Patrick Taillandier
email: patrick.taillandier@gmail.com

Mathieu Bourgais
email: mathieu.bourgais@insa-rouen.fr

Philippe Caillou
email: caillou@lri.fr

Carole Adam
email: carole.adam@imag.fr

Benoit Gaudou
email: benoit.gaudou@ut-capitole.fr

A BDI agent architecture for the GAMA modeling and simulation platform

Keywords: agent-based simulation, BDI architecture, GAMA platform

With the increase of computing power and the development of user-friendly multi-agent simulation frameworks, social simulations have become increasingly realistic. However, most agent architectures in these simulations use simple reactive models. Indeed, cognitive agent architectures face two main obstacles: their complexity for the field-expert modeler, and their computational cost. In this paper, we propose a new cognitive agent architecture based on the BDI (Belief-Desire-Intention) paradigm integrated into the GAMA modeling platform and its GAML modeling language. This architecture was designed to be simple-to-use for modelers, flexible enough to manage complex behaviors, and with low computational cost. An experiment carried out with different profiles of end-users shows that the architecture is actually usable even by modelers who have little knowledge in programming and in Artificial Intelligence.

Introduction

Agent-based simulations are widely used to study complex systems. When the goal is to understand or even predict the behaviour of such complex systems, the simulation requires an accurate agent architecture to lay valid results. However, designing the agents is still an open issue, especially for models tackling social issues, where some of the agents represent human beings. In fact, designing complex agents able to act in a believable way is a difficult task, in particular when their behaviour is led by many conflicting needs and desires.

A classic paradigm to formalize the internal architecture of such complex agents in Agent-Oriented Software Engineering is the BDI (Belief-Desire-Intention) paradigm. This paradigm, based on the philosophy of action [START_REF] Bratman | Intentions, plans, and practical reason[END_REF], allows to design expressive and realistic agents, yet it is still rarely used in social simulations, for two main reasons. First, most agent architectures based on the BDI paradigm are too complex to be understood and used by non-computer-scientists. Second, they are often very time-consuming in terms of computation and therefore not adapted to simulations with thousands of agents.

In a previous paper [START_REF] Caillou | A Simple-to-use BDI architecture for Agent-based Modeling and Simulation[END_REF], we proposed a first architecture to overcome these obstacles. This architecture is now fully integrated into the GAMA platform, an open-source modeling and simulation platform for building spatially explicit agent-based simulations [START_REF] Grignard | GAMA 1.6: Advancing the art of complex agent-based modeling and simulation[END_REF][START_REF] Drogoul | Gama: multi-level and complex environment for agent-based models and simulations[END_REF]2]. GAMA's Modeling Language (GAML) and the integrated development environment support the definition of large scale models (up to millions of agents) and are usable even with low-level programming skills. Our BDI architecture was implemented as a new GAMA plug-in, and enables the straightforward definition of BDI agents through the GAML language. Thus, modelers have a wide range of ways to define their agents behaviors, by mixing classic GAMA primitives with BDI concepts. Indeed, we state that there is no unique way of defining the behavior of agents that can fit all the possible application contexts, types of agents to define (level of cognition), and modelers' backgrounds (programming skills, modeling habits...). This paper describes some major improvements for the architecture, as well as a usability study carried out with modelers. The paper is structured as follows: Section 2 proposes a state of the art of BDI architectures and their use in simulations. Section 3 is dedicated to the presentation of our architecture, and more particularly the novelties from the previous work. In Section 4, we present an experiment in which we asked modelers with different profiles to test the architecture. Finally, Section 5 provides a conclusion and some perspectives.

State of the art

Agent architectures for social simulations

Balke and Gilbert [START_REF] Balke | How do agents make decisions? a survey[END_REF] cite Sun as remarking that "social simulation researchers frequently only focus on agent models custom-tailored to the task at hand. He calls this situation unsatisfying and emphasises that it limits realism and the applicability of social simulation. He argues that to overcome these short-comings, it is necessary to include cognition as an integral part of an agent architecture." There are several options to endow agents with complex cognitive capabilities: cognitive architectures based on cognitive sciences, such as SOAR or ACT-R, or BDI architectures based on the philosophy of action.

BDI agents in AOSE frameworks

The BDI approach has been proposed in Artificial Intelligence [START_REF] Bratman | Intentions, plans, and practical reason[END_REF] to represent the way agents can do complex reasoning. It has first been formalized using Modal Logic [START_REF] Cohen | Intention is choice with commitment[END_REF] in order to disambiguate the BDI concepts (Belief, Desire and Intention; detailed in Section 3.1) and the logical relationships between them.

In parallel, the Agent-Oriented Software Engineering (AOSE) field designed BDI operational architectures to help the development of Multi-Agent Systems embedding BDI agents. Some of these BDI architectures are included in frameworks allowing to directly use them in different applications. A classic framework is the Procedural Reasoning System (PRS) [START_REF] Myers | User guide for the procedural reasoning system[END_REF], that has been used as a base for many other frameworks (commercial, e.g. JACK [START_REF] Howden | JACK intelligent agentssummary of an agent infrastructure[END_REF], or open-source, e.g. JADE [START_REF] Bellifemine | JADE-A FIPA-compliant agent framework[END_REF] with its add-on Jadex [START_REF] Pokahr | Jadex: A BDI reasoning engine[END_REF] offering an explicit representation of goals).

BDI agents in ABMS platforms

BDI architectures have been introduced in several agent-based modelling and simulation (ABMS) platforms. For example, Sakellariou et al. [START_REF] Sakellariou | Enhancing NetLogo to simulate BDI communicating agents[END_REF] have proposed an education-oriented extension to Netlogo [START_REF] Wilensky | Netlogo. center for connected learning and computer based modeling[END_REF] to allow modellers to manipulate BDI concepts in a simple language. This very simple architecture is inspired by PRS: agents have a set of beliefs (information obtained by perception or communication), a set of intentions (what they want to execute), and ways to manage these two sets.

Singh and Padgham [START_REF] Singh | OpenSim: A framework for integrating agent-based models and simulation components[END_REF] went one step further by proposing a framework acting like a middleware to connect components such as an ABMS platform and a BDI framework (e.g. JACK [START_REF] Howden | JACK intelligent agentssummary of an agent infrastructure[END_REF] or Jadex [START_REF] Pokahr | Jadex: A BDI reasoning engine[END_REF]). They demonstrated their framework on an application coupling the Matsim platform [START_REF] Balmer | Matsim-t: Architecture and simulation times[END_REF] and the GORITE BDI framework [START_REF] Rönnquist | The goal oriented teams (gorite) framework[END_REF] for a bushfire simulation; but it aims at being generic and can be extended to couple any kind of ABMS platforms and BDI frameworks, only by implementing interfaces to plug each new component to the middleware. This approach is very powerful but remains computer-scientist-oriented, as it requires high programming skills to develop bridges for each component and to implement agents behaviours without a dedicated modelling language.

Previous work: BDI agents in GAMA

First attempts already exist to integrate BDI agents into the GAMA platform [START_REF] Grignard | GAMA 1.6: Advancing the art of complex agent-based modeling and simulation[END_REF]. Taillandier et al. [START_REF] Taillandier | A new BDI agent architecture based on the belief theory[END_REF] proposed a BDI architecture where the choice of plans is formalized as a multi-criteria decision-making process: desires are represented as criteria, and decisions are made by evaluating each plan w.r.t. each criterion according to the agent's beliefs. However, this architecture was tightly linked to its application context (farmer's decision making), did not offer any formalism to model the agent's beliefs, and was rather limited w.r.t. how plans were carried out (e.g. no possibility to have complex plans with sub-objectives).

Le et al. [START_REF] Le | A new BDI architecture to formalize cognitive agent behaviors into simulations[END_REF] proposed another architecture dedicated to simulation with a formalized description of beliefs and plans and their execution. However, the desires and plans have to be written in a very specific and complex way that can be difficult to achieve for some application contexts, in particular for noncomputer scientists. In addition, this architecture has a scalability problem: it does not allow to simulate thousands of agents.

Finally, we proposed in [START_REF] Caillou | A Simple-to-use BDI architecture for Agent-based Modeling and Simulation[END_REF] a first simple architecture. This generic architecture proved its interest through an application to simulate agricultural parcels dynamics in Vietnam. However, the use of this architecture required to write many lines of code, in particular to manage the perceptions of agents and the inference of desires. Moreover, it did not allow to manage plans with different temporal scales. All these limitations made this architecture not usable for some applications and more complex to use by non-computer scientists. The next section therefore presents the improvements that we propose for this architecture.

Presentation of the Architecture

Overview

Our architecture is defined as a GAMA species architecture. The modeler is only required to define simpleBDI as the agent's architecture, and to define at least one plan and one initial desire to make it operational. After that the modeler mostly defines plans (what the agent can do), desires (what the agent wants) and (usually one) perception(s) (how the agent perceives its environment). Many keywords and functions are defined to help the user updating and managing Beliefs, Desires and Intentions bases, and creating and managing predicates.

The architecture and the vocabulary can be summarized with this simple Gold Miner example: the Miner agent has a general desire to find gold. As it is the only thing it wants at the beginning, it is its initial intention (what it is currently doing). To find gold, it wanders around (its plan is to wander). When it perceives some gold nuggets, it stores this information (it has a new belief about the existence and location of this gold nugget), and it adopts a new desire (it wants to extract the gold). When it perceives a gold nugget, the intention to find gold is put on hold and a new intention is selected (to extract gold). To achieve this intention, the plan has two steps, i.e. two new (sub)intentions: to choose a gold nugget to extract (among its known gold nuggets) and to go and take it. And so on.

This vocabulary is explained in more details in the next section.

Vocabulary

Knowledge. Beliefs, Desires and Intentions are described using predicates. A predicate has a name, may also have values (a set of name-value pairs), and can be true (by default) or false. Predicates can be easily defined through the GAML language. For example, in Figure 1, we define a new predicate location_gold, that represents the fact that a gold nugget is present at location (20,50). Note that when comparing two predicates, if one of them has no value for a specific key, the values corresponding to this key are not compared. For instance, to test if the predicate ("location_gold") is present in the belief base will return true if the predicate ("location_gold", ["location_value"::20,50]) is in the base. BDI agents have 3 databases: Intentions can be put on hold (for example when they require a sub-intention to be achieved). For this reason, there is a stack of intentions: the top one is the current intention, all others are on hold.

Behavior In addition to the basic behavior structures available for all GAMA agents, our simpleBDI architecture has three available types of behavior structures (which are new compared to [START_REF] Caillou | A Simple-to-use BDI architecture for Agent-based Modeling and Simulation[END_REF]):

-Perception: A perception is a function executed at each iteration to update the agent's Belief base, to know the changes in its environment (the world, the other agents and itself). The agent can perceive other agents up to a fixed distance or inside a specific geometry. For example, a miner can perceive all the gold nuggets at a distance of 10 meters and for each of them add a new belief concerning the gold's location (see Figure 2). Like for many other GAMA statements, several facets (most of them optional) can be used to tune the perceptions:

• target (mandatory): the list of agents (or species of agents) to perceive.

• when (optional): perception condition -can be a dynamic expression.

• in (optional): can be either a radius (float) or a geometry (for instance, definition of a cone of perception or something more complex). -Rule: A rule is a function executed at each iteration to infer new desires or beliefs from the agent's current beliefs and desires, i.e. a new desire or belief can emerge from the existing ones. For example, when the miner has a belief about some gold nuggets at a specific location, the rule creates the desire to extract gold (see Figure 3). Of course, if the new desire (or belief) is already in the base, nothing is added. Rules facets are the following:

• belief (optional): required belief to activate the rule.

• desire (optional): required desire to activate the rule.

• when (optional): required condition to activate the rule.

• new_belief (optional): the new belief to be added to the agent's base.

• new_desire (optional): the new desire to be added to the agent's base.

-Plan: The agent has a set of plans, which are behaviors defined to accomplish specific intentions. Plans can be instantaneous and/or persistent, and may have a priority value (that can be dynamic), used to select a plan when several possible plans are available to accomplish the same intention. For example, when the miner has the intention to f ind_gold, it can activate its letsW ander plan (see Figure 4) that makes it randomly move. Concerning the facets of plans:

• intention (optional): intention the plan try to fulfill.

• when (optional): activation condition.

• finished_when (optional): termination condition.

• priority (optional): priority of the plan, 1 by default.

• instantaneous (optional): if f alse, no other plan can be executed afterwards during the current simulation step; otherwise, (at least) one other plan will be executed during the same step. By default f alse.

Thinking process

At each step, the agent applies the process described in Figure 5. Roughly, the agent first perceives the environment, then i) continues its current plan if it is not finished, or ii) if the plan is finished and its current intention is not fulfilled, it selects a plan, or iii) if its current intention is fulfilled, it selects a new desire to add to its intention stack. This process is similar to the one proposed in [START_REF] Caillou | A Simple-to-use BDI architecture for Agent-based Modeling and Simulation[END_REF], however it introduces the perception and rule steps. In addition, it allows to apply several plans in the same time step in case of instantaneous plans. More precisely:

-Do I keep the current intention?:

To take into account the environment instability, an intention-persistence coefficient ip is applied: with the probability ip, the current intention is removed from the intention stack. More details about this coefficient are given in section 3.4.

5 -Do I have a current plan?: If I have a current plan, just execute it. As for the current intention stability, the aim is both persistence (I stick to the plan I have chosen) and efficiency (I don't choose at each step). Similarly to intentions, a plan-persistence coefficient pp is defined: with a probability pp, the current plan is just dropped.

6 -Choose a desire as new current intention: If the current intention is on hold (or the intention base is empty), choose a desire as new current intention. The new selected intention is the desire with higher priority among those not already present in the intention base.

-Choose a plan as a new current plan:

The new current plan is selected among the plans compatible with the current intention (and if their activation condition is checked) and with the highest priority.

-Execute the plan:

The current plan is executed.

-Is my plan finished?:

To allow persistent plans, a plan may have a termination condition. If it is not reached, the same plan will be kept for the next iteration.

10 -Was my plan instantaneous?: Most multi-agent based simulation frameworks (GAMA included) are synchronous frameworks using steps. One consequence is that it may be useful to apply several plans during one single step. For example, if a step represents a day or a year, it would be unrealistic for an agent to spend one step to apply a plan like "choose a destination". This kind of plans (mostly reasoning plans) can be defined as instantaneous: in this case a new thinking loop is applied during the same agent step.

Properties

Persistence and priority In our architecture, the persistence of plans and desires, as well as their priority can be dynamically defined (i.e. modified by the agents during the simulation and/or computed from a function).

Flexibility

The architecture aims at being simple-to-use and as flexible as possible for the modeler. More advanced modelers can use its full potential (for example dynamic coefficients as presented before), while others can rely on the default values of most parameters to easily specify agents using only some of the available features. In addition, in order to manipulate the different elements of the architecture, we provide modelers with built-in actions directly usable in their models, among which (desire related actions are not listed here):

-add_belief: add a new belief to the belief base.

-remove_belief: remove a belief from the belief base.

-has_belief: test if a belief is in the belief base.

-replace_belief: replace a belief by a new one in the belief base.

-get_belief: return the first belief corresponding to a given predicate.

-get_beliefs: return all beliefs corresponding to a given predicate.

-remove_all_beliefs: remove all beliefs corresponding to a given predicate.

-clear_beliefs: remove all beliefs.

-get_current_intention: return the current intention.

-clear_intentions: remove all intentions.

-add_subintentions: add a sub-intentions to a predicate -current_intention_on_hold: put the current intention on hold and add as its activation condition all its sub-intentions.

Experiments

Modelers feedback analysis

Context of the experiment In order to validate our claim that our architecture is simple to use and is adapted to several types of users, we carried out an experiment with six modelers with different backgrounds: 3 computer scientists, and 3 geographers, all of whom knew GAMA (at least the basic concepts). Two of them have a low level in programming (2 geographers) and four a higher level (1 geographer and 3 computer scientists). All of them have at least a low (2) or medium (3) level in GAMA (they know at least the basic concepts), the last one being an expert. Only the 3 computer scientists know the BDI paradigm. This experiment consisted in a short lesson (45 minutes) about the simpleBDI architecture, followed by an exercise which required to use this architecture (2 hours). At the end of the exercise, each participant answered a short survey to assess the architecture (with both open questions and closed scaled assessments). All these documents (introduction course, exercise, models developed by the different modelers, and their answers to the questionnaire) are available on the project website [1].

The presentation of the simpleBDI architecture used a simple gold miner model to present the underlying concepts. Then the exercise theme was the evacuation of the city of Rouen (France) (see Figure 6). A technological hazard is simulated in one of the buildings of the city center. Drivers can perceive the hazard at a distance of 400 meters. Those who know that there is a hazard try to reach one of the evacuation sites (shelters). A driver who sees (in a radius of 10 meters) another driver trying to evacuate has a small chance to understand that a hazard is happening.

The participants were given a first basic model of that situation, containing four species of agents: road, hazard, evacuation site and driver. The behavior of the driver agents is defined by a single reflex executed at each simulation step: the agent moves towards a random target (any point on the road network), and if it reaches its destination, it chooses a new random target. A weighted graph is used for the movement of drivers: they first compute the shortest path between their location and their target, then use this path to move. The weights of the edges of the graph (roads) are updated every 10 simulation steps to take into account the number of drivers on each road. The agent speed on each road is a function of the number of drivers on this road and its maximum capacity. If a driver already has a computed path, it will not recompute it even if the weights of that path change. At the initialization of the model, the roads (154), evacuation sites [START_REF] Caillou | A Simple-to-use BDI architecture for Agent-based Modeling and Simulation[END_REF] and the hazard (1) agents are created and initialized using shapefiles; then 500 driver agents are created and randomly placed on the roads. The GAML code for the driver species is given Figure 7. In this first basic model, drivers do not perceive the hazard and do not try to reach evacuation sites; they just keep moving randomly.

The exercise was composed of two steps of increasing difficulty:

-Step 1: modification of the behaviors of the driver agents, in order to:

• make them aware of the hazard: via its direct perception (with a probability of 0.2 if they are in a radius of 400m) or indirectly when they see other driver agents trying to reach an evacuation site (with a probability of 0.01 in a radius of 10m); • make them try to reach the closest shelter (euclidean distance) when they know that there is a hazard.

-Step 2: modification of the behaviors of the driver agents and of the evacuation site, so that:

• if a driver agent trying to reach a shelter thinks that its road is blocked (speed coefficient lower than 0.5), then it should test a new path (recomputation of the path according to the current weights of the graph); • drivers take into account the maximum capacity of the evacuation sites (50 drivers each). To know that an evacuation site is full, drivers have to be less than 20 meters away from it.

Possible solution of the exercise

We present here a possible solution. We first add the simpleBDI architecture to the driver species and 3 new variables (Figure 8):

-escape_mode: boolean to indicate if the agent is evacuating or not -recompute_path: boolean to indicate if the agent has to recompute its path or not. -current_shelter: the current evacuation site to reach. as well as 4 predicates that will be used as the content of mental attitudes:

-at_target: the agent is at its target used as the desire to go to a specific target. -in_shelter: the agent is in an evacuation site used as the desire to go to an evacuation site. -has_target: the agent has a defined target used as the desire to define a target. -has_shelter: the agent has selected an evacuation site used as the desire to choose an evacuation site.

We define 4 perceptions (Figure 9):

-hazard: perceives the hazard in a radius of 400m with a probability of 0.2.

-driver: perceives the drivers in escape mode in a radius of 10m with a probability of 0.01. -road: perceives the blocked roads in a radius of 1.

-shelter: perceives the full evacuation sites in a radius of 20m.

We define two rules to infer the desire to go toward an evacuation site (if the agent perceived the hazard directly or through the observation of escaping drivers) and an action to switch to escape mode (Figure 10):

Finally, we define four plans for the agents (Figures 11 and12): Results of the experiment After the exercise, each participant answered a short survey about the BDI architecture. The survey was composed of 7 closed questions using a 1-5 scale, 2 yes/no questions and 9 open questions/commentary sections. The first three questions were used to assess the participant background and the others to assess the simpleBDI architecture and the exercise performance. Due to the low number of participants, the survey results are used as a qualitative evaluation, and not as a statistically significant quantitative assessment.

A first analysis of the results shows that all the participants find the proposed architecture clear (answer values of 4 or more for that question). Furthermore, the three participants that have a background in BDI architectures find that our architecture translates the BDI paradigm well (3+ answers). Concerning the simplicity of use of the architecture inside GAMA, three of the participants find it good (4) or very good [START_REF] Bellifemine | JADE-A FIPA-compliant agent framework[END_REF], and two pretty good [START_REF] Balke | How do agents make decisions? a survey[END_REF].

Five out of the six participants succeeded in implementing Step 1, but none of them achieved Step 2. A tentative explanation is that two hours was a too short time to understand what was asked in the exercise, formalize the behavior of the agents using the simpleBDI architecture, write the GAML code, and test it. In addition, only one of the participants knew GAMA very well, so as mentioned in the survey, most of the others lost a lot of time searching for specific GAMA operators (not linked with the simpleBDI architecture). Nevertheless, four of the participants were very close to succeed in the second step of the exercise (the three computer scientists and one of the geographers).

Concerning the comparison to other architectures, one of the modelers (the BDI expert) preferred simpleBDI to the others, while two of them found sim-pleBDI to be complementary to the existing ones, and mentioned that it allows to define the behavior in a simpler way, avoiding to write many complex reflexes. Only one modeler mentioned that they preferred the reflex architecture as they were more used to it. The last two participants did not answer this question.

An interesting remark is that some of the participants mixed the simpleBDI and reflex architectures, using the BDI architecture to define perceptions and objectives (especially the agents target), and reflexes for the repetitive operational behaviors (moving).

To conclude, this first experiment showed very promising results: all the participants found the simpleBDI architecture clear and easy to use. After a short 45 minutes lesson, they were able to apply it to a real model previously unknown to them. To achieve a more complete and efficient use of the simpleBDI architecture, they would however have needed more time, better programming skills in GAMA, and/or a better knowledge of the BDI paradigm.

Architecture scalability

In order to test the architecture scalability, we used the two previous models (gold miners and city evacuation) with an increasing numbers of agents. As GAMA is often used by modelers with old computers (for social simulation in developing countries), we chose to carry out the experiment with a 5-year old Macbook pro (2011) with an i7 processor and 4Go of RAM.

The gold miner model was tested with 10 000 miners, 1000 golds and a square environment of 10 x 10 kilometers. The simulation was stopped when all the gold nuggets had been returned to the base. The average duration of a simulation step (without any graphical display) was 140ms.

The evacuation model was tested with 1000 drivers (due to the road network used and how the capacity of roads was defined, it was not possible to test the model with more driver agents -all the road would have been blocked) and a capacity for each of the evacuation sites of 200 driver agents. We stopped the simulation when all the drivers reached an evacuation site. The average duration of a simulation step (with no graphical display) was 70ms. The results obtained in terms of performance show that the architecture can already be used with medium-scale real world problems. However, the architecture will still be continuously optimized and we plan to compare the results with other GAMA architectures (especially the reflex one). We also plan in the future to test the architecture with more complex agents having many possible desires, sub-desires and plans.

Conclusion

In this paper, we have presented a new BDI architecture dedicated to simulations. This architecture is integrated into the GAMA modeling and simulation platform and directly usable through the GAML language, making it easily usable even by non-computer scientists. We have presented a first experiment that was carried out with modelers with different profiles (geographers and computer scientists). This first experiment showed that our plug-in can be used even by modelers that have little knowledge in programming and Artificial Intelligence, and that it allows to simulate several thousands of agents.

If our architecture is already usable, some improvements are planned. First, we want to improve the inference capabilities of our architecture: when a new belief is added to the belief base, desire and intention bases should be updated in a efficient way as well. Second, we want to make it even more modular by adding more possibilities concerning the choice of plans and desires (beyond just choosing that with the highest priority): user-defined selection, multi-criteria decision process, etc. Finally, we want to add the possibility to use high performance computing (distribute the computation on a grid or cluster) to decrease computation time.

Figure 1 .

 1 Figure 1. predicate definition

Figure 2 .

 2 Figure 2. perception definition: when the agent perceives a gold nugget at a distance lower than 10 meters, a new belief is added concerning the location of this gold nugget.

Figure 3 .

 3 Figure 3. rule definition: when the agent has a belief about a gold nugget somewhere, a desire is added to extract gold

Figure 4 .

 4 Figure 4. plan definition: if the agent has the intention to f ind_gold, it can use the letsW ander plan, which makes the agent moves randomly (uses of the wander built-in action of GAMA)

Figure 5 .

 5 Figure 5. Activity diagram

Figure 6 .

 6 Figure 6. Snapshot of the city evacuation model: the red circle is the hazard perception area, the green circles the evacuation sites and the blue triangles the drivers.

Figure 7 .

 7 Figure 7. Initial GAML code for the driver species using reflex architecture, provided to the modelers.

Figure 8 .

 8 Figure 8. Possible solution GAML code for the driver species and variables

Figure 9 .

 9 Figure 9. Possible solution GAML code for the driver perceptions

Figure 10 .Figure 11 .

 1011 Figure 10. Possible solution GAML code for the driver rules and action

Figure 12 .

 12 Figure 12. Possible solution GAML code for the evacuation and choose_shelter driver plans

Acknowledgement

This work is part of the ACTEUR (Spatial Cognitive Agents for Urban Dynamics and Risk Studies) research project funded by the French Research Agency (ANR).


Stefano Bosi

Patrice Fontaine

Cuong Le Van
email: levan@univ-paris1.fr.

Stefano Bosi Epee

Interest rates parity and no arbitrage as equivalent equilibrium conditions in the international financial assets and goods markets

Keywords: general equilibrium, no arbitrage, returns on financial assets, exchange rates, Law of One Price, Uncovered Interest rate Parity. JEL Classification: D53, F31, G11, G15

In this paper, we consider a two-period consumption model with many financial assets. In the spirit of Hart [5], consumers purchase financial assets in period 0 and consume in period 1. We differ from Hart by considering that each agent is a country. We provide conditions for the existence of an equilibrium in both international financial assets and goods markets.

First, we introduce a weaker notion of Uncovered Interest (rate) Parity (UIP) called Weak Uncovered Interest (rate) Parity (WUIP), and we show its equivalence to the no-arbitrage condition in the international financial markets.

Second, we introduce the concept of common no arbitrage and we show its equivalence to UIP.

These results bridge concepts of no arbitrage in general equilibrium theory and financial microeconomics, and of interest parity in international financial macroeconomics. In a multi-country model with many currencies and only one good, we introduce a country-specific conversion rate which transforms the returns on assets valued in local currency into units of physical good. We the define also the exchange rates between currencies of different countries. The UIP condition is required for the existence of an equilibrium in both international financial assets and goods markets and for the existence of the Law of One Price.

Interest rates parity and no arbitrage as equivalent equilibrium conditions in the international financial assets and goods markets 1 Introduction In this paper, we revisit a two-period financial model by Hart [START_REF] Hart | On the existence of an equilibrium in a securities model[END_REF] to introduce the international trade. The first period will be called period 0, the second, period 1. In period 0, agents buy or sell financial assets to diversify their portfolios and maximize an expected utility function. They know the states of nature of period 1 and have beliefs on the occurrence of these states. In period 1, they buy or sell consumption goods with their endowments and the gains from financial investments. These gains are the returns on the financial assets the agents receive in each state of the nature.

In our paper, we have many assets but only one good for consumption purposes. There are many countries which have access to the same financial market. In one-country models, it is equivalent to express the returns in terms of consumption good or currency (money), while in multi-country models as ours, the prices may differ from one country to another because of the exchange rates. Therefore, we require an independent numeraire. The returns on financial assets are initially valued in local currency. However, each agent forms expectations on the conversation rates between the currency of her country and the consumption good. We introduce also the exchange rates between the currency of her country and a reference country, say country 1. We assume that agents use their expectations on the gains of period 1 valued in terms of consumption good to purchase financial assets in period 0. Each agent maximizes a utility function in period 0 depending on her consumption in any state nature. We define the conditions for the existence of a general equilibrium of both the trade balance and the international financial market. First, we address the issue of equilibrium existence in the international financial market. We apply the usual no-arbitrage condition of general equilibrium financial literature, which refers to the existence of appropriate financial assets prices that prevent agents from making gains without cost. In the spirit of Allouch, Le Van and Page [START_REF] Allouch | The geometry of arbitrage and the existence of competitive equilibrium[END_REF], Dana, Le Van and Magnien [START_REF] Van | On the different notions of arbitrage and existence of equilibrium[END_REF], Page and Wooders [START_REF] Page | A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium[END_REF], Werner [START_REF] Werner | Arbitrage and the existence of competitive equilibrium[END_REF] among others, we show that, under the strict concavity of utility functions, no-arbitrage conditions are necessary and sufficient to ensure the existence of an equilibrium in the international financial market.

In a second stage, we introduce two conditions. The first one is the Uncovered Interest rate Parity (UIP) which postulates the existence of exchange rates under which the returns on the financial assets of all the countries become equal in each state of nature if they are expressed in terms of consumption good. A weaker notion of UIP, called Weak Uncovered Interest rate Parity (WUIP), postulates that (i) there exists a system of exchange rates and (ii) all the countries share the same probability distribution over the states of nature such that the expected values of the returns on financial assets in terms of consumption good are equal. Our first main result is the equivalence of WUIP and the noarbitrage condition in the international financial market and the existence of an equilibrium in this market.

We introduce the notion of common no-arbitrage financial prices which postulates the existence of S financial price vectors q s such that, if the gains generated by the purchases of financial assets are positive in state s for any country, then the values of these assets calculated with the common no-arbitrage prices q s will be positive. Our second main result is to prove that UIP is equivalent to the existence of a common no-arbitrage condition.

These results bridge concepts of no arbitrage in general equilibrium theory and financial microeconomics, and of interest parity in international financial macroeconomics.

Our third result is the proof that if UIP holds then there exists an equilibrium in both financial and goods markets where the prices of financial assets are no-arbitrage prices and the prices of consumption goods in terms of numeraire satisfy the Law of One Price (LOP). Moreover, the trade balance holds in terms of nominal value and in terms of physical goods as well.

UIP implies WUIP while the converse is not true. 1 Then, one may wonder what aspects of equilibrium are preserved under WUIP. While UIP entails the equilibrium in both markets, under WUIP, the equilibrium of trade balance fails. However, under the probability distribution of WUIP, the expected value of the trade balance turns out to be zero. This result is the last of the paper and sheds light on the equilibrium implications of UIP and WUIP, and their interplay.

In order to understand better the role of UIP, we provide three examples where this parity fails. In the first one, the no-arbitrage condition holds in the international financial market and an equilibrium exists but it is no-trade. In particular, the LOP is not verified. In a second example, the no-arbitrage condition holds in the international financial market and an equilibrium exists as well, but the goods market is imbalanced. In the last example, the sets of noarbitrage prices for the international financial market differ across the countries and do not intersect: thus, the no-arbitrage condition fails and there exists no equilibrium.

The paper is organized as follows. In Section 2, we introduce the main assumptions of the model, while, in Section 3 we define the equilibrium. The noarbitrage conditions in the international financial market are defined in Section 4, while some notions of parity are introduced in Section 5. In Section 6, we link the notions of no-arbitrage and UIP, and we prove two other main results of the paper, that is the equivalences (1) between the existence of a no-arbitrage financial assets price and WUIP, and (2) between the existence of common no-arbitrage condition and UIP. The existence of equilibrium is eventually proved in Section 7 through two equivalent theorems. In Theorem 1, UIP is a sufficient condition to the existence of an equilibrium. In Theorem 2, UIP is replaced by the existence of a common no-arbitrage financial assets price. In Section 8, we compare the equilibrium implications of UIP and WUIP. The implications of UIP failure are reconsidered in Section 9 through three examples.

In Example 1, the only equilibrium we obtain is a no-trade equilibrium for both markets. In Example 2, UIP also fails but the no-arbitrage condition holds for the international assets market. There exists an equilibrium in the financial market but no equilibrium in the goods market. In Example 3, the no-arbitrage condition fails and no equilibrium exists in both markets. Section 10 concludes.

Model

We consider a pure exchange economy where financial assets and goods are traded in international markets. Before addressing the equilibrium issue, we introduce notations and assumptions, and we describe saving diversification and consumption.

Focus on a two-period exchange economy with many countries. Financial assets are traded in period 0 and goods are consumed in period 1. The consumption good is supposed to be perfectly tradable. The representative agent of country i ∈ {1, . . . , I} purchases K financial assets in period 0 to smooth consumption in period 1 across S states of nature. At the beginning of period 1, this agent is endowed with an amount of consumption good. This amount depends on the state of nature. Since we are in presence of many currencies, we assume the existence of a numeraire such as gold or one of the currencies. The role of this numeraire is to value the assets purchases and the consumption goods in every country.

Notations

For the sake of simplicity and readability, we present all the variables of the model at the beginning and currency i should be understood as currency of country i.

We introduce the financial side of the economy through a compact notation for asset prices and quantities.

q ≡ (q 1 , . . . , q K ) is a row of financial assets prices where q k denotes the price of asset k in terms of numeraire in period 0.

x ≡ x i k is the K × I matrix of portfolios where x i k denotes the amount of financial asset k in the portfolio of agent i. Column x i ≡ x i 1 , . . . , x i K T is the portfolio of the representative agent i of country i. R i ≡ R i sk is the S × K matrix of returns2 where R i sk ≥ 0 denotes the return on financial asset k in the state of nature s. R i s is the sth row of the matrix. Returns R i sk correspond to returns on asset k in country i and state s and are valued in currency i. They are expected in period 1 with a probability π i s .

≡ i s is the I × S matrix of conversion rates where i s denotes the conversion rate between currency i and the physical good in the state of nature s. Actually, one unit of physical good, in state s, equals i s units of currency i. Therefore, i s > 0 for any i and any s. We denote i ≡ i 1 , . . . , i S the ith row of the matrix. These conversion rates are actually the expected prices of commodity in terms of currency i. The matrix i s is exogenously given. We obtain the expected exchange rates between currency 1 and currency i by the expression τ i s = 1 s / i s . τ ≡ τ i s is the I × S matrix of exchange rates where τ i s denotes the exchange rate between currencies 1 and currency i in the state of nature s. Actually, one unit of currency i, in state s, equals τ i s units of currency 1. Obviously, τ i s > 0 for any i and any s. We denote τ i ≡ τ i 1 , . . . , τ i S the ith row of the matrix. Of course, the first row is a vector of units: τ 1 s = 1 for any s. 3Given the system of conversion rates , let R i s denote the returns on financial assets from the country i valued in physical good. More explicitly, R i s ≡ R i s / i s . These returns are also expected returns in period 1.

We introduce now a compact notation for beliefs, prices and quantities on the real side of the economy.

c ≡ c i s is the S ×I matrix of consumptions where c i s denotes the amount of good consumed by agent i in the state of nature s in period 1.

c i ≡ c i 1 , . . . , c i S T
is the consumption column of agent i. e ≡ e i s is the S ×I matrix of endowments where e i s denotes the endowment nature provides to agent i in the state s in period 1. e i ≡ e i 1 , . . . , e i S T is the endowment column of agent i. π ≡ π i s is the I × S matrix of beliefs where π i s denotes the belief of agent i about the occurrence of state s. The individual row of beliefs π i ≡ π i 1 , . . . , π i S lies in the S-unit simplex. p ≡ p i s is the I × S matrix of good prices, in period 1. The price p i s is the required quantity of numeraire for the purchase of one unit of good, in country i and at state s.

Notice that vectors q, R i s , τ i , p i , π i are rows, while vectors x i , c i , e i are columns.

In the article, i , s , k will denote unambiguously the explicit sums I i=1 , S s=1 , K k=1 .

Portfolios

The agents' behavior comes down to a saving diversification to finance future consumption. In period 1 and state s, agents exchange their endowments according to their portfolio:

c i s = e i s + R i s x i (1)
where R i s ≡ R i s / i s . We recall that the returns R i s on portfolio are valued in terms of physical good.

Preferences of agent i are rationalized by a Von Neumann-Morgenstern utility function weighted by subjective probabilities: s π i s u i c i s , where u i is the utility function and c i s is the quantity of her consumption good. As a consumption amount, c i s is required to be nonnegative and the utility function is defined on the nonnegative orthant.

Therefore, the set of portfolios is

X i ≡ x i ∈ R K : for any s, e i s + R i s x i ≥ 0 (2)
We introduce the set of consumption allocations Y i + generated by the purchase of a portfolio x i of financial assets:

Y i + ≡ c i ∈ R S : there is x i ∈ R K such that c i s = e i s + R i s x i ≥ 0 for any s
In period 0, given the prices of financial assets (q), any agent i chooses a portfolio x i * which maximizes her preferences taking into account the financial budget constraint: max

x i ∈X i s π i s u i e i s + R i s x i (3)
qx i ≤ 0
The RHS of the budget constraint is zero since we consider agents' net purchases.

Assumptions

In order to prove and characterize the existence of a general equilibrium with financial assets, we introduce some mild assumptions. The first triplet of hypotheses specifies the financial fundamentals (returns); the second triplet specifies the real fundamentals (endowments and preferences).

Assumption 1 For any country i and any state s, k R i sk > 0. Assumption 2 For any country i and any financial asset k, s R i sk > 0. When Assumption 1 fails, there is a country i and a state s where any financial asset k yields R i sk = 0. In this case, the representative agent of country i will consume her endowment in the state s.

When Assumption 2 fails, there is an asset k yielding R i sk = 0 in any state of nature s in country i: the representative agent i will refuse to buy this financial asset. The following assumption is stronger and implies Assumption 2.

Assumption 3 For any country i and any portfolio x i = 0, the portfolio return is nonzero: R i x i = 0.

Assumption 3 means that there are no nonzero portfolios with a null return in any state of nature. In other terms, whatever the country i we consider, the matrixes R i and R i are full rank with rankR i = rankR i = K. 4Assumption 4 Endowments are positive: e i s > 0 for any agent i and any state s.

Assumption 5 Beliefs are positive: π i s > 0 for any agent i and any state s.

This assumption means that any representative agent considers each state possible.

Eventually, preferences are required to satisfy regular assumptions. Assumption 6 For any agent i, the utility function u i is concave, continuous, strictly increasing from R + to R.

Definition of equilibrium

Let us provide a general definition of equilibrium, then to distinguish the equilibrium in the assets markets and in the goods markets. Finally, we introduce an important and new notion of quasi-equilibrium in the sense that trade is balanced in expectations but not state of nature by state.

Definition 1 (equilibrium) Given exchange rates, beliefs and endowments (τ, π, e), returns and preferences R i , u i for any country i, an equilibrium is a list of financial assets prices and allocations (p, q, c, x) * , with p * >> 0, q * >> 0, and, for any i, s p i * s = 1 and j q * j = 1 such that: individual plans are optimal (points (1) and (2) below) and markets clear (points (3) and ( 4)).

(1) Portfolios are optimized given the financial assets prices q * : in period 0, any agent i chooses a portfolio x i * which maximizes her utility function under the financial budget constraint:

max x i ∈X i s π i s u i e i s + R i s x i (4)
q * x i ≤ 0

(2) In period 1, for any agent i, c i * = e i + R i x i * solves the program

max c i ∈Y i + s π i s u i c i s ( 5
)
p i * c i ≤ p i * e i
(3) The international financial assets markets clear:

i x i * = 0. (4)
The trade balance is satisfied in any state of nature:

i c i * s = i e i s for any s.

We recall that the price p i s is the quantity of numeraire necessary for the purchase of one unit of good, in country i and at state s. We will see later that, in equilibrium, p i * R i x i * = q * x i * = 0. Therefore, in period 1, we can actually replace the Arrow-Debreu contingent constraint p i * c i ≤ p i * e i by the constraint involving the income obtained with the purchase of assets in period 0:

p i * c i ≤ p i * e i + p i * R i x i *
An equilibrium in our economy is composed by two equilibria, one in the international assets markets and a second one in the international consumption goods market. The definitions of these equilibria are given below.

The program max

x i ∈X i s π i s u i e i s + R i s x i subject to qx i ≤ 0 is equivalent to the program max x i ∈X i s π i s u i e i s + R i s x i subject to p i e i + R i x i ≤ p i e i
and e i +R i x i ≥ 0. With this two-stage approach, we can link the asset prices (q) to the consumption good prices (p). Also, this allows us to make a distinction between the two equilibria, one for the international assets market, the other for the international goods market. Definition 2 (equilibrium in assets markets) A list of financial assets prices and asset portfolios (q, x) * with q * >> 0 and j q * j = 1, is an equilibrium for the international financial assets markets if:

(1) portfolios are optimized given the financial assets prices q * , that is any agent i chooses her portfolio x i * which satisfies (15);

(2) the international financial assets markets clear:

i x i * = 0.
Definition 3 (equilibrium in consumption goods markets) A list of consumption good prices and consumption allocations (p, c) * with p * >> 0, ∀i, s p i * s = 1, is an equilibrium for the international consumption goods markets if:

(1) consumption utilities are optimized given the consumption good prices p * , that is any agent i chooses her consumptions c i * which satisfies (16);

(2) the international consumption goods markets clear: i c i * s = i e i * s for any s.

Condition [START_REF] Jeanblanc | Financial Markets in Continuous Time[END_REF] in the definition of equilibrium in the goods market rests on a strong requirement: the (goods) market clearing. It seems more reasonable to consider instead an international goods market which clears "at average":

s δ s i c i * s -e i s = 0
, where the weight δ s is just the positive probability of the state of nature s. In other words, the expectation of the trade balance under the probability distribution δ is null: E δ i c * i -e i = 0. This accounts for the following notion of δ-equilibrium. 5Definition 4 (δ-equilibrium) Let δ >> 0 be a probability distribution on the set of states of nature. Given the exchange rates, beliefs and endowments (τ, π, e), returns and preferences R i , u i for any country i, a δ-equilibrium is a list of financial assets prices and allocations (p, q, c, x) * , with p * , q * = 0 such that: individual plans are optimal (points (1) and ( 2) below) and markets clear (points (3) and ( 4)).

(1) Portfolios are optimized given the financial assets prices q * : in period 0, any agent i chooses a portfolio x i * which maximizes her utility function under the financial budget constraint:

max x i ∈X i s π i s u i e i s + R i s x i q * x i ≤ 0
(2) In period 1, for any agent i, c i * = e i + R i x i * solves the program

max c i ∈Y i + s π i s u i c i s p i * c i ≤ p i * e i
(3) The international financial assets markets clear:

i x i * = 0. (4)
The expectation under δ of the trade balance is satisfied:

E δ i c * i = E δ i e i .
We now provide conditions to obtain successively an equilibrium in the assets markets and an equilibrium in the international goods markets. For assets markets, these conditions are usually called no-arbitrage conditions.

For the international goods markets, we require well-known conditions in the theory of international finance and trade: the Uncovered Interest rate Parity (UIP) and the Law of One Price (LOP). These concepts are well-defined in the case of no uncertainty. We generalize them to the stochastic case.

No arbitrage in the international financial markets

We first define the useful portfolios for every agent. The useful portfolios, called useful vectors in the general equilibrium literature, were introduced by Werner [START_REF] Werner | Arbitrage and the existence of competitive equilibrium[END_REF].

Definition 5 (useful portfolio) w i is a useful portfolio for agent i if, for any µ ≥ 0 and any x i ∈ X i , one has:

(1)

x i + µw i ∈ X i , ( 2
) s π i s u i e i s + R i s x i + µw i ≥ s π i s u i e i s + R i s x i . Let V i ≡ v i ∈ R K : R i s v i ≥ 0 for any s = v i ∈ R K : R i s v i ≥
0 for any s be the set of portfolios with nonnegative returns in any state of nature and W i denote the set of useful portfolios for agent i.

Proposition 1 Let Assumption 6 hold. For any agent i, V i = W i .

Proof : See Appendix.

We introduce a No-Arbitrage (NA) condition for financial assets markets. Definition 6 A vector q is a no-arbitrage financial assets price system (or, more compactly, a NA price) for agent i if qw i > 0 for any w i ∈ W i \ {0}.

Condition 1 (NA) There exists a vector q that is a no-arbitrage financial assets price system for any agent i.

Let S i denote the cone of no-arbitrage financial assets prices for agent i. The following corollary characterizes NA.

Proposition 2 NA is equivalent to ∩ i S i = ∅. Proof : See Appendix.
A more useful characterization of NA is given in the following proposition.

Proposition 3 Suppose Assumptions 1, 3, 6 hold. Then q ∈ R K is NA if and only if (i) there exists a I × S matrix β ≡ β i s with β i s > 0 for any i and s such that q = s β i s R i s for any i, (ii) or, equivalently, there exists a I × S matrix β ≡ β i s with β i s > 0 for any i and s such that q = s β i s R i s for any i.

Proof : See Appendix.

The coefficients β i s can be interpreted as the prices of a unit of currency i, in terms of numeraire, in country i and state s, while the coefficients β i s can be interpreted as the prices of a unit of good, in terms of numeraire, in country i and state s. It is obvious that

β i s = β i s i
s . We will introduce no-arbitrage financial assets prices relative to a state of nature.

Definition 7 (useful portfolio in state s) w i is a useful portfolio for agent i in the state s if, for any µ ≥ 0 and any x i ∈ X i , one has:

(1)

x i + µw i ∈ X i , ( 2
)
u i e i s + R i s x i + µw i ≥ u i e i s + R i s x i .
Proposition 4 w i is useful in the state s if and only if R i s w i ≥ 0.

Proof : See Appendix.

Definition 8 A vector q ∈ R K is a no-arbitrage financial assets price system for an agent i in the state s if qw i > 0 for any portfolio w i which satisfies R i s w i > 0 or equivalently qw i > 0 for any useful portfolio w i = 0 in state s.

This definition relies on the no-free-lunch condition. The financial assets price q is such that, if the return yielded by a portfolio w is strictly positive, then its value calculated with q is strictly positive.

Proposition 5 A vector q is a no-arbitrage financial assets price system in the state s, if and only if q = µ i s R i s with µ i s > 0.

Proof : See Appendix.

Here µ i s is the quantity of good one obtains in country i and state s with one unit of currency i. Definition 9 q s ∈ R K is a common no-arbitrage financial assets price system in the state s if it is a no-arbitrage system for any i in this state, i.e., for any i, if R i s w > 0 then q s w > 0. We say that common no-arbitrage holds for the economy if, for any state s, there exists a common no-arbitrage financial assets price system q s in this state.

Proposition 6 (1) If there exists a common no-arbitrage financial assets price system in the state s, then there are exchange rates τ i s such that τ i s R i s = R 0 s . (2) Conversely, if there are exchange rates τ i s such that τ i s R i s = R 0 s for any i, then there exists a common no-arbitrage financial assets price system in state s.

Proof : See Appendix.

Remark 1 If a common no-arbitrage financial assets price holds then there exists a NA financial assets price. Indeed, let (q s ) be the list of S common no-arbitrage financial assets price systems. Then for any i, q s = µ i s R i s . Let q = s q s . From Proposition 3, q is NA.

Uncovered interest rate parities and the law of one price

We consider two notions of parity usually applied in the theory of international finance and trade (UIP and LOP).

If we have two countries (a domestic and a foreign one), the UIP holds when the returns on domestic financial assets are equal to the returns on foreign financial assets valued in domestic currency.

Definition 10 (UIP) The Uncovered Interest rate Parity holds if there exists a system of exchange rate τ such that, for any agent i and any state s, we have

τ i s R i s = R 1 s .
In other words, we have

R i s = R 1 s .
UIP means that, the returns on the financial assets of all countries valued in consumption good, are equal for any state s. This condition may be considered as very stringent since it imposes that UIP holds state by state. We weaken it with the notion of Weak Uncovered Interest rate Parity (WUIP).

We introduce a convenient notation. If (δ s ) S s=1 is a probability distribution over the states of nature and then we denote its expected value by

ρ i ≡ τ i s R i
E δ ρ i ≡ S s=1 δ s τ i s R i s , that is a K- dimensional row.
Before stating the definition of WUIP, let us introduce a characterization.

Proposition 7

The following two statements are equivalent.

(1) There exists a probability distribution δ >> 0 and a system of exchange rates τ with τ 1 s = 1 for any s such that E δ ρ i = E δ ρ 1 for any i with

ρ i ≡ τ i s R i s S s=1 .
(2) There exists a probability distribution δ >> 0 and a system of conversion rates such that

E δ R i = E δ R 1 where E δ R i = s δ s R i s and R i s ≡ R i s / i s .
Proof : See Appendix.

This proposition allows us to introduce two equivalent definitions of Weak UIP.

Definition 11 (WUIP) (1) The Weak Uncovered Interest Rate Parity holds if there exist a common probability distribution (δ s ) S s=1 with δ s > 0 for any s, and a system of exchange rate τ such that τ 1 s = 1 for all s, and

E δ ρ i = E δ ρ 1 for any i with ρ i ≡ τ i s R i s S s=1 .
(2) Equivalently, the Weak Uncovered Interest Rate Parity holds if there exist a common probability distribution (δ s ) S s=1 with δ s > 0 for any s, and a system of conversion rates such that E δ R i = E δ R 1 for any i.

Proposition 8 UIP ⇒ WUIP.

Proof : Apply the definitions.

Remark 2 UIP is equivalent to WUIP when there is no uncertainty.

Definition 12 (LOP) A system of consumption good prices p satisfies the Law of One Price if

p i s = p 1 s ( 6
)
for any s and any i. The price of the consumption good in any country is the same when valued in currency 1.

6 No-arbitrage conditions and uncovered interest rate parities: two sides of the same coin?

The following proposition links the different notions of UIP and the noarbitrage conditions. Another main result of our paper is given just below.

Proposition 9 NA is equivalent to WUIP.

Proof : See Appendix. Remark 3 Assume any agent i has a linear utility: u i c i s = c i s . The equilibrium financial assets price system is NA since it satisfies q * = λ i π i R i with λ i > 0 for any i. It is interesting to observe that we have WUIP with specific probability distributions: the beliefs of the agents.

Proposition 10 If the return matrices R i and the system of exchange rates τ satisfy UIP, then NA holds.

Proof : Since UIP implies WUIP and, from Proposition 9, WUIP implies NA, the result is immediate.

Proposition 9 states the equivalence between existence of a NA financial assets price and WUIP. The following proposition claims that existence of common no-arbitrage financial assets price for the economy is equivalent to existence of exchange rates τ i s such that the return matrices R i and the system of exchange rates τ satisfy UIP. This proposition gives sufficient and necessary conditions to obtain a system of exchange rates for which UIP holds. In other words it gives sufficient and necessary conditions to endogeneize the exchange rates in order to get UIP. It is another main result of our paper.

Proposition 11 (1) If the common no-arbitrage financial assets price holds for the economy, then there are exchange rates τ i s such that the return matrices R i and the system of exchange rates τ satisfy UIP.

(2) Conversely, if there are exchange rates τ i s such that the return matrices R i and the system of exchange rates τ satisfy UIP, then common no-arbitrage financial assets price holds for the economy.

Proof : The proof follows from Proposition 6.

In this section, we prove the existence of an equilibrium in economies with international financial assets and goods markets.

Equilibrium in the international assets markets

We first describe a property of the equilibrium prices q * of financial assets markets.

Proposition 12 Let Assumptions 1, 3, 5, 6 hold. Let (q, x) * be an equilibrium in the financial assets markets. Then, q * satisfies the following equivalent properties.

(1) q * is NA.

(2) For any i, q * = s β i s R i s = s β i s i s R i s with β i s > 0 for any i and s.

Proof : See Appendix.

We now prove the existence of an equilibrium in the international financial assets market.

Proposition 13 Let Assumptions 1, 3, 4, 5, 6, hold.

(1) If NA (Condition 1) holds, then there exists an exchange rate system such that (q, x) * is an equilibrium in the international financial assets markets with q * k > 0 for any k. (2) Conversely, assume there exists an equilibrium (q, x) * in the international financial assets market associated with an exchange rate system. Then NA financial assets price holds.

Proof : See Appendix.

Remark 4 Observe that Proposition 13 states actually the equivalence

NA ⇔ There is an equilibrium in the international financial market Remark 5 Combining Remark 4 and Proposition 9 we have the equivalence WUIP ⇔ NA ⇔ There is an equilibrium in the international financial market 16 Documents de travail du Centre d'Economie de la Sorbonne -2016.63

Equilibrium in both markets

We come now to one of our main results.

Theorem 1 Let Assumptions 1, 3, 4, 5, 6, hold. Assume that the returns R i and the system of exchange rates τ satisfy UIP.

(1) There exists an equilibrium (p, q, c, x) * (Definition 1) with q * = λp i * R i , λ > 0.

(2) The consumption good prices p i * satisfy LOP, that is p i * s = p 1 * s for any country i and any state s, and there exists λ > 0 such that q * = λp i * R i for any agent i.

Proof : See Appendix.

From Proposition 11, Theorem 1 is equivalent to the following theorem.

Theorem 2 Let Assumptions 1, 3, 4, 5, 6, hold. Assume the common noarbitrage financial assets price holds for the economy. Then, there exists a system of exchange rates τ for which the following statements are true.

(1) There exists an equilibrium (p, q, c, x) * (Definition 1) with q * = λp i * R i and λ > 0.

(2) The consumption good prices p i * satisfy LOP, that is p i * s = p 1 * s for any country i and any state s, and there exists λ > 0 such that q * = λp i * R i for any agent i.

Remark 6 (i) It is worthwhile to notice that if the returns R i and the system of exchange rates τ satisfy UIP then, in equilibrium, the balance of consumption goods is satisfied also in currency 1. Explicitly, for any state s, i p i * s c i * s = i p i * s e i s and i c i * s = i e i s . (ii) Suppose the I countries are in the same monetary zone. They use the same currency. In this case, τ i s = 1 for any i and s. UIP implies R i s = R 1 s and R i s = R 1 s for any i and s.

Matrix of returns R i are given, but prices (q, p) are endogenous. The rigidity of returns R i s ≡ R i s / i s depends on that of the conversion rates. The equilibrium in the financial markets is ensured by the flexibility of the prices (q, p), while that in the goods market depends on the conversion rates because the difference between the consumption demand and endowment is precisely given by the portfolio return. If the exchange rates satisfy UIP, then the equilibrium in the goods market is ensured.

According to Theorem 1 and Remark 6, we know that, under UIP, there exists an equilibrium and the equilibrium trade is balanced, i.e., i c i * s = i e i s in any state of nature s. WUIP means that there exist a probability distribution δ and a system of exchange rates τ such that s δ s R 1 s = s δ s τ i s R i s for any i. By definition, WUIP is less demanding than UIP. We know that WUIP preserves the equilibrium in the international assets markets (Proposition 9 and 13). One may wonder what becomes the balanced trade under WUIP. While UIP implies that the value of trade balance is zero, WUIP entails instead that the expected value of trade balance under δ is zero. We say that the returns R i and the system of exchange rates τ satisfy WUIP under a probability distribution δ if s δ s R 1 s = s δ s τ i s R i s for any i.

Theorem 3 Let Assumptions 1, 3, 4, 5, 6 hold and assume that the returns R i and the system of exchange rates τ satisfy WUIP under a probability distribution δ >> 0. Then, a δ-equilibrium exists.

Proof : See Appendix.

Theorem 3 bridges the equilibrium implications of UIP and WUIP. Under WUIP, trade may be imbalanced in some state, but its expected value is zero.

A δ-equilibrium is a weak notion of disequilibrium, because, precisely, trade may be imbalanced even if its expected value is zero and a financial equilibrium holds. The very reason is that, considering the expected interest parity E δ ρ i = E δ ρ 1 instead of the interest parity ρ i = ρ 1 , we obtain the expected trade balance E δ i c i * = E δ i e i instead of the trade balance i c i * s = i e i s . To understand why this happens, focus on the proof of Theorem 3: since the difference between consumption and endowment is given by the portfolio return (c i s -e i s = R i s x i ), if the (expected) aggregate portfolio return is zero because of the (Weak) UIP, then the (expected) aggregate net demand is zero, that is the (expected) trade balance holds.

Examples with UIP failure

In order to better understand the role of UIP, we provide three examples where UIP fails. In the first one, the no-arbitrage condition holds in the international assets market and an equilibrium exists but it is no-trade and the consumption goods prices don't satisfy LOP. In a second example, the noarbitrage condition holds in the international financial assets market and an equilibrium exists as well but the goods market is imbalanced. In a third example, the sets of no-arbitrage prices for the international assets market differ across the countries and do not intersect: thus, the no-arbitrage condition fails and there exists no equilibrium.

Example 1

In this example, where the UIP fails, there exists a unique equilibrium which is no-trade. We consider an exchange economy with one consumption good, two countries: i = 1, 2, two assets: k = 1, 2, and two states of nature: s = 1, 2. The matrices of returns on assets, in terms of local currencies, are given:

R 1 = R 1 sk ≡ R 1 11 R 1 12 R 1 21 R 1 22 = 1 0 1 2 (7)
R 2 = R 2 sk ≡ R 2 11 R 2 12 R 2 21 R 2 22 = 0 1 2 1 (8)
In this economy, UIP is violated because

τ 2 1 R 2 1 = R 1 1 implies τ 2 1 (0, 1) = (1, 0), a contradiction.
Individuals share the same beliefs and the states are considered equiprobable:

π ≡ π 1 1 π 1 2 π 2 1 π 2 2 = 1 2 1 1 1 1 (9)
The utility functions are also the same across the countries:

s π i s u i c i s = 1 2 s c i s (10)
but the initial endowments differ:

e ≡ e 1 1 e 2 1 e 1 2 e 2 2 = 1 1 1 2 (11)
The sets of useful portfolios

W 1 = x 1 1 , x 1 2 : x 1 1 ≥ 0, x 1 1 + 2x 1 2 ≥ 0 W 2 = x 2 1 , x 2 2 : x 2 2 ≥ 0, 2x 2 1 + x 2 2 ≥ 0
determine the cones of no-arbitrage prices:

S 1 = -int W 1 0 = p 1 1 , p 1 2 : p 1 1 > 0, p 1 2 > 0, 2p 1 1 -p 1 2 > 0 S 2 = -int W 2 0 = p 2 1 , p 2 2 : p 2 1 > 0, p 2 2 > 0, 2p 2 2 -p 2 1 > 0
We see immediately that (1, 1) ∈ S 1 ∩ S 2 . From Proposition 13, there exists a pair (q, x) * with q * k > 0 for any k such that, for any agent i, x i * is a solution of program (3) and x * is a net trade, that is the financial assets markets clear:

2 i=1 x i * = 0.
In Appendix, we prove that there exists an equilibrium which is no-trade and LOP does not hold.

Example 2

In this example, UIP fails and there is no equilibrium. As above, we consider an exchange economy with one consumption good, two countries: i = 1, 2, two assets: k = 1, 2, and two states of nature: s = 1, 2. The matrix of returns R 1 remains unchanged and is given by [START_REF] Rogoff | The purchasing power parity puzzle[END_REF], while R 2 is now replaced by a slightly different matrix:

R 2 = R 2 sk ≡ R 2 11 R 2 12 R 2 21 R 2 22 = 0 1 2 2 (12)
In this economy, UIP is violated because

τ 2 1 R 2 1 = R 1 1 implies τ 2 1 (0, 1) = (1, 0), a contradiction.
The other fundamentals remain the same as in Example 1 (beliefs, preferences and endowments are given by ( 9), ( 10) and (11) respectively).

Clearly, the set of useful portfolio W 1 and the cone of no-arbitrage financial assets prices S 1 do not change (R 1 is the same), while the corresponding set for country 2 becomes now

W 2 = x 2 1 , x 2 2 : x 2 2 ≥ 0, 2x 2 1 + 2x 2 2 ≥ 0 S 2 = -int W 2 0 = p 2 1 , p 2 2 : p 2 1 > 0, p 2 2 > 0, p 2 2 -p 2 1 > 0
We see immediately that (2, 3) ∈ S 1 ∩ S 2 . From Proposition 13, there exists a pair (q, x) * with q * k > 0 for any k such that, for any agent i, x i * is a solution of program (3) and x * is a net trade, that is the assets markets clear:

2 i=1 x i * = 0.
In Appendix we prove that the trade cannot be balanced. Hence equilibrium does not exist.

Example 3 Again, we consider an economy with two countries i = 1, 2, two financial assets k = 1, 2 and two states of nature s = 1, 2. The matrices of returns on financial assets are given: The sets of useful portfolios

R 1 = R 1 sk ≡ R 1 11 R 1 12 R 1 21 R 1 22 = 1 0 1 2 (13) R 2 = R 2 sk ≡ R 2 11 R 21 12 R 2 21 R 2 22 = 1 4 1 3 (14)
W 1 = x 1 1 , x 1 2 : x 1 1 ≥ 0, x 1 1 + 2x 1 2 ≥ 0 W 2 = x 2 1 , x 2 2 : x 2 1 + 4x 2 2 ≥ 0, x 2 1 + 3x 2 2 ≥ 0
determine the cones of no-arbitrage financial assets prices:

S 1 = -int W 1 0 = p 1 1 , p 1 2 : p 1 2 > 0, 2p 1 1 -p 1 2 > 0 S 2 = -int W 2 0 = p 2 1 , p 2 2 : -4p 2 1 + p 2 2 < 0, -3p 2 1 + p 2 2 > 0
Assume there exists (p 1 , p 2 ) ∈ S 1 ∩ S 2 . In this case we have

p 1 > 0, 2p 1 -p 2 > 0, -4p 1 + p 2 < 0, -3p 1 + p 2 > 0
These inequalities lead to 0 < 3p 1 < p 2 < 2p 1 which is impossible. Thus S 1 ∩ S 2 = ∅ and NA financial assets price does not exist. In this case WUIP does not hold and hence UIP as well.

Remark 7

We want to show that when any country wants also to consume in period 0 then, under NA and UIP, the economy still has an equilibrium on both markets and LOP holds for any period, any state. The results concerning UIP, WUIP still hold.

Consider the model where any agent i solves in period 0: max

(c i 0 ,x i )∈(R + ×X i ) u i 0 (c i 0 ) + s π i s u i e i s + R i x i p i 0 (c i 0 -e i 0 ) + qx i ≤ 0
and, in period 1, we have c i s = e i s + R i s x i for any s. The functions u i 0 and u are strictly concave and increasing. An equilibrium of this economy is a list (p, q, c, x) * with p * , q * >> 0 such that individual plans are optimal (points (1) and ( 2) below) and markets clear (points (3) and ( 4)).

(1) Portfolios and consumption in period 0 are optimized given the financial assets prices and consumption price q * and p * 0 : in period 0, any agent i chooses a consumption c i * 0 and a portfolio x i * which maximizes her utility function under the budget constraint:

max (c i 0 ,x i )∈(R + ×X i ) u i 0 (c i 0 ) + s π i s u i e i s + R i s x i (15)
p i * 0 (c i 0 -e i 0 ) + q * x i ≤ 0 21 Documents de travail du Centre d'Economie de la Sorbonne -2016.63

(2) In period 1, for any agent i, c i * = e i + R i x i * solves the program max

c i ∈Y i + s π i s u i c i s (16) p i * c i ≤ p i * e i
(3) The international financial assets markets clear: i x i * = 0. (4) The trade balance is satisfied in period 0 and in any state of nature of period 1:

i c i * 0 = i e i 0 i c i * s = i e i
s for s = 1, . . . , S Claim 1 Assume UIP hold.

(1) There exists an equilibrium (p, q, c, x) * with q * = p i * R i , for any i.

(2) The consumption good prices p i * satisfy LOP, that is (a) in period 0, p i * 0 = p 1 * 0 , for any country i, (b) in period 1, p i * s = p 1 * s for any country i and any state s, Proof. Define p i 0 = p 0 for any i, where p 0 ≥ 0. At period 0 agent i solves max

(c i 0 ,x i )∈(R + ×X i ) u i 0 (c i 0 ) + s π i s u i e i s + R i s x i p 0 c i 0 -e i 0 + qx i ≤ 0
Recall that S i denotes the set of no-arbitrage asset prices for agent i. Then, the set of no-arbitrage prices for period 0 is σ = R ++ × ∩ i S i . In addition, ∩ i S i = ∅ ⇔ σ = ∅. From e.g. Dana, Le Van and Magnien, 1999, there exists p 0 , q, c i 0 , x i i * with (p 0 , q) * >> 0, such that any agent i solves her/his consumption problem with prices (p 0 , q) * and i c i * 0 -e i * 0 = 0, i x i * 0 = 0. For period 1, the proof of Theorem 1 applies.

We can easily check that we still have NA ⇔ WUIP since these results are relative only to the second period.

Conclusion

Our paper considers a two-period model à la Hart [START_REF] Hart | On the existence of an equilibrium in a securities model[END_REF] with consumption good and financial assets. The consumption and the assets are internationally tradable. Our main results are the following. (1) We introduce a Weak UIP condition and we show that it is equivalent to No-Arbitrage Condition financial assets price. Moreover we have the equivalences:

NA holds ⇔ Weak UIP holds ⇔ An equilibrium exists in the international financial markets (2) We introduce the common no-arbitrage financial assets price and show Existence of a common no-arbitrage financial asset price ⇔ UIP holds (3) When the UIP holds, we have an equilibrium in both the international financial assets and goods markets. At this equilibrium, we obtain trade balance in value and LOP.

(4) When the UIP fails, we show by means of three examples that (i) the no-arbitrage financial assets price condition holds and an equilibrium exists as well but it is no-trade and LOP is not satisfied (Example 1); (ii) the noarbitrage condition price holds in the international financial assets market and an equilibrium exists as well but there exists no balance in the goods markets (Example 2); (iii) the no-arbitrage condition financial assets price fails and there exists no equilibrium (Example 3).

Modelling the coexistence of an equilibrium in the financial assets markets jointly with a disequilibrium in the goods markets through the failure of parities remains an interesting question. Most of financial papers (Rogoff [START_REF] Rogoff | The purchasing power parity puzzle[END_REF] among others) consider that the parities are not respected in the short run. As suggested by Frenkel and Mussa [START_REF] Frenkel | The efficiency of foreign exchange markets and measures of turbulence[END_REF] in a monetary model, trade balance disequilibria seem plausible under a regime of pegged rates because relative price adjustments are achieved through slow changes in the goods markets, while financial markets are mobile and integrated. This point is tackled in the paper through three examples.

Appendix

Proof of Proposition 1 First, we want to prove that

V i ⊆ W i . If v i ∈ V i , R i s v i ≥ 0 for any s.
According to (2), for any x i ∈ X i and any µ ≥ 0, one has e

i s + R i s x i + µv i = e i s + R i s x i + µR i s v i ≥ µR i s v i ≥ 0, that is Definition 5, point (1)
. From

e i s + R i s x i + µv i = e i s + R i s x i + µR i s v i ≥ e i s + R i s x i (17)
and the increasingness of u i (Assumption 6), we obtain also Definition 5, point

.

Conversely, we want to show that W i ⊆ V i . Let w i ∈ W i . Then, point (1) in Definition 5 requires e i s + R i s x i + µw i ≥ 0 for any s. Dividing both the sides of this inequality by µ > 0 and letting µ go to infinity, we get R i s w i ≥ 0 for any s, that is

w i ∈ V i . Finally, observe R i s w i ≥ 0 ⇔ R i s w i ≥ 0.
Proof of Proposition 2 We observe that S i = -int W i 0 where W i 0 is the polar6 of W i . Under Assumption 3, the sets W i do not contain lines and the sets S i are nonempty (see Dana, Le Van and Magnien [START_REF] Van | On the different notions of arbitrage and existence of equilibrium[END_REF] among others). Let ∩ i S i be the intersection of all the cones of no-arbitrage prices. Thus, if NA holds, q belongs to ∩ i S i , and, if ∩ i S i is nonempty, NA holds.

Proof of Proposition 3

(1) Let q be NA. Let w i ∈ W i \ {0}. From Assumption 3, R i s w i ≥ 0 for any s and R i s w i > 0 for some s. From Dana and Jeanblanc [START_REF] Jeanblanc | Financial Markets in Continuous Time[END_REF], there exists a I × S matrix β ≡ β i s with β i s > 0 for any i and s such that q = s β i s R i s for any i. (2) Conversely, assume q = s β i s R i s for any i with β i s > 0 for any i and s. If w i ∈ W i \ {0}, then, from Assumption 3, R i s w ≥ 0 for any s and R i s w > 0 for some s. Hence, for any i, qw i > 0 for any w i ∈ W i \ {0} or, equivalently, q is NA.

Proof of Proposition 4

Sufficiency. R i s w i ≥ 0 implies u i e i s + R i s x i + µw i = u i e i s + R i s x i + µR i s w i ≥ u i e i s + R i s x i because u i is increasing. Necessity. u i is concave and w i is useful in the state s. Then,

u i e i s + R i s x i µR i s w i ≥ u i e i s + R i s x i + µR i s w i -u i e i s + R i s x i ≥ 0
Dividing both the sides of this inequality by µu i e i s + R i s x i > 0, we obtain

R i s w i ≥ 0. It is equivalent to R i s w i ≥ 0.
Proof of Proposition 5 R i s w i ≥ 0 implies that w i is useful in the state s and, since q is a no-arbitrage asset price system in the state s, qw i ≥ 0. According to the Farkas' Lemma, R i s w i > 0 ⇒ qw i > 0 implies q = µ i s R i s with µ i s > 0. The converse is obvious.

Proof of Proposition 6 (1) Since

q s = µ i s R i s = µ 1 s R 1 s for any i, we can define τ i s ≡ µ i s /µ 1 s in order to obtain τ i s R i s = R 1 s .
(2) Consider a useful portfolio w i = 0 which satisfies R i s w i > 0. Define q s ≡ R 1 s = τ i s R i s . Then, q s w i > 0 for any i. Apply Definitions 8 and 9.

Proof of Proposition 7 (1) ⇒ (2). Assume we have [START_REF] Allouch | The geometry of arbitrage and the existence of competitive equilibrium[END_REF]. Take (

1 s ) s >> 0. Define R i s ≡ R i s / i s , i s = 1 s /τ i s and δ s ≡ δ s 1 s σ δ σ 1 σ
We get s δ s R i s = s δ s R 1 s , that is (2). ( 2) ⇒ (1). Define τ i s = 1 s / i s and

δ s ≡ δ s / 1 s σ δ σ / 1 σ Then E δ R i = E δ R 1 becomes s δ s τ i s R i s = s δ s R 1 s , that is (1).
Proof of Proposition 9 (1) If a vector q is NA, then from Proposition 3, there exists β i s > 0 for any i and s such that q = s β i s R i s for any i. Define δ s ≡ β 1 s / σ β 1 σ and τ i s ≡ β i s /β 1 s . Clearly, the probability distribution δ is well-defined. Consider the random vectors ρ 1 ≡ R 1 (2) Conversely, if E δ ρ i = E δ ρ 1 for any i, then the vector

q = E δ ρ 1 = E δ ρ i = S s=1
δ s τ i s R i s , for any i is NA from Proposition 3.

Proof of Proposition 12 (1) First observe that we have q * x i * = 0 for all i. Let ε = (ε, . . . , ε) T ∈ R K with ε > 0. Let w ∈ W i . We then have This implies q * w i > 0 and q * ∈ S i for any i. Equivalently, q * is NA.

(2) The second statement comes from Proposition 3.

Proof of Proposition 13 (1) Assume that q * is NA. In this case, the proof is provided by Werner [START_REF] Werner | Arbitrage and the existence of competitive equilibrium[END_REF], Page and Wooders [START_REF] Page | A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium[END_REF], and Dana, Le Van, Magnien [START_REF] Van | On the different notions of arbitrage and existence of equilibrium[END_REF] among others. The strict positivity of q * results from the strict increasingness of u i jointly with Assumptions 1 and 3.

(2) Refer to Proposition 12.

Proof of Theorem 1 (1) Since we have UIP, the price q = s R i s is NA since it equals s R 1 s . It follows from Proposition 13 that we have an equilibrium (q, x * ) in the international assets markets.

From Proposition 12, q * = s β i s R i s for any i, with β i s > 0 for any i and any s. Define for any i βi s ≡

β i s σ k β i σ R i σk
We claim that, for any i, s βi s R i s = s β1 s R 1 s . Indeed, for any k and any i, we have s β i s R i sk = s β 1 s R 1 sk . Hence, for any i,

σ k β i σ R i σk = σ k β 1 σ R 1 σk . Let λ ≡ σ k β 1 σ R 1 σk = σ k β i σ R i σk for any i. Hence, s β i s R i s λ = s β 1 s R 1 s λ Equivalently, s β i s R i s σ k β i σ R i σk = s β 1 s R 1 s σ k β 1 σ R 1 σk
Our claim is true. Now, define q * ≡ s βi s R i s . We have j q * j = 1. Setting pi s ≡ βi s , for all i, all s, we find

q * = s pi s R i s = s pi s R 1 s = s p1 s R 1 s (18)
for any i. Let Z ≡ z ∈ R S : s z s R 1 s = 0 and observe that Z = {0} in the case of complete assets markets (rankR 1 = S).

From (18), we get, for any i, pi s = p1 s + z i s with z i ∈ Z. Define p1 s ≡ p1 s and pi s ≡ pi s -z i s = p1 s for any i and any s. Let c i * s = e i s + R i s x i * for any i and any s. q * x i * = 0 implies that pi c i * = pi e i + q * x i * = pi e i : the budget constraint is satisfied in any country (Definition 1, point (2)).

Noticing that, for any i and s, c i * s = e i * s +R i s x i * = e i * s +R 1 s x i * and summing over i, we obtain for any s Let λ ≡ 1/ σ pi σ . Then, q * = λp i * R i for any i. (2) Finally, it is clear that the system p * satisfies the LOP.

Proof of Theorem 3

Since the returns R i and the system of exchange rates τ satisfy WUIP under a probability distribution δ >> 0, no-arbitrage condition (NA) holds. Hence an equilibrium in the international assets markets (q, x) * exists. If c * denotes the associated consumptions in period 1, we have c i * s = e i s + τ i s R i s x i * for any i and s. We can easily check that they satisfy the consumers problems (16). We now check that the expectation under δ of the trade balance is satisfied. Indeed, we have since i x i * = 0.

Example 1

Trade balances are satisfied in any state of nature s if i c i * s = i e i s . Let

τ ≡ τ i s = 1 1 τ 1 1 τ 1

  s S s=1 is the random vector of returns, 13 Documents de travail du Centre d'Economie de la Sorbonne -2016.63

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  s = E δ ρ i that is WUIP.

s

  We have proved that (p, c * ) satisfies point (2) of Definition 1. Observe that p >> 0. We now normalize the equilibrium consumption prices by defining i and any s.

Country-specific returns change over the states of nature in the case of equities while they don't in the case of bonds. Therefore, WUIP implies UIP in the case of bonds.

Documents de travail du Centre d'Economie de la Sorbonne -2016.63

Returns are also designed as payouts or cash flows.

We have τ i s i s = 1 s for any i. This means that, ex-ante, the commodity prices satisfy the LOP if the currency 1 is the numeraire. 6 Documents de travail du Centre d'Economie de la Sorbonne -2016.63

Market completeness means that the columns of R i span the whole space R S (rankR i = S) and implies that a full insurance is possible. Redundancy of assets means that dim ker R i > 0, that is K > rankR i . When markets are complete and assets are not redundant, we have K = S = rankR i . In this case, the return matrix is square and invertible.

In general, a δ-equilibrium is not an equilibrium because trade may be imbalanced. When there is no uncertainty, δ-equilibrium is equivalent to equilibrium. 10 Documents de travail du Centre d'Economie de la Sorbonne -2016.63

The polar cone of a set X ⊆ R K is defined as X 0 ≡ y ∈ R K : y T x ≤ 0 for any x ∈ X .

Referees for their thoughtful remarks and questions. We are also grateful to Saqib Jafarey for valuable comments and to the participants of the NBER General Equilibrium Conference held in Bloomington on September 2012. This research has been conducted as part of the project LABEX MME-DII (ANR11-LBX-0023-01).