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International audience; Electricity systems are quitting monopoly for an operation framed by competitive markets. It therefore questions how the operation of nuclear plants capable of load-following is expected to change at the transition stage in a challenging economic environment for nuclear producers given the high penetration of renewables and the increase of nuclear production costs for renovation and safety reasons. To answer this question, we are focusing on a month-by-month management horizon of the nuclear fuel given the technico-economical features of the operation of flexible nuclear plants. The resolution of the month-by-month profit maximization problem contributes to identify potential difficulties and limits of production optimization and provides us with an equilibrium which proves the existence of an optimal production path. This benchmark could serve as a reference providing valuable insight on the resolution of more complex optimisation problems.

The road towards the decarbonization of electricity leads to high deployment of low-carbon power sources including intermittent energy sources. In this context, flexible nuclear power plants could play a significant role because they do not produce CO2 emissions and under certain conditions flexible operation is necessary to ensure the stability of the electricity grid. Flexible nuclear reactors have the ability to load-follow the predicted fluctuations in demand. However, high fixed costs of nuclear production, tighter regulations since the Fukushima accident and the extensive participation of renewable sources in the energy mix challenge the economic profitability of nuclear production. Consequently, a question that arises is how nuclear power producers can manage flexible nuclear production in order to maximize their profits. We proved that optimal production behaviour is not characterized by constant nuclear production unless further investments in thermal capacity are realized. On the contrary, both nuclear and thermal production are flexible at the optimum level.

We give a generalized version of the well-known Borsuk’s antipodal fixed point theorem for a large class of antipodally approachable condensing or compact set-valued maps defined on closed subsets of locally convex topological vector spaces. These results contain corresponding results obtained in the literature for compact set-valued maps with convex values.

In the spirit of Michael’s selection theorem [6, Theorem 3.1”’], we consider a nonempty convex-valued lower semicontinuous correspondence φ : X → 2 Y {\varphi:X\to 2^{Y}} . We prove that if φ has either closed or finite-dimensional images, then there admits a continuous single-valued selection, where X is a metric space and Y is a Banach space. We provide a geometric and constructive proof of our main result based on the concept of peeling introduced in this paper.

International audience; Flexible nuclear power plants can adjust their electricity production to the predicted evolution of demand. Under certain conditions, flexible operation is necessary to ensure the stability of the electricity system. However, despite the potential advantages of nuclear energy including the flexibility of nuclear reactors, the social acceptance of nuclear has reduced after the Fukushima accident, leading some countries to reduce or even phase out nuclear (e.g. Germany). So, a question that arises is how flexible nuclear power plants have to be operated in order to maximize social welfare. The French nuclear fleet gives an illustration of flexible management while social acceptance of nuclear is questioned; this was reflected in the new French Energy Transition law. Theoretically and numerically, we found that the production behavior that maximizes social welfare is characterized by a constant thermal production and a totally flexible nuclear production given sufficient nuclear capacity.Energy

In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, introduces conditions which, when applied to the convex case, give for Banach commodity spaces the well-known result of decentralization by continuous prices of Pareto-optimal allocations under an interiority condition. In this paper, in order to prove a different version of the Second Welfare Theorem, we reinforce the conditions on the commodity space, assumed here to be a Banach lattice, and introduce a nonconvex version of the properness assumptions on preferences and the total production set. Applied to the convex case, our result becomes the usual Second Welfare Theorem when properness assumptions replace the interiority condition. The proof uses a Hahn-Banach Theorem generalization by Borwein and Jofre (in Joper Res Appl Math 48:169–180, 1997) which allows to separate nonconvex sets in general Banach spaces

International audience; In this paper, we study a discrete-time version of the Lucas model with externality but without physical capital. We give conditions for which the optimal human capital sequences are increasing. When the instantaneous utility function is isoelastic and the production function is Cobb-Douglas, we prove that the optimal human capital sequences grow at constant rate. Moreover, there exists a unique equilibrium which, under an additional assumption on the human capital technology, is also the unique competitive equilibrium.

In this paper, we establish an equivalent version of the hairy ball theorem in the form of a fixed point theorem. By using a version of Mas-Colell theorem [6] and by applying homotopy and approximation methods, we obtain our main result., Nous présentons dans cet article, une nouvelle version équivalente au théorème de la boule chevelue sous la forme d'un théorème de point fixe. Afin de montrer le résultat principal, on a eu recours au théorème de Mas-Colell theorem [6] ainsi que des techniques d'approximations et d'homotopies.

In the case of a large nuclear set (like the French set), nuclear production needs to be flexible to adjust to the predicted evolutions of the energy demmand. Consequently, the dominant position of nuclear in the national energy mix makes it responsible for the overall equilibrium of the electricity system which is directly intertwined with social welfare. In a previous work, we looked at producers own profits (short-term, inter-temporal) considering the equality between supply and demand. Here, we proceed with a full optimization of the social welfare in an identical framework. Theoretically, the optimal production behaviour that maximizes social welfare is characterized by a constant thermal production and a totally flexible nuclear production given that the nuclear capacity is sufficient. Numerically, the significant amount of nuclear capacities compared with thermal capacities in the French electricity market leads to the same “paradoxical” production behaviour. Therefore, we conclude that social optimum is ensured within our model by investing sufficiently in nuclear capacity. The optimal production scheduling determined by the social welfare maximization problem and the optimal inter-temporal production problem are totally opposite.

This paper addresses existence of equilibria for an overlapping generations exchange economy with incomplete markets, one-period real assets and bounded short sales, defined on an infinite event-tree where there is a continuum of branches at each node of the tree. The demographic structure used in the paper is quite general and allows for infinite-lived as well as finite-lived agents.

We look at the optimal inter-temporal management of the fuel reservoir of nuclear units in a liberalized electricity market. We use the assumption that nuclear fuel works as a “reservoir” of energy due to the periodical shutdown of nuclear units to reload their fuel. In the medium-term, how a producer sets the nuclear fuel of the reservoir to respond to the variations of seasonal demand in order to maximize its production value on a multi-annual basis? The dynamic nature of the nuclear fuel reservoir highlighted the discontinuity of the price which complicates the resolution of the optimal inter-temporal production problem and even leads to a lack of solutions. Theoretically, at the optimum, nuclear is used to serve baseload and thermal follows demand's variations. Numerically, both nuclear and thermal units operate in load-following mode. Solutions characterized by a constant nuclear production do not exist which shows that the significant share of nuclear in the energy mix does not permit to produce at a constant rate unless further investments in thermal capacity are done. Inter-temporal optimization shows the role of nuclear for ensuring the equilibrium between supply and demand.

In many countries, the electricity systems are quitting the vertically integrated monopoly organization for an operation framed by competitive markets. It therefore questions how flexible nuclear plants capable of load-following should be operated in an open market framework. A number of technico-economical features of the operation of flexible nuclear plants drive our modelling complex which makes difficult to determine the optimal management of the nuclear production within our model. In order to examine the existence of an equilibrium and calculate it, we focus on a short-term (monthly) management horizon of the fuel of nuclear reactors. The marginal cost of nuclear production being (significantly) lower than that of thermal production induces a discontinuity of producer's short-term profit. The problem of discontinuity makes the resolution of the optimal short-term production problem extremely complicated and even leads to a lack of solutions. That is why it is necessary to study an approximate problem (continuous problem) that constitutes a “regularization” of our economical problem (discontinuous problem). Its resolution provides us with an equilibrium which proves the existence of an optimal production trajectory.

The general equilibrium model with incomplete markets is here extended to infinite horizon economies populated by a finite number of infinitely lived agents. the crucial issue that divides the infinite horizon setting from the finite horizon setting is in the nature of borrowing constraints, which added to spot constraints, define a plausible budget set for individual agents. the paper relates seven alternative definitions of equilibrium and states corresponding equilibrium existence theorems when assets are one-period and purely financial.

We consider a general equilibrium model of an economy with increasing returns to scale or more general types of nonconvexity and without ordered preferences. Firms are instructed to set their prices according to general pricing rules which may depend on the production plans of other firms. We suppose, moreover, that the pricing rule of the firms verifies a condition of weak bounded losses. This includes the case of profit maximizing, average cost pricing and marginal (cost) pricing, thanks to a transformation used by Bonnisseau. The tastes of the consumers may depend both on other consumptions and on the prices This paper reports a general existence result in this model which extends the results of Bonnisseau and Cornet.

A very simple proof is given for the existence of stochastic equilibria in a T -period economy with an incomplete financial structure.

In electricity markets where competition has been established for a long time, a nuclear operator familiar with the operation of such markets could be interested in the optimal long-term management of a flexible nuclear set (like the French) in a competitive market. To obtain a long vision of the optimal management of a nuclear set, we realize a full inter-temporal optimization of the production which results from the maximization of the value of generation over the whole game. Our model takes into consideration the periodical shut-down of nuclear units to reload their fuel, which permits to analyze the nuclear fuel as a stock behaving like a reservoir. A flexible nuclear reservoir permits different allocations of the nuclear fuel during the different demand seasons of the year. Our analysis is realized within a general deterministic dynamic framework where perfect competition is assumed and two flexible types of generation exist : nuclear and thermal non-nuclear. The marginal cost of nuclear production is (significantly) lower that the one of non-thermal production, which induces a discontinuity of producers' profit. In view of this price discontinuity, a "regularization" of the merit order price is achieved within our numerical model which leads to an alternative optimization problem (reglarized problem) that constitutes a good approximation of our initial problem. We also prove that in the absence of binding productions constraints, solutions are fully characterized by a constant nuclear production. However, such solutions do not exist within our numerical model because of production constraints that are active at the optimum. Finally, we study the maximization of social welfare in an identical framework. Similarly, we demonstrate that in the absence of binding production constraints a constant non-nuclear thermal production is a characteristic property of solutions of the social welfare maximization problem., Dans les marchés de l'électricité où la concurrence a été établie depuis longtemps, un exploitant nucléaire familiarisé avec le fonctionnement de marchés concurrentiels pourrait être intéressé par la gestion optimale d'un parc nucléaire flexible (comme le français) à long terme. Notre modèle prend en considération les arrêts périodiques des unités nucléaires pour recharger leur combustible, ce qui permet d'analyser le combustible nucléaire comme un stock que l'on gère comme un réservoir. Notre analyse est réalisée dans un cadre dynamique déterministe général, où la concurrence est supposée parfaite et deux types de production flexibles existent : le nucléaire et le non-nucléaire thermique. Le coût marginal de production nucléaire est (significativement) inférieur à celui de production non-thermique, ce qui induit une discontinuité de profit des producteurs. Compte tenu de cette discontinuité des prix, une "régularisation" du prix merit order est réalisée au sein de notre modèle numérique qui conduit à un problème d'optimisation alternative (problème régularisé) qui constitue une bonne approximation de notre problème initial. Nous montrons aussi que quand les contraintes de production ne sont pas saturées, les solutions sont entièrement caractéris&eaute;es par une production nucléaire constante. Toutefois, de telles solutions n'existent pas au sein de notre modèle numérique en raison des contraintes de production qui sont actives à l'optimum. Enfin, nous étudions la maximisation du bien-être social dans un cadre identique. De la même façon, nous démontrons qu'en l'absence de saturation des contraintes de production, une production non-nucléaire thermique constante est une propriété caractéristique des solutions du problème de maximisation du bien-être social.

In many countries, the electricity systems are quitting the vertically integrated monopoly organization for an operation framed by competitive markets. In such a competitive regime one can ask what the optimal operation/management of the nuclear generation set is. We place ourselves in a medium-term horizon of the management in order to take into account the seasonal variation of the demand level between winter (high demand) and summer (low demand). A flexible nuclear set is operated to follow a part of the demand variations. In this context, nuclear fuel stock can be analyzed like a reservoir since nuclear plants stop periodically (every 12 or 18 months) to reload their fuel. The operation of the reservoir allows different profiles of nuclear fuel uses during the different seasons of the year. We analyze it within a general deterministic dynamic framework with two types of generation : nuclear and non-nuclear thermal. We study the optimal management of the production in a perfectly competitive market. Then, we build a very simple numerical model (based on data from the French market) with nuclear plants being not operated strictly as base load power plants but within a flexible dispatch frame (like the French nuclear set). Our simulations explain why we must anticipate future demand to manage the current production of the nuclear set (myopia can not be total). Moreover, it is necessary in order to ensure the equilibrium supply-demand, to take into account the non-nuclear thermal capacities in the management of the nuclear set. They also suggest that non-nuclear thermal may remain marginal during most of the year including the months of low demand., Dans de nombreux pays, les systèmes d'électricité sont en train de quitter l'organisation de monopole verticalement intégré pour une opération encadrée par des marchés concurrentiels. Dans un tel régime concurrentiel, on peut se demander ce qu'est l'opération/la gestion optimale du parc de génération nucléaire. On se place dans un horizon "moyen terme" de la gestion afin de prendre en compte les variations saisonnières du niveau de demande entre l'hiver (forte demande) et l'été (basse demande). Un parc nucléaire flexible permet de suivre une partie des variations de la demande. Dans ce cadre, le stock de combustible nucléaire peut être analysé comme un réservoir puisque les centrales nucléaires s'arrêtent périodiquement (tous les 12 ou 18 mois) pour recharger leur combustible. La gestion de ce réservoir permet des profils différents d'usages de combustible nucléaire au cours des différentes saisons de l'année. On se place dans un cadre dynamique déterministe général avec deux type de génération : nucléaire et thermique non-nucléaire. Nous étudions la gestion optimale de la production dans un marché parfaitement concurrentiel. Ensuite, nous construisons un modèle numérique très simple (basé sur les données du marché français) où les centrales nucléaires ne sont pas opérées à production constante, mais dans un cadre de placement flexible (comme le parc nucléaire français). Nos simulations expliquent pourquoi il faut anticiper la demande future pour gérer la production actuelle du parc nucléaire (la myopie ne saurait être totale). Il faut de plus pour assurer l'équilibre offre-demande prendre en compte les capacités thermiques non nucléaires dans la gestion du parc nucléaire. Elles suggèrent que le thermique non-nucléaire peut rester marginal pendant la majeure partie de l'année, y compris durant les mois de basse demande.

In many countries, the electricity systems are quitting the vertically integrated monopoly organization for an operation framed by competitive markets. In such a competitive regime one can ask what the optimal management of the nuclear generation set is. We place ourselves in a medium-term horizon of the management in order to take into account the seasonal variation of the demand level between winter (high demand) and summer (low demand). A flexible nuclear set is operated to follow a part of the demand variations. In this context, nuclear fuel stock can be analyzed like a reservoir since nuclear plants stop periodically (every 12 or 18 months) to reload their fuel. The operation of the reservoir allows different profiles of nuclear fuel uses during the different seasons of the year. We analyze it within a general deterministic dynamic framework with two types of generation: nuclear and non-nuclear thermal. We study the optimal management of the production in a perfectly competitive market. Then, we build a very simple numerical model (based on data from the French market) with nuclear plants being not operated strictly as base load power plants but within a flexible dispatch frame (like the French nuclear set). Our simulations explain why we must anticipate future demand to manage the current production of the nuclear set (myopia can not be total). Moreover, it is necessary in order to ensure the equilibrium supply-demand, to take into account the non-nuclear thermal capacities in the management of the nuclear set. They also suggest that non-nuclear thermal could stay marginal during most of the year including the months of low demand.

ED EPS; International audience; This Note presents a theorem of the existence of the Nash equilibrium for discontinuous games in a topological vector space. We will use an assumption of better reply secure which is stronger then that of Reny. If the payoff function is upper semi-continuous, the two assumptions coincide. Our proof is simple, independent and based on a version of Fan-Browder theorem of existence of maximal element due to Deguire and Lassonde, which is extended to the non-Hausdorf case.; Nous présentons un théorème d'existence d'équilibre de Nash pour un jeu discontinu et dans un espace vectoriel topologique. On utilise une hypothèse de meilleure réponse sécurisée plus forte que celle de Reny. Si la fonction de paiement est semi-continue supérieurement, les deux hypothèses de meilleure réponse sécurisée coïncident. Notre preuve est simple, indépendante et basée sur une version du théorème de Fan-Browder d'existence d'élément maximal dû à Deguire et Lassonde, dont nous démontrons une extension au cas des espaces non séparés.

Cahiers de la Maison des Sciences Economiques ; b04063; International audience; This paper studies the existence of solutions in continuous time optimization problems. It provides a theorem whose conditions can be easily checked in most models of the optimal growth theory, including those with increasing returns and multi-sector economies.

We present an extension of Borsuk's antipodal theorem (existence of a zero) for antipodally approachable correspondences without convex values. This result is a generalization of Borsuk-Ulam Theorem and has a fixed-point equivalent formulation., Nous présentons dans cet article une extension du théorème de Borsuk antipodal (existence de zéro) aux correspondances antipodalement approchables à valeurs non convexes. Ce résultat est aussi une généralisation du théorème de Borsuk-Ulam et admet une formulation en termes de point fixe.

We present some application of the generalized Ky Fan's Matching Theorem stated by Chebbi, Gourdel and Hammami in minimax and variational inequalities using a generalized coercivity type condition for correspondences defined in L-space., On s'intéresse dans cet article à l'application de théorème de FKKM généralisé de Chebbi, Gourdel et Hammami aux inégalités minimax et variationnelle. Ces résultats présentent un affaiblissement des hypothèses de compacité et de quasi-concavité.

We present a theorem on the existence of a maximal element for a correspondence which is upper hemi-continuous in some variables and which satisfies with respect to the other ones one the following conditions : (i) lower semi-continuous if the space has a finite dimension, (ii) lower semi-continuous if the space is complete, (iii) open fibers. This theorem generalizes the result of Gale and Mas-Colell (1975-1979) and the one of Bergstrom (1975) and extend to the infinite dimensional setting the result of Gourdel (1995)., Nous présentons ici un théorème d'existence d'éléments maximaux pour une correspondance dont les composantes sont hémi-continues supérieurement par rapport à une partie des variables et qui vérifie par rapport aux autres l'une des conditions suivantes : (i) semi-continues inférieurement si l'espace est de dimension finie, (ii) semi-continues inférieurement et à valeurs fermées si l'espace est complet. (iii) à images inverses ouvertes. Ce théorème généralise les résultats de Gale and Mas-Colell (1975-1979), celui de Bergstrom (1975) et étend à un cadre de dimension infinie celui de Gourdel (1995).

In this paper, we present a more simple and independent proof of Reny's theorem (1998), on the existence of a Nash equilibrium in discontinue game, with a better-reply secure game in a Hausdorff topological vector space stronger than Reny's one. We will get the equivalence if the payoff function is upper semi-contineous like in the second Reny's exemple. Our proof is based on a new version of the existence of maximal element of Fan-Browder given by Deguire and Lassonde (1995). Reny's proof used a lemma of approximation of payoff function by a continuous sequence and show the existence of Nash equilibrium by the existence of equilibrium in mixed strategy proved in continuous game by the classical result., Nous présentons une preuve plus simple et indépendante du théorème 3.1 de Reny [1] qui montre l'existence d'un équilibre de Nash dans un jeu discontinu et dans un espace vectoriel topologique séparé. On utilise une hypothèse de meilleure réponse sécurisée plus forte que celle de Reny, mais qui coincide si la fonction de paiement est semi-continue supérieurement comme dans le deuxième exemple de Reny [1]. Notre preuve est basée sur une version du théorème de Fan-Browder d'existence d'élément maximal du à Deguire et Lassonde [2], tandis que Reny a utilisé une approximation de la fonction de paiement discontinue par une suite de fonctions continues (voir lemme 3.5[1]). Reny montre l'existence d'un équilibre de Nash à l'aide d'une suite d'équilibres en stratégie mixte obtenus dans le cas continu par les résultats classiques.

We consider an extension of a general equilibrium model with incomplete markets that considers cash-in-advance constraints. The total amount of money is supplied by an authority, which produces at no cost and lends money to agents at short term nominal rates of interest, meeting the demand. Agents have initial nominal claims, which in the aggregate, are the counterpart of an initial public debt. The authority covers its expenditures, including initial debt, through public revenues which consists of taxes and seignorage, and distributes its eventual budget surpluses through transfers to individuals, while no further instruments are available to correct eventual budget deficits. We define a concept of equilibrium in this extended model, and prove that there exists a monetary equilibrium with no transfers. Moreover, we show that if the price level is high enough, a monetary equilibrium with transfers exists., Nous considérons une extension d'équilibre général avec marchés incomplets incluant de la monnaie. Une autorité fait face à une dette publique qu'elle rembourse en produisant de la monnaie sans frais, et en la prêtant aux agents à un taux d'intérêt à court terme. Elle perçoit également des taxes que les agents doivent payer. Les agents reçoivent un transfert de l'autorité si cette dernière accumule un excédent (ou déficit) budgétaire. Nous introduisons la notion d'équilibre monétaire dont nous démontrons l'existence dans ce modèle.

International audience; In this paper we adapt a discrete time version of the Lucas model with social protection where part of the total production is devoted to the health expenditures. The output is produced by labor and the technomogy exhibits externalities. The rate of growth of human capital depends on the ratio of health expenditures over GDP. We give conditions for which the optimal human capital sequences are increasing. When the instantaneous utility function is isoelastic and the production function is COBB-DOUGLAS, we prove that the optimal human capital sequences grow at constant rate. Moreover, we prove there exists a unuique equilibrium in the sense of LUCAS [1988] or ROMER [1986]

This paper addresses existence of equilibria for a classical overlapping generations exchange economy with incomplete markets, one-period real assets and bounded short sales, defined on an infinite event-tree where there is a continuum of branches at each node of the tree.

A financial equilibrium existence theorem is proved under very mild assumptions on a two-period exchange model with bounded portfolios. We also characterize the asset trading that supports allocations, prices and financial transfers at the limit, when bounds are relaxed to infinity.