicable to the problem, and also by the publication of a paper by G. C. Evans (Proc. Nat. Acad. Sci., Jan. 1925, pp. 90-95) touching upon the subject.

It appears that Hotelling clearly regarded his generic project-i.e., the theory of exhaustible assets-as an extension of his theory of depreciation. Interestingly, another draft from the same time, March 1925, indicates that he had meanwhile started to investigate the more concrete and real-world side of the problem of exhaustible resources, with special attention to minerals, oil and natural gas in the case of an oligopoly. This confirms not only the entanglement of the two research lines but also their synchrony: 17 The problem is inderteminate because after one mine is exhausted the other has a monopoly; and since we have assumed the demand inelastic the profits of the monopoly are unlimited.

In summary, Hotelling's economics of exhaustible resources in the 1920s was composed of two levels of analysis. Because the 1931 article mixed the two levels, it became difficult for readers to identify the purposes and limits of the different arguments, including the area of validity of the Hotelling rule. The archival material helps provide a clearer understanding: the apparent discrepancy between Hotelling's concrete subject (non-renewable resource depletion) and abstract proposals (e.g., the Hotelling rule) is due to the existence of two initially distinct lines of work developed in parallel, and later merged in the same contribution. For the treatment of real-world resources, Hotelling needed more information on the actual dynamics and issues pertaining to extractive sectors. Archival materials helped us identify where he found this information. 17 HHP, Box 42, Exploitation of Irreplaceable Assets. Gaudet (2007) states Hotelling wrote his 1931 paper in reaction "to the demand for regulation of the exploitation of exhaustible resources by the conservation movement, which had been particularly strong in the U.S. during the period 1890 to 1920" (p. 1034). Livernois (2009) asserts, "Hotelling's seminal article was a response to the conservation movement's claims that unregulated private markets would lead to the overexploitation of non-renewable resource stocks" (p. 23). Their conclusion is likely based on the opening paragraph of "The Economics of Exhaustible Resources," where Hotelling textually frames his research scope in relation to the "conservation movement" (p. 137). However, he mentions the movement only one more time (p. 143), in section 3.

Conservation (or not) and energy policy

The reference to conservation is intriguing because, in the 1920s, among economists, the movement was not as popular as in the 1900s and 1910s, when Lewis C. Gray (1913, 1914) and Richard T. Ely (1918) wrote on the subject. 18 Of course it was still an important trend in the reflections about natural resources (e.g., Ise 1925). However, the importance given by Hotelling to the conservation movement as a counterpoint to his analysis and as a source of information about on-going issues in extractive sectors is not clear at all and deserves to be examined in detail. Contemplation of the world's disappearing supplies of minerals, forests, and other exhaustible assets has led to demands for regulation of their exploitation, by taxation or otherwise. On the other hand the existence of monopolies and quasi-monopolies has raised the same problems of excessively curtailed production here as in other industries, but in a more complex form. The economics of the exploitation, that is, the study of the rates of exploitation which on the one hand tend to take place under commercial conditions, and of those which on the other hand ought in the public interest to take place, has however received no serious attention [our emphasis].

This provisional introduction reveals that conservation was not so directly decisive in Hotelling's undertaking.

If not in the conservation literature (especially the economic literature), where did Hotelling find information about the concrete dynamics of extractive sectors (minerals, oil and gas)? Several allusions disseminated in the 1931 article find justification in issues of the 1920s not directly related to conservation:

The prohibitions against oil and mineral development and cutting timber on certain government lands have this justification, as have also closed seasons for fish and game and statutes forbidding certain highly efficient means of catching fish (p. 137).

On the other hand, certain technical conditions most pronounced in the oil industry lead to great wastes of material and to expensive competitive drilling, losses which may be reduced by systems of control which involve delay in production (p. 138).

The government of the United States under the present administration has withdrawn oil lands from entry in order to conserve this asset, and has also taken steps toward prosecuting a group of California oil companies for conspiring to maintain unduly high prices, thus restricting production (p. 138).

Daniel Yergin's 1991 work on oil history

19 HHP, Box 42, Exploitation of Irreplaceable Assets.

illuminates the events Hotelling refers to. The 1920s in the United States were marked by fast growing oil demand brought by gasoline-fuelled cars (Yergin 1991, p. 208). Meanwhile, very few new oil discoveries had been registered on the 1917-1920 period, leading oil prices to triple in the period. Within this context took place in the early 1920s the "Teapot Dome Scandal." This episode of corruption was about the exploitation of several oil fields in Wyoming initially preserved by the US government as naval oil reserves under the Taft and Wilson administrations (Yergin 1991, p. 211). In 1924, when Hotelling started to work on exhaustible resources, the presidential election was marked by the scandal. The mention of government ownership over lands and resources in the 1931 paper (above quotes) matches this episode.

Hotelling's allusion to technical conditions leading to waste can also be related to legal debates in the 1920s. The so-called "Rule of Capture" attributed ownership over oil resources to the owners of the land immediately above it. When several properties spanned a single reservoir, operators would rush to produce as rapidly as possible in order to avoid letting the neighbours drain the volumes under their properties through their wells. This ultimately led to an excessive number of wells and lowered the ultimate recovery of oil from the reservoir. Debates took place mostly under the Coolidge administration (1923-1929), leading to the establishment of the Federal Oil Conservation Board, in 1924 (Yergin 1991, pp. 220-23).

Therefore, the remarks on minerals, oil and gas disseminated by Hotelling in his final article, not fully related to the academic debates on conservation, actually correspond to various affairs linked to the US federal policy. His allusions to practical cases were not thought experiments, or pretexts to explore fictive situations; they were based on real political and economic experiences that marked American public life in the 1920s.

In this context, the Hotelling rule, although essential for the generic project on exhaustible assets, and which supposed perfect competition and no State intervention for achieving the social optimum, was clearly established as inadequate for the analysis of concrete, specific natural resources (minerals, oil and gas), as -Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -7 demonstrated by the closing of section 2: ...[T]here are in extractive industries discrepancies from our assumed conditions leading to particularly wasteful forms of exploitation which might well be regulated in the public interest (pp. 143-44; our emphasis).

The word "discrepancies" is significant as a strong indicator of the inoperability of the basic model for describing concrete situations. As soon as these situations are involved, the "real economic world" (Hotelling 1931, p. 171) matters, up to assigning a role to regulators in the extractive sector, in contrast to what is usually retained from Hotelling's analysis in terms of State intervention. 20 Theoretically speaking, empirical realities require further developments beyond the basic result.

An insight emerges from the realization that Hotelling did not see what Solow (1974) labelled "the Hotelling rule" as a rule for non-renewable resource economics because it could not describe what took place in reality in extractive sectors, as Hotelling found in concrete experiences from the 1920s. The area of validity of the rule was limited to finitely available assets, in the absence of some realities he proceeded to analyze. Among these realities, apart from alternate market structures and taxation issues, we discovered Hotelling paid close attention to geological constraints, which we will show he concluded to be necessary conditions to depict natural resources.

Hotelling and geological contraints

Geological constraints in the 1931 article

"The Economics of Exhaustible Resources" is known for its basic model, which is described in sections 2 and 3 of the 1931 article. The final paper, nonetheless, contains 15 sections. Archive research revealed the basic result dates back to the first drafts, circa 1924/1925. Subsequent drafts deal with more substantial models, which one can find in the less-known sections of the 20 Regarding Hotelling's conception of State intervention, some insight can be found in Franco et al. (2019).

1931 article. Slade and Thille (2009) acknowledge that "Hotelling derived several variants of his model" (p. 241; our emphasis). These "variants" actually represented the core of Hotelling's work, at least as reflected in the archives.

From sections 8 to the end of his analysis, Hotelling deals with his specific research line on concrete mineral resources, directing his attention to the realities of mining extraction. 21 There he leaves aside the basic equation and turns to the issue of "cumulative production affecting price" (p. 152). The "net price" function becomes p(x(t),q(t),t): cumulative production x(t) = ∫ 0 t q(t) dt is now a variable, affecting the formation of net prices. Hotelling justifies the need for this variable on the cost side. It turns out resources are located under the ground, in varying depths. A first geological constraint (GC1) other than finiteness thus appears: "the cost of extraction increases as the mine goes deeper" (p. 152).

In section 12, "The Need for Steadiness in Production" (p. 162), Hotelling further expands his net price function to portray the rate of change in production q'(t) as an additional variable. The cost of increasing production, Hotelling argues, depends on the magnitude of the increase. In developing a mine, producers have to make capital investments, which makes sudden increases in production more expensive than gradual ones. This is due to the uneven distribution of concentrations of resources under the ground-a second geological constraint (GC2) other than finiteness, which imposes a cost for drawing the labour force to or from the site of the mine, making large variations in output significantly costlier than small variations. Hotelling closes the section by noting:

The cases considered in the earlier part of this paper all led to solutions in which the rate of production of a mine always decreases. By considering the influence of fixed investments and the cost of accelerating production at the beginning, we may be led to production curves which rise continuously from zero to a maximum, and then fall more slowly as exhaustion approaches. Certain production curves of this type have been found statistically to exist for whole industries of the extractive type, such as petroleum production (p. 164; our emphasis).

In section 15, the closing section of the paper, Hotelling comes back to the discussion between perfect competition and monopoly, presenting the case of duopoly, in which a few competing sellers can share the market's demand. In this model, Hotelling avoids the complex formulation of section 12, excluding for simplicity the influence of the rate of change in production as a determinant of the price. However, at this point, Hotelling makes reference to a third geological constraint (GC3), other than finiteness, namely the uncertainty on the actual size of reserves in the beginning and the need to acquire geological information through drillings: 22 The problems in exhaustible resources involve the time in another way besides bringing on exhaustion and higher prices, namely, as bringing increased information, both as to the physical extent and condition of the resource and as to the economic phenomena attending its extraction and sale (p. 174).

Therefore, aside from market structure and taxation, Hotelling's 1931 displays another, often overlooked, clear axis of progression from his basic result, towards the incorporation of GC1, GC2 and GC3, notably through x(t) and q'(t), as fundamental variables. While the number of firms that operate on a given market or its tax regime can in principle be affected by 22 In all mineral activities, the underground location of mines is initially unknown, and each mine or well drilled brings increased geological knowledge that facilitates the discovery of the next underground occurrence. Costs are not known with certainty from the start, which carries substantial investment risks. Only upon drilling wells and evaluating their output or how they respond after another well is drilled into the same oil reservoir, it it possible to infer engineering properties such as reservoir pressure and porosity and, as a result, determine the total number of wells needed to optimally exploit the resource. policy, geological constraints are something no one can do anything about. Therefore, they should not be considered as "variants" or "possibilities," as mentioned by Slade and Thille (2009), nor as additional factors, but as fundamental inalterable characteristics with the same status as finiteness, affecting all mineral resources. In addition to the already mentioned closing of section 2, in the opening of section 8, Hotelling states the influence of x(t) on the exploitation of resources (which results from GC1) is a certainty, not a mere possibility: "[t]he net price...depends...on past production,", i.e., it depends on it, not may depend on it (p. 152; our emphasis). 23 Therefore, sections 8 to 15 of Hotelling's article discuss concrete real-world factors that define the category of mineral resources within the class of non-renewable resources, itself a subset of non-renewable (exhaustible) assets. Once more, in contrast to modern interpretations that keep the basic rule as a central principle to be sophisticated, it clearly emerges that this theoretical principle does not apply here. The rule governs only generic finite assets in the absence of geological constraints other than finiteness (GC1, GC2 and GC3). It therefore does not govern mineral resources (minerals, oil and gas), as presently overstated.

Hotelling's preparatory work in geology and engineering

By contrasting the archival materials with the final paper, we discovered the relative degree of effort Hotelling devoted to the topic of geological constraints. Two drafts from November 1928 are dedicated entirely to the issue of "strong variations," in which Hotelling includes for the first time cumulative production x(t) as a fundamental variable, and devote significant effort to understanding the implications to the overall equilibrium. 24 Furthermore, in the draft abstract written in 1929, Hotelling does not present the basic model at all, starting with the model with x(t), using If the price obtainable varies with the amount which has been extracted, as with gold and diamonds, or if the cost of production increases as the shaft sinks deeper, the problem is to maximize ∫ 0 T e^(-γt) p(x,q,t) qdt.

By 1930, Hotelling continued to focus on how geological constraints other than finiteness could affect the net price function. A noticeable preparatory work is his correspondence with Stanley C. Herold, professor of petroleum engineering at Stanford University-Hotelling was still at Stanford in 1930. 26 Herold wrote to Hotelling on July 7, 1930, in a friendly tone, providing empirical data on how, in the case of petroleum, costs increase as shafts sink deeper (GC1); see table 1.

Table 1 shows a non-linear (quadratic) relationship between cost and depth; costs increase more than proportionately with well depth (figure 1).

GC2 and GC3 are also addressed in the letter, 25 and some remarks can be traced to Hotelling's statements in his final article. For production speed at the beginning of the extraction, Herold writes:27

The cost mounts more rapidly than the footage on account of the fact that deeper holes require heavier rigs and equipment, also more time is consumed in running in and out the holes.

Indeed, because different oil and gas reservoirs vary in depth and many other geological conditions, their unit production costs vary. Producers seek to developthe lowest cost ones first (Herfindahl 1967), although sometimes they are unable at first to discover the best reservoirs when only limited geological information is available (Uhler 1976). This leads to Hotelling's remark on GC2, when he mentions "the cost of accelerating production in the beginning" (1931, p. 164).

Still in the same letter, Herold describes the spatial element affecting costs, thus insisting on the heterogeneity of wells:28 ...[H]oles at Kettleman Hills are costing a little more than shown here on account of transportation of equipment and general operating expenses, longer delays on replacement of equipment, etc. This explains Hotelling's remarks in section 12 about costs related to the relocation of production factors, in particular the labour force: Under the term "capital" might possibly be included the costs, both to employers and to laborers, in drawing laborers to the mine from other places and occupations (Hotelling 1931, p. 163).

The example of Kettleman Hills is significant because it also highlights the variability of costs resulting from new discoveries: when promising wells are discovered-the Kettleman Hills oil fields, in particular around the North Dome, were a major discovery in California in the late 1920s-they attract capital and labour, with -Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -10 potentially abrupt variations in both production and costs.

Therefore, Hotelling's work between 1928 and 1930, as reflected in the archives, dealt almost exclusively with the impact of geological constraints on his basic model from 1925. The importance given to the incorporation of x(t) as a variable is unrivalled, comparable only to the distinction between monopoly and perfect competition. In addition to Herold's 1930 letter to Hotelling, the 1928/1929 drafts, and how these documents were reflected in the final paper, demonstrate that geological constraints are a necessary factor to describe real-world minerals and fossil fuels. All such resources are subject to geological constraints.

Towards an alternative basic Hotelling model?

The By considering the influence of fixed investments and the cost of accelerating production at the beginning, we may be led to production curves which rise continuously from zero to a maximum, and then fall more slowly as exhaustion approaches. Certain production curves of this type have been found statistically to exist for whole industries of the extractive type, such as petroleum production (p. 164).

More intriguing is precisely what Hotelling retained from his reading of Van Orstrand. In the archival material, we found a short memo, entitled "Mine Economics," that contained a full reference to Van Orstrand's paper as well as a handwritten graph, quickly drawn, representing a bell-shaped curve with a long tale (figure 2). Unlike Van Orstrand, who specified on the axes of his graphs the dates and volumes of production from real data on West Virginia, Ohio and Pennsylvania oil fields and so on, Hotelling sketches only the general slope, without the original figures, merely noting "production of petroleum" next to the vertical axis and "t" (for time) next to the horizontal axis. This is not anecdotal and suggests that his vision of the bell-shaped curve was generic and not contingent on this or that field. This type of approach-to transform specific quantified cases into generic stylized facts-reminds us of Cournot, who was the first in the history of economics (1838) to draw stylized curves from specific cases, in particular for demand functions. We know that Cournot played an important role in Hotelling's mathematical economics training (Hotelling 1929, p. 51; Darnell 1990, pp. 12 and 14), and we found references to Cournot in drafts and notes from the 1920s. 29 Hotelling's ambition with the bellshaped curve therefore can be related to the description of a stylized fact for mineral resources, in Cournot's tradition.

The fact that "fixed investments and the cost of accelerating production at the beginning...

-Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -11 may...le[a]d [us]" to trajectories like the one sketched in Hotelling's hand-drawn graph suggests he knew that the incorporation of q' on the net price function would change the model towards a bell-shaped equilibrium. We were able to reproduce it ourselves through a simulation exercise, adjusting the linear net price function in section 8 to p = a-bq-cq'-dq'', where q is the cumulative production at time t, assuming Hotelling's original parameters and d = 10.000 (see box 1 and figure 3). 30 Results indicate a bell-shaped equilibrium for production q' and an U-shaped equilibrium for market prices p = a-cq'.

In 1931, Hotelling did not have at his disposal mathematical tools such as optimal control theory (developed in the 1950s), allowing him to propose an alternative model to fit with this kind of observations. The hand- 30 We make g = 0 in order to avoid masking the U-shaped price path in equilibrium by an increasing demand as a function of time.

drawn graph, however, confirms his awareness of bell-shaped supply trajectories for the case of extractive industries, such as petroleum. His stylized-fact drawing, supplemented by his quotation, confirms that his basic model leading to the Hotelling rule was not relevant for concrete mineral resources. To the best of our knowledge, the suggested model in section 12, as described in box 1, has not been depicted in the literature to date. While its complete characterization might be insightful, it is beyond the scope of the present paper.

Conclusion

Based on unprecedented archival inquiry, this paper has reconstructed Hotelling's project on non-renewable resources in the 1920s leading to the publication of "The Economics of Exhaustible Resources" in 1931. We showed that Hotelling actually worked along two lines of research: a generic one on asset valuation, in the line of his theory of depreciation, and a specific one on concrete mineral natural resources, requiring more empirical insights from the real world. In contrast to what is often argued, his source of information for implementing concrete situations in his framework was not the conservation movement but rather specific experiences in US policy throughout the 1920s.

We found Hotelling carried out substantial preparatory work to circumscribe the geological conditions of mineral resource extraction. This is hardly noticeable to the naked eye in the 1931 article. However, after consulting the archives, we presented a new reading of the article where the importance of geological constraints for the treatment of minerals and fossil fuels is significantly increased. After considering the implication of their presence, Hotelling became aware of, notably, bell-shaped trajectories linked to U-shaped price trajectories.

In addition to presenting these novel findings in the history of economic thought, our investigation allowed us to derive theoretical implications. After nearly 90 years of developments since Hotelling's 1931 article, we still attribute the title "Hotelling rule" to his basic result p t = p 0 • e r•t and consider it to be the starting-point principle of non-renewable 2), 800-820 -12 resource economics. This seems problematic given the rule lacks empirical validity and requires strong amendments in order to describe the long-term, aggregate behaviour of its target object. Our reconstruction of Hotelling's research project sheds light on this theoretical matter.

In particular, we established that Hotelling never saw his basic result p t = p 0 • e r•t as a rule for non-renewable resource economics. Instead, he stated his basic result referred to a hypothetical stylized asset marked exclusively by finiteness, which is not the case of mineral resources. For this category, elaborating on his preparatory work in geology and engineering, Hotelling concluded it was necessary to consider as fundamental factors at least the influences of cumulative production x(t), because costs increase as mines go deeper (GC1), and the rate of change in production q'(t), because strong variations in output are costlier than small ones (GC2).

By sketching the implied model, we were able to confirm it is possible to obtain bellshaped trajectories for supplies and U-shaped ones for prices, as described by Hotelling, at least for given values of the parameters. Future research should investigate the formal mathematical demonstration of this result. If confirmed, two inherent geological constraints (GC1 and GC2), in addition to exhaustibility, would be sufficient to reflect the long-term equilibrium of mineral resources. Hotelling's 1931 contribution would thus remain the starting point of non-renewable resource economics, albeit in a form different from that advocated to date.

-

-

  Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -6 1924/1925: 19

-

  Authors's post-print --published in the Canadian Journal of Economics (2020), 53(2), 800-820 -9 empirical examples to justify his formulation: 25

  constraints GC1 and GC2 are at the foundation of the technical understanding of what we see today as bell-shaped supply trajectories insofar as they explain the specific dynamics of extraction in relation to reserve degradation and to uncertainty about reserve size. We know that Hotelling was aware of C. E. Van Orstrand's 1925 article "On the Empirical Representation of Certain Production Curves" because Hotelling refers to it in his own 1931 article (p. 164). In his contribution, Van Orstrand pays particular attention to extractive sectors and depicts their dynamics by drawing three bell-shaped graphs (1925, pp. 26 and 32), including two with a long tale, depending on estimations. Hence, Hotelling knew of the existence of bell-shaped representations of the exploitation of exhaustible resources. Hotelling's description of extractive industries in his 1931 article corresponds to this kind of production trajectory, as shown in the earlier quotation:

  s post-print --published in the Canadian Journal of Economics (2020), 53(

Letter from Hotelling to Fisher, July 23, 1930. Online digitized version. RAFP: Ronald A. Fisher Papers, Special Collections Library, University of Adelaide, Australia. Catalogue available at digital.library.adelaide.edu.au/ dspace/handle/2440/3860.

Letter from Mills to Hotelling, Jan. 17, 1931. HHP, Box 1, Mills, Frederick C.

HHP, Box 10, AMS Reports and Correspondence (3).

HHP, Box 42, Exploitation of Irreplaceable Assets.

HHP, Box 42, Exploitation of Irreplaceable Assets.

For instance, in the draft "Exhaustible Assets under Competition" from 1924, Hotelling states his problem as "applying Euler's equation." HHP, Box 10, AMS Reports and Correspondence (3).

HHP, Box 42, Exploitation of Irreplaceable Assets.

HHP, Box 10, AMS Reports and Correspondence. For a full inquiry into the intertwining of Hotelling's projects on depreciation and exhaustible assets, see[START_REF] Missemer | Depreciation and Exhaustible Resources, Exploring Harold Hotelling's Twin Projects" #BNREproject Working Paper[END_REF].

Monthly Report from Hotelling to the Directors of the FRI. Mar. 2, 1925. HHP, Box 41, Agriculture III.

The archives reveal that early drafts make no mention of the conservation movement, even along the specific, more empirical research line. In a preliminary introduction of the paper, in December 1929, and which was very close to the final version, the catchphrase is already written (apart from the last addition on taxation), but the part on conservation is missing. Instead, Hotelling insists on competitive settings and on exhaustible assets as a specific case of a wider issue, in line with his previous drafts starting in18 On the economic thought of the first conservation movement, see for instance[START_REF] Smith | Natural Resource Economic Theory of the First Conservation Movement (1895-1927)[END_REF][START_REF] Ramos Gorostiza | Ethics and Economics: From the Conservation Problem to the Sustainability Debate[END_REF],[START_REF] Kula | History of Environmental Economic Thought[END_REF] and[START_REF] Missemer | Depreciation and Exhaustible Resources, Exploring Harold Hotelling's Twin Projects" #BNREproject Working Paper[END_REF]. Gray worked on conservation when he was at the University of Saskatchewan (Canada), initiating a long tradition of Canadian research in resource and environmental economics (see[START_REF] Copeland | Environmental and Resource Economics: A -Authors's post-print[END_REF]. On Gray's contributions, see[START_REF] Crabbé | The Contribution of L. C. Gray to the Economic Theory of Exhaustible Natural Resources and Its Roots in the History of Economic Thought[END_REF].

The word "mine(s)" is also employed a few times in the sections dedicated to the basic model, but to designate stylized mines, for which finiteness is the only characteristic considered. Hotelling makes clear from section 8 onwards that concrete mines do not fall under the same analysis.

In other parts of the article (e.g., section 12), Hotelling uses less clear-cut vocabulary for the occurrence of geological factors, but he always argues they have to be of the utmost importance.

HHP, Box 42, Exploitation of Irreplaceable Assets.

Letter from Herold to Hotelling, July 7, 1930. HHP, Box 42, Exploitation of Irreplaceable Assets.

Letter from Herold to Hotelling, July 7, 1930. HHP, Box 42, Exploitation of Irreplaceable

"Talk on 'Mathematical Economics' before Math. Club, May 1928." HHP, Box 26, Mathematical Economics. We thank Marion Gaspard for drawing our attention to the connection between Cournot and Hotelling.

Acknowledgements

This paper benefited from the support of the European Society for the History of Economic Thought (ESHET), through the project Bifurcations in Natural Resource Economics (1920s-1930s). We thank Marion Gaspard, Marco P. Vianna Franco, Thomas M. Mueller and Franck Nadaud for stimulating discussions in the working-group. Thanks also to the participants in the Facts in Environmental and Energy Economics, Models and Practices, Past and Present workshop (Paris, October 2018) for feedback. We are grateful to the staff of the Rare Book and Manuscript Library of Columbia University for their help during our visit to Hotelling papers archival collection, to the Hotelling familiy for encouraging our work and to three anonymous referees and the editor of the journal, who substantially helped us to improve the paper. The views, thoughts and opinions expressed in the text belong solely to the authors and not necessarily to their employers, organizations, committees or other groups to which they are affiliated. Any errors or omissions are ours alone.

-

Supporting information

Supporting information is available in the online version of this article (publisher's website).

-