Renaud Bourlès
email: renaud.bourles@centrale-marseille.fr

Juliette Rouchier
email: juliette.rouchier@univ-amu.fr

Evolving informal risk-sharing cooperatives and other-regarding preferences

Keywords: Agent-based, risk-sharing, other-regarding preferences, learning, segregation

In this paper we present a model of formation and destruction of informal cooperatives in a population of agents who perform a risky activity and who are heterogeneous in terms of success in their actions. Although some agents have high-risk and others low-risk, our model displays a dynamics with cooperatives in which agents share equally their income with a certain stability. We are interested in studying at the same time the existence of cooperatives, their ability to integrate a large proportion of agents and the degree of segregation of these cooperatives. Three factors can explain the existence, stability and lack of segregation. First, we show that the classical explanation in economics holds within the framework of our model: when agents are risk averse, high success agents can share with low success agents so that to stabilize the value of their income -the higher the risk aversion, the more stable the cooperatives and the lower the segregation. Learning can explain in a small proportion the existence of cooperatives: we designed agents so that they have to learn whether they are high or low-risk, and while they are learning, they tend to create cooperatives that can last. Eventually we worked on the integration of other-regarding preferences in the model, with two different definitions. As expected, the influence of other-regarding preferences is to increase stability and decrease segregation, and the two models of rationality react differently to the type of network in which the agents are immersed. This paper, mainly exploratory, presents our model and shows the influence of the definition of network as well as all other factors presented before. In that sense, although we have mainly done a rough exploration of its relevant parameters for the moment, it exposes different insights that can be gained by its study.

Introduction

In this paper we present a model with which we want to study repeated risk-sharing among agents who are heterogeneous in terms of risk. More precisely, we would like to know why less risky agents would accept to share risk with more risky ones. To do so, we model risksharing through cooperatives in which agents share risk equally, in a population which is homogeneous in risk aversion, but heterogeneous in risk. 1 We are more precisely interested in how cooperatives evolve (that is how they are created and destroyed) and to what extend people with different risk exposures accept to share (equally) risk between each other. In our model, a cooperative can be seen as a coalition which is formed by one agent who proposes others to join, and from which agents2 can choose to leave individually. The dynamics of formation and destruction of the cooperative is not highly dynamical or complex, since all decisions of agents are coordinated as once every step.

In this paper, we try to analyze three ways of explaining how and why people with different risk exposures share risk. We use Agent-Based modeling to establish their credibility and limits. The first explanation of this behavior is rather usual in economics: agents can have a negative attitude towards risk and, when given the choice, prefer a secure income to a stochastic one. It is then said that these agents are risk-averse and it has been shown that this attitude towards risk can explain the existence of pooling in a group. Another point can be their lack of knowledge: it is not obvious that individuals know their own risks precisely when they perform an activity, and it seems logical that they should know others' risks even less clearly. In that case, could imperfect rationality be the explanation for the fact that some lucky people are ready to share with unlucky ones on a regular basis? Eventually it is possible to imagine that individuals are not just selfish (and that others' income are also of interest to them) or that participating to a community is as important as having a high income. There is no positive model concerning what is called in economics "other-regarding preferences", where others' are considered in an agent's utility, so we propose and test two different models of rationality and see if (and why) they increase the stability of sharing behaviors.

We present the main results showing that our three different modelling features can explain, with more or less strength, an increase of stability of our cooperatives. This paper is rather exploratory in the sense that it mainly shows how our platform can be re-used to test more specifically the influence of network or the definition of other-regarding preferences. It also helped us to stabilize some of the observation indicators.

The paper is built in four parts. The first part motivates our research and gives some elements on informal cooperatives, equal sharing, their link to risk aversion, and other-regarding preferences. In a second part we present the assumptions we used to build the model and give the equations on which it is based, as well as the simulation and observation protocol for running the model. In a third part we present the dynamics of the model, depending on parameters that are presented previously. We eventually discuss our results and conclude in the last part.

2 Attitude towards risk, learning, and other-regarding preferences

The questions of the functioning and the stability of risk-sharing agreements have been relatively well documented in economic literature during the last decade. This literature starts from empirical studies in village economies analyzing to what extend villagers share risk.

Testing the empirical relevance of equal sharing of wealth in India, [START_REF] Townsend | Risk and insurance in village india[END_REF] found that even though equal sharing is statistically rejected, it is a surprisingly good benchmark. Moreover, [START_REF] Fafchamps | Risk-sharing networks in rural philippines[END_REF] argue, using data from Filipino, that this finding may be due to the fact that risk-sharing takes place at a lower level than the entire village (i.e. across communities). Based on this result, [START_REF] Genicot | Group formation in risk-sharing arrangements[END_REF] propose a model describing how such communities form. More precisely, they analyze the stability of cooperatives (or coalitions) under limited commitment, that is when transfers are not enforceable (and therefore occur only if the agents have a long term interest to do so). [START_REF] Bloch | Informal insurance in social networks[END_REF] add a network component to the model and analyze the stable transfer schemes depending on punishment strategies. The literature on risk-sharing also points out the important role played by risk aversion. This question goes back to [START_REF] Arrow | Aspects of the Theory of Risk-Bearing[END_REF] in which efficiency gains from risk-sharing increase with risk and with aversion towards risk. This is confirmed in a model with two homogeneous agents by [START_REF] Kimball | Farmers' cooperatives as behavior towards risk[END_REF] that states that the higher risk aversion, the higher the discount rate below which an equal sharing agreement is sustainable. However, considering limited commitment, [START_REF] Fafchamps | Risk sharing and quasi-credit[END_REF] argues that this might no more be the case as an increase in risk aversion might make an agent leave the agreement. Still [START_REF] Lazcó | Riskiness, risk aversion, and risk sharing[END_REF] shows that as soon as there is no aggregate risk, an increase in risk aversion increases risk-sharing.

We differ from this literature by adding heterogeneity across agents, imperfect information and altruistic preferences. Few papers on risk-sharing model heterogeneous agents. One exception is [START_REF] Genicot | Does wealth inequality help informal insurance[END_REF] who analyzes sharing agreement between agents that are heterogeneous with respect to their level of wealth. She finds that most of the time, more inequality increases the likelihood of equal sharing of risk. As we focus here on risk heterogeneity, a closer paper is [START_REF] Bourlès | Risk-sharing contracts with asymmetric information[END_REF] that analyzes the optimal risk-sharing agreement between two agents who differ about their risk exposure. This paper shows that equal sharing is optimal if risk aversion is high and heterogeneity is low. Regarding the formation of cooperative or coalition, [START_REF] Moizeau | Heterogeneity and the formation of risk-sharing coalitions[END_REF] show that, under perfect information and selfish preferences, agents form groups with those who have pretty close risk exposure. We analyze here to what extend imperfect information and altruistic preferences can decrease this "homophily" (that we measure with a segregation index). Indeed, altruism (in particular in the case of kinship ties) has been shown to be key in risk-sharing agreement in both empirical [START_REF] Fafchamps | The formation of risk sharing networks[END_REF]) and theoretical [START_REF] Alger | Kinship, incentives, and evolution[END_REF]) literature.

Learning about one's own risk

One element that is rarely addressed in the classical economic literature on risk-sharing is the one of information. It is always assumed that agents know well their own probability of success and that they also have access to the probabilities of success of the other agents or, at least, the expected income they will get from their belonging to the cooperative.

Learning in agent-based model is a central concept, and is even the main justification for using such a modelling techniques [START_REF] Brenner | Agent learning representation: advice on modelling economic learning[END_REF]). Indeed it is a very specific characteristics to have autonomous agents, each of them getting different information and hence creating their own representations of their environment. In economics, the most usual learning is made on actions -it is called reinforcement learning in a generic way, and has been made very popular in the form of the Roth-Erev algorithm [START_REF] Roth | Learning in extensive form games: Experimental data and simple dynamic models in the intermediate run[END_REF]). Agents learn to choose the action which brings them the highest profit among all the possible actions belonging to a fixed set. Information that is used is only personal past actions, but one can expect some kind of optimization at the population level, where everyone converges towards Nash strategy. Another usual reinforcement learning is genetic algorithm, where the set of possible actions evolves in time (see for example [START_REF] Vriend | An illustration of the essential difference between individual and social learning, and its consequences for computational analyses[END_REF])

On top of the learning on action, agents can also learn about the characteristics of the environment, which is called "belief learning". For example, an agent can learn which other agent he should interact with and construct a map of good and bad partners [START_REF] Rouchier | A multi-agent model for describing transhumance in north cameroon: Comparison of different rationality to develop a routine[END_REF]). He can also learn to link context to relevant actions, with classifier systems [START_REF] Moulet | The influence of sellers' beliefs and time constraint on a sequential bargaining in an artificial perishable goods market[END_REF]). One integration of belief learning that is very well known is the one of Camerer and Ho, EWA [START_REF] Camerer | Experience-weighted attraction learning in normal form games[END_REF]), which mixes reinforcement and fictitious play, so that to create a representation of the best actions which is not just linked to one's own past actions.

Here we use a learning about the belief of the agent, what he believes to be his risk exposure. The issue we want to address is the impact of informational blindness on the stability of cooperatives, that is how would cooperatives evolve if agents only knew that there is heterogeneity of risks among them but do not know which level of risk they themselves undergo. We also add a noise in the transmission of information among agents and test its impact on our dynamics.

Other-regarding preferences

The term other-regarding preferences is rather generic, since it represents any situation where agents are biased in their individual choice because of the presence of others.3 It does not mean just positive attitudes towards others, but any attitude that transforms decision process in presence of others: aversion to inequality, for example, can make agents avoid situation where they earn more than others, but also where others earn more than they do. These preferences are so strong that in some cases individuals are ready to lose money so that to punish those who deviated from a behaviour they consider as acceptable [START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF]).

Empirical evidence are often taken in village economy [START_REF] Townsend | Risk and insurance in village india[END_REF]). Concerning risk-sharing issues, people pay more attention to family members or friends [START_REF] Fafchamps | Risk-sharing networks in rural philippines[END_REF]) than other villagers or even other members of a community. Compassionusually translated into patience or willingness to accept to loose some money for the sake of sharing -has a huge impact on participation to cooperatives.

To the best of our knowledge, almost none has proposed positive representation of otherregarding preferences, by integrating them in the decision process of an agent based models. Janssen and Ahn propose an interesting, but complex, model for integrating relation to inequality in a learning model in a context of public good provision games [START_REF] Janssen | Learning, signalling and social preferences in public-good games[END_REF]), taking inspiration in the equation proposed by [START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF]). In this paper we propose two different ways to represent the sensitivity of agents to the presence of others in their choices.

Assumptions for the model

We want to test three dimensions of agents' rationality that can have an impact on risk-sharing among agents who are heterogeneous in their probability of success: the risk aversion of agents, an incomplete information and when they have other-regarding preferences (here designated as compassion and friendship). In our model, the size of the network certainly has an impact and will also be tested.

Heterogeneity and similarities of agents. The basis of our questioning is the heterogeneity of agents in terms of risk. We divide the population in two levels of risk: high and low. We represent this risk as a probability of losing some fixed amount of money for each agent, those probabilities being uncorrelated in the group. For other parameters, such as the level of risk aversion or the size of the network they belong to, agents are homogeneous. When drawn, the network is made following the same algorithm for all agents but each of them of course has a different network.

Leaving a cooperative. At each step, an agent who belongs to a cooperative calculates if he wants to stay in the cooperative or not. If not, he becomes a lonely agent.

Sharing. Within a cooperative, agents use the equal sharing rule (meaning that the aggregate wealth in the cooperative is equally shared among its members), which is the most documented in the literature.

Building a cooperative. At each step, an agent who does not belong to a cooperative can choose to build one. Before deciding, he gathers information about the risk of agents who belong to his network, going to level two (friends and friends of friends), and who are lonely, i.e. not part of a cooperative. He roughly evaluates the risk, as will be seen in the description of model algorithms, and decides if he wants to belong to the cooperative he could be creating. If the cooperative is created, agents who have been contacted have no choice but to accept to share the income at this time-step. They can decide to leave the cooperative at the following step.

Aversion to risk. To represent the attitude of the agents toward risk, we choose to use a CRRA (for Constant Relative Risk Aversion) utility function. This means that this attitude is summarized by a parameter, the relative risk aversion. Such a formulation has two main advantages. First, it is consistent with the evidence of decreasing absolute risk aversion (namely, the richer an agent, the lower his risk aversion). Second, most of the papers estimating aversion towards risk, estimate the coefficient of relative risk aversion (see for example [START_REF] Chetty | A new method of estimating risk aversion[END_REF] or [START_REF] Meyer | Relative risk aversion: What do we know?[END_REF]). Based on this utility function, agents can choose among the options they face: either to stay in a cooperative or leave; either to create a cooperative or not.

Agents do not know their type. Agents use Bayesian learning to know which type they belong to (they assign a probability to each existing type). Therefore they use their past performance only to estimate their level of risk. The algorithm is such that the more difference there is between success probabilities (of high-risk and low-risk types), the easiest it is for agents to identify their type. For example if the difference in probabilities of success between both types is 0.20, it takes an agent 85 steps to know his own type with a strength of 95%; whereas if the difference between probabilities is 0.70, it takes him 5 time-steps.

Agents do not know others' type. At each step, agents have to evaluate the income they get by staying in a cooperative. Since they have no information about others' types, they evaluate the cooperative performance using ex-post results, i.e. the income they got from belonging to the cooperative. Learning algorithms classically include forgetting, and it is particularly relevant in our context, since the cooperatives evolve a lot and expected consumption is not stable: with our forgetting rate, agents put value mainly on the most recent information and give almost no weight to events older than 6 time-steps.

Agents like others. We use two ways to model the fact that agents are not just driven by income and reduction of risk, but that they also have a sense of belonging to a group. One aspect is the responsibility agents feel towards the others. When deciding if he wants to leave a cooperative, an agent sees if his gain in utility would be significant compared to the loss others would incur from his departure. This issue can make a low-risk agent stay because his presence raises the average income of his cooperative (we will refer to such a mechanism as "compassion"). Another aspect is an individual pleasure that agents have when belonging to the same cooperatives as members of their network. This pleasure is directly integrated in the utility function (friendship). Both attitudes are modeled with the use of a parameter that can take a range of values.

Model

The model is described in several steps. We define the "basic" model as the choice process of agents who are defined just by their risk-type and their attitude towards risk (that varies with the value of risk aversion).

Basic economic model

Agents are producers who work and see the success of their activity after their work is over. They can have two types of result: success or failure, which makes them gain two different values: y ∈ {y + ; y -}. Each agent is characterized by his probability of failure, which determines the history of his income. We assume that agents are heterogeneous in this probability of failure and more precisely that it can either be low (noted p) or high (p). In our setting, half of the agents will be of low-risk type, the other half being of high-risk type.

When an agent belongs to a cooperative, he shares his gain equally with others: all participating agents put their income in a common pool, and the value is redistributed equally to all. This constitutes what is called the consumption, c, which is a characteristic, at a given time-step, of the cooperative. If a cooperative is made of agents {a 1 , ...a n } with income {y 1 , ...y n },

c = j y j n (1)
Agents are also defined by their behavior towards risk, which can push them to prefer to be in the cooperative even if it decreases their expected consumption. To represent this behavior we use a classical utility function, which has already been used to describe the behaviors of populations in farmers' societies [START_REF] Kimball | Farmers' cooperatives as behavior towards risk[END_REF]: CRRA (for Constant Relative Risk Aversion). It is based on a parameter ρ that describes the relative risk aversion. Based on previous works [START_REF] Kimball | Farmers' cooperatives as behavior towards risk[END_REF], [START_REF] Chetty | A new method of estimating risk aversion[END_REF] or [START_REF] Meyer | Relative risk aversion: What do we know?[END_REF]) we allow ρ to vary in [1.1; 6]. For an agent of consumption c, the utility is:

u(c) = c 1-ρ -1 1 -ρ (2)
The shape of u(•) then represents agents' behavior towards risk and the index of concavity: -u (•)/u (•) is called index of absolute risk aversion. Agents are said risk averse if this index is positive, which means that they are averse to a mean preserving spread of consumption (in other words they always prefer E( x) to x, where x is a random variable). The main property of the CRRA class chosen here is to exhibit a constant index of relative risk aversion: -cu (c)/u (c) = ρ ∀c, which is notably consistent with the observation that the index of absolute risk aversion is (in real life) generally decreasing with revenue. It turns out that the higher ρ, the more the agents are ready to sacrifice (expected) consumption to reduce risk (i.e. variance in consumption).

When belonging to a cooperative with high-risk agents, agents with low risk (i.e. a low probability of failure) will not get as high an expected consumption as if they were alone, but they will get a more stable consumption since the risk is shared among the members of the cooperative. In our context of equal sharing, a low value of ρ pushes agents with low risk to stay out of a cooperative, and a high value of ρ makes them prefer to be in the cooperative.

Bayesian learning

Our agents have cognitive limitations: they do not know their own risk (i.e. their own probability of failure) and have to learn it along the time, building beliefs. They know that there exists two different probabilities of failure, high (p) and low (p), and that each group of agents represents half of the population. At each time-step t, they have beliefs (subjective probabilities) of belonging to each category: π t of being low-risk (that is of having a probability of failure of type p) and (1-π t ) of being high-risk (that is of having a probability of failure of type p). They initially believe that they have probability of 50% to be of each category (they know the real distribution of the population). Each agent then updates his belief in time following a Bayesian learning: at time t, if he has experienced k losses among the t first periods, his belief on his probability of being of low-risk type writes:

π t = p k (1 -p) t-k p k (1 -p) t-k + p k (1 -p) t-k (3)
Therefore, at time t, the belief of agent i that he belongs to the low-risk type is such that π 0 = 0.5 and the update depends on the achievement of the previous period:

• if y i (t) = y - π t = pπ t-1 pπ t-1 + p(1 -π t-1 ) (4) • if y i (t) = y + π t = (1 -p)π t-1 (1 -p)π t-1 + (1 -p)(1 -π t-1 ) (5)
The interest of this learning is that agents keep track of the quality of their belief, which is expressed as a probability of belonging to one group or another. Depending on the difference between p and p, the time to learn is different, i.e: if p -p = 0.20, it takes an agent 85 steps to know his type with a strength of 95% whereas if p -p = 0.70 it takes him 5 steps.

Staying or leaving

A cooperative applies equal sharing, which means that, at each period t, the consumption of each belonging agent corresponds to the average income of the cooperative. Formally, in a cooperative made of agents {a 1 , ...a n } with income {y 1 (t), ...y n (t)}, the consumption of every agent is:

c(t) = n j=1 y j (t) n (6)
is the consumption of each agent belonging to the cooperative.

In the basic model, an agent stays in the cooperative he belongs to if:

E πt (u(y)) -u(c) ≤ 0 (7)
that is if, based on his belief and on the history of the cooperative, he his better off outside than inside the cooperative. More precisely:

u(c) = t<=T δ (T -t) u(c(t)) Z (8) with Z = 1 -δ 1 -δ T (9)
meaning that our agent computes a weighted average of the utility levels he had inside the cooperative, putting more weight to the near past (this allows to take into account the dynamic of the cooperative).

And:

E πt (u(y)) = π t u + + (1 -π t )u - (10)
where u + (resp. u -) is the expected utility of low-(resp. high-) risk agents:

u + = pu(y -) + (1 -p)u(y + ) (11) u -= pu(y -) + (1 -p)u(y + ) (12)
Therefore, E πt (u(y)) represents the utility the agent expects alone, according to his belief π t on his probability of being low-risk.

Compassion, friendship

We include altruism ("other-regarding preferences") through two mechanisms: compassion and friendship, which are translated in the algorithm thanks to parameters comp and f that are constant over the simulation and the same for all agents. Compassion makes the agent matter for his marginal impact on common welfare: he is more reluctant to leave if it is bad for the (n -1) other agents belonging to the cooperative. He stays if:

E πt (u(y)) -u(c) ≤ comp. u(c) -u nc -E πt (y) n -1 ( 13
)
where c = t<=T δ (T -t) c(t)/Z represents his "estimation" of individual consumption in the cooperative (based on past levels of consumption, with forgetting) and

E πt (y) = π t [py -+ (1 - p)y + ] + (1 -π t )[py -+ (1 -p)y + ]
represents the expected individual income according to his belief π t . These two terms enable the agent to calculate the individual utility of others in the cooperative if he stays compared to their individual utility if he goes; if the gain he adds to others by staying is high enough (using factor comp) compared to his own interest of leaving the cooperative (on the left hand side), he will stay.

The utility of agents is directly transformed by friendship, and the more friends are present in the cooperative, the higher the agent gets from being part of it. He stays if:

E πt (u(y)) -(1 + rf )u(c) ≤ 0 ( 14
)
where r is the share of friends in the cooperative. Under this formulation, the agent takes into account directly the utility of his friends in the cooperative (recall that they all have the same level of consumption in the cooperative).

Building a cooperative

At start, there is no cooperative. At each step, one agent who does not belong to a cooperative is randomly chosen to test if he wants to create a cooperative. For this he contacts all single agents belonging to his network at level 2 (all agents who do not belong to a cooperative and with whom he has a direct link (friends) and all their direct friends). This is the pool of people who will be included in the cooperative if it is created. The agents answer their own belief on their probability to be of low-risk type. The building agent trusts his direct friends (that is ∀j ∈ r 1 ) at a level of 90% (with probability 0.1 he assigns them the average probability in the population: 0.5) and the friends of his friends (that is ∀l ∈ r 2 ) at (90%) 2 . The agent who wants to form a cooperative then aggregates the information so that to create one virtual agent with whom he decides to create the cooperative or not. First, he calculates the probability of this virtual agent g being in the low-risk group: π t .

π t =
j∈r 1 (0.9 * π t,j + 0.1 * 0.5) + l∈r 2 (0.81 * π t,l + 0.19 * 0.5)

card(r 1 ) + card(r 2 ) (15)
The agent building the cooperative then infers the risk of the group seen as one agent g at the next step as:

p g = π t p + (1 -π t )p (16)
He already knows his own risk at the next step:

p i = π t p + (1 -π t )p (17)
and the expected utility when associating with the fictitious agent g is then:

E pg (u(c)) = p i p g u(y -) + (1 -p i )p g u y + + y - 2 +p i (1 -p g )u y + + y - 2 + (1 -p i )(1 -p g )u(y + ) ( 18
)
which has to be higher than his own expected utility (E πt (u(y))) for the cooperative to be created.

One can note several elements about this model. First, the agent has a pessimistic view of the association, since he aggregates all agents' income in a way that overestimates risk. Moreover, there is no link creation associated to the building of cooperative. Even if agents belong to the same cooperative and share their income, they do not get linked and our network is thus exogenous and stable. All agents can belong to one cooperative only since they are not chosen to built one or to participate if they are already in a cooperative. The cooperative cannot increase in size either -nobody gets in after it has been created, and it can only loose agents.

Simulations 4.1 A typical run

At time 0:

• N (usually, 200) artificial agents are created;

• Half of the agents are given probability of failure p and other half p;

• Network is created (we only use random network in this paper) Afterwards:

• All agents check if they want to stay in the cooperative they belong to;

• One lonely agent (that is an agent who does not belong to a cooperative) is chosen to check if he wants to create a cooperative;

• We run for 50 steps

Impact of risk aversion

As can be expected, the value of risk aversion has a huge impact in our system. When risk aversion increases, the number of cooperatives that are created increases and then decreases, and at the same time the number of cooperatives existing in the system on average increases up until risk aversion is around 2.5 and then is stable (see Figure 1). It means that up until a certain level of risk aversion, more and more cooperatives get created when risk aversion increases. When risk aversion is higher than 2.5, the created cooperatives last and hence there are less and less available agents to create cooperatives along the simulation. At the end, less cooperatives are created because there is no one to create them. At the same time, the higher risk aversion the more stable the cooperatives: their average lifetime increases almost linearly with risk aversion, from around 9 time-steps when risk aversion is 1.1 to 20 when risk aversion equals 4.

[Figure 1 about here.]

We have used two different segregation measures in our model (see Figure 2). The first one is supposed to be used to compare situations where a constant population is used. However, as we saw before, our model does not involve a constant number of agents in cooperatives. When risk aversion is low, few agents are in cooperatives, and when watching in details, they are mostly high-risk agents (since low risk agents tend to leave fast from cooperatives and do not create that many). Hence the basic segregation index is low because the population of "agents belonging to cooperatives" is low with respect to the entire population. When risk aversion increases, and the population is mostly involved in cooperatives, then the basic segregation index decreases. On the other hand, the revised segregation index ("segregation 2" on figures) incorporates free agents as extremely segregated cooperatives (of segregation 1) and that's why this index is much higher than the basic one, but is decreasing in risk aversion.

Combining both in the analysis shows the influence of risk aversion on our model again. In some settings (low risk aversion) cooperatives are not segregated with basic segregation index (using the population of agents belonging to cooperatives only, who are mainly highrisk agents) but low-risk agents do not share much because they stay alone. In other settings (when risk aversion is higher than 2.5), most agents are integrated in cooperatives and then, an increase in risk aversion makes cooperatives more and more mixed. This value of 2.5 for risk aversion in the basic model corresponds to a size of population within cooperatives that reaches about 120 agents. We note that for all cases that are given later, this value of 120 agents is always the one where our indicators change.

[Figure 2 about here.]

In that sense, the size of network has a direct impact on the number of cooperatives that are created, although rather limited since it just changes this value of 2.5. The larger the number of friends in the network, the quicker many agents are involved in a cooperative, since at each run, the agent who creates the cooperative can reach more agents. Hence, the value of 120 is reached for slightly lower values of risk aversion, and as a consequence the tendency to decrease of all indicators which have a concave curve (total number of cooperatives and basic segregation) takes place for lower values of risk aversion, as can be seen in Figures 3. The average number of cooperatives decreases once this turning point is attained for networks of size 10 and 15, which is not true for size 5, but over all, the dynamics is the same. Segregation also has different values when size of network varies, but is similarly impacted by the value of risk aversion: the larger the network the smaller segregation.

[Figure 3 about here.] The probability of success, that are set in the basic simulation to p = 0.9 and p = 0.7 does not have much an impact in terms of qualitative result (as long as the difference between the two are the same). After a change in p and p, the shape of the curves remain similar, even though the turning point (when basic segregation is at its maximum or the number of cooperatives reaches its peek) changes. For example, when p = 0.8 and p = 0.6 this peek is for risk aversion close to 2. The dynamics is similar.

Impact of information

We test the system in three different situations regarding the access of information for agents: in the first case they follow the mechanism that is described above (they learn and received information with noise from their neighbors -WLNI = With Learning and Noise Information), in the second case they know their type but there is noise in the transmission of information (NLNI -No Leaning Noise Information), and in the third case they know their type and get the right information from their neighbors (NLPI -No Learning Perfect Information). For most of our indicators, there is no difference after 50 steps. The results that are synthesized in Figure 4 show that there is no difference among these cases for the average number at each time-step indicators. The mean lifetime of cooperatives is also similar. The total number of cooperative that are created in the simulation is different when risk aversion is low. Observing simulation runs, one can explain the difference by the dynamics of the first steps, during which more cooperatives get formed when agents have no information about their type: low-risk agents build cooperatives although they have no interest to do so, as the case without learning shows. There is also a small difference in segregation for high value of risk aversion, which is also due to the first steps where low-risk share more and hence the number of agents who are included in a cooperative is higher. The more simulation goes, the less low-risk agents form cooperatives and the more segregation increases. It is important to note that the noise in communication has no impact on all the indicators we are interested in. This was tested with various values of noise and this result holds when the noise increases.

[Figure 4 about here.]

Impact of compassion

We studied the impact of compassion: for this we ran simulations for 50 steps, with 5 friends per agent, varying compassion from 1 to 5. On figures, one can see that there is a similarity for compassion 0 and 1, and that when the value increases to 5 there is a qualitative difference in the dynamics. The number of cooperatives (Figure 5) is higher on average with compassion 1 than 0, but the curve is pretty similar, showing an increase and almost a stabilization when risk aversion increases, as seen before. When compassion is equal to 5, the number of cooperatives at each step is higher for low risk aversion than high risk aversion. But what has to be noticed is that the number of cooperatives in the population at each step is equal to the total number of cooperatives created, which means that almost all cooperative that is created stays until the end of the simulation. lifetime also increases when compassion goes from 0 to 1, and hence the number of cooperatives that are created in total decreases with compassion. We indeed stabilize cooperatives by adding compassion in our model. In parallel, at least 160 agents are on average in a cooperative when compassion is 5, whereas the value increases for compassion 0 from 40 to 140 when risk aversion increases. With compassion 1, there is no qualitative change compared to 0 (lifetime and number of agents belonging to cooperatives is similarly related to risk aversion) but all indicators indicate a stabilization of cooperatives.

[Figure 5 about here.] Segregation also shows that compassion allows cooperatives to be stable and mixed even for low values of risk aversion. The curve of basic segregation when compassion is 5 (Figure 6) is very different from the others. Many agents are in cooperative and this is why the mix of population is more clearly linked to risk aversion. Above the already witnessed turning point (risk aversion 2.5), compassion 5 shows that it creates cooperatives that are much more mixed than other compassions. This is rather logical in our setting, since the interest for others does not depend on their success, and the desire for sharing that compassion creates is so strong that any cooperative survives. Interestingly, there is almost no difference in segregation when compassion is 0 and 1, which confirms the fact that a small level of compassion does not change the dynamics qualitatively.

[Figure 6 about here.]

Impact of friendship

When friendship is higher than 0, its influence is very strong. When increasing its value, it increases this tendency, without changing the impact qualitatively. The number of cooperatives is very low as soon as risk aversion is above 1.5 and these cooperatives are very stable. This can be seen as the number of cooperatives and the number of cooperatives created in total are very close (Figure 7). Their mean lifetime are also very long as soon as risk aversion is higher than 1.5. With friendship, the segregation is lower than in any other situation we have built before (Figure 8). This is due to the fact that agents do not leave the cooperatives at all, contrary to all other cases. Even with compassion, cooperatives are stable but lose some of their members who are mainly low-risk agents. The other sign is that the number of agents in cooperatives and the number of low-risk agents in cooperatives reach almost the maximum, for any value of risk aversion. Hence, as there is no possibility to create cooperatives anymore, their number is low.

The impact of the size of network is very strong in this setting, since it determines the size of cooperatives, and hence the number that are created. When risk aversion is higher than 1.5 and agents have 5 friends this value is 20, but it drops to 12 when they have 10 friends and to 8 when they have 15 friends. However, there is no such difference in segregation, which stays lower than in any other setting, between 0.2 and 0.3 for almost any value of risk aversion.

[Figure 7

Conclusions

In this paper, our aim was to explore a rather generic framework in which agents who are heterogeneous in success in an action can create cooperatives to share and reduce their risk in time. In economics, the only usual explanation for the possibility to perform risk-sharing is risk aversion: by sharing, agents expect to be helped on the days they would need it. One author who says that redistribution is also existing because individuals have to follow social norms or are interested in the consumption of others on top of their own, could put this element forward quite recently. The point is that up until now, the inclusion of otherregarding preferences in utility-based models has not been so general, and there is no largely accepted robust model to represent this tendency. Our model enabled us to test several representations of this type of preferences, especially because they both gave very different types of results. We could assess the dynamics of the model thanks to the cross analysis of several indicators, a work which was most of the time necessary, even with indicators that could seem redundant (like our two versions of segregation).

As expected, risk aversion has a strong impact on the system. Learning effect is not so clear when running the simulation for 50 steps. The variation of some of our parameters, like the probability of success, does not change the dynamics much. Surprisingly, the density of the network only has a small impact on stability and segregation. Compassion has a strong impact on the stability of cooperative, but not so strong on the reduction of segregation. Only when it gets high, it can almost neutralize the influence of risk aversion by increasing a lot the stability and mixing of our population. Friendship stabilizes our cooperatives and reduces segregation even more, although it is not true for very low values of risk aversion.

One critic has been done several times already to this model by economists who do not use agent-based. The building of cooperatives has been criticized because it is a one-way autocratic process. The agent who decides on building the cooperative gives no chance of refusal to the ones who come in the cooperative. If one considers the building as a one timestep process, this critic is very logic. However, since we consider that the judgement of a cooperative is made ex-post by agents, and that they can leave after just one time-step, one can consider that the process is just in two steps: agents who have been chosen to belong to the cooperative participate in one step and can decide if they want to stay. The alternative would be to have all agents decide if they accept to be in the cooperative. But in the case when one agent does not accept, all agents have to recalculate their own decision; and each time an agent decides to drop out, all agents calculate again. This process implies endless calculation, which we transform in learning in our model. We do not think this choice is so illogical within the agent-based framework.

These results are preliminary, since we need to perform a better statistical analysis, but also to decide if we want to simplify the model (like keeping learning or not, keeping the two-step network for the creation of cooperatives, etc...) so that to make it maybe easier to compare to analytic results. For this we might have to check more precisely the dynamics of the model for the first steps with and without learning more precisely. Another step would be to integrate heterogeneity not only in the probability of success, but also in risk-aversion, compassion and friendship, which have been proven to be idiosynchratic and have a massive impact on reciprocity or participation to common good [START_REF] Anh | Trust in two person games: game structures and linkages[END_REF]. (5,10,15). The size of network does impact on the number of cooperatives but there is no qualitative difference among the dynamics.

Figure 4: Segregation for situations where agents learn their type, know their type, and know everyone's type with certainty. This knowledge has a very small impact on the sharing behavior of agents -this is due to the fact that they learn pretty quickly their actual risk.

  about here.] [Figure 8 about here.]

Figure 1 :

 1 Figure1: The impact of risk aversion in the basic model: the number of cooperatives on average at one step and the number of cooperatives that are created for the case when agents have 5 friends in their network and there is no compassion nor friendship.

Figure 2 :

 2 Figure 2: The impact of risk aversion in the basic model: basic segregation and segregation2 for the case when agents have 5 friends in their network and there is no compassion nor friendship.

Figure 3 :

 3 Figure 3: Number of cooperatives on average and created in total depending on risk aversion with different size of network (5, 10, 15). The size of network does impact on the number of cooperatives but there is no qualitative difference among the dynamics.

Figure 5 :

 5 Figure5: The number of cooperatives that are created and those that exist in the population at each step, for network with 5 friends and different values of compassion. When compassion increases, cooperatives are more stable, and especially when compassion is 5, the value of risk aversion has less impact on stability.

Figure 6 :

 6 Figure 6: Basic segregation and segregation2 for network with 5 friends and different values of compassion. Compassion reduces segregation.

Figure 7 :

 7 Figure 7: The number of cooperatives that are created and those that exist in the population at each step, for network with 5 friends and different values of friendship. As soon as friendship is positive, the number of cooperatives created and staying are almost the same and are equivalent for all values of risk aversion above a certain threshold.

Figure 8 :

 8 Figure 8: Basic segregation and segregation2 for network with 5 friends and different values of friendship. Friendship reduces segregation except for very low values of risk aversion.

Even though statistically rejected, equal sharing of wealth have been shown to be a relatively good benchmark by[START_REF] Townsend | Risk and insurance in village india[END_REF].

2 The meaning we give to the notion of "agents" is closer to economist approach than multi-agent modeling, in that they deal only with little information about their own utility and the belief they have on the utility of others, without actually having notion of collective action.

It should be noticed that in our model, preferences are not clearly defined here. They are only summarized by the "objective" function

Acknowledgements

We wish to thank Victorien Barbet and Simon Venturi, both students at École Centrale Marseille, for their work in the process of writing, correcting, and studying this model.

Parameters

The values of the parameters used in the simulations are given in Table 1 • Segregation index characterizing the mix of agents of high-risk and low-risk type in each of the J cooperatives. Calling n l (resp. n h ) the number of low-risk (resp. high-risk) agents in the population and n l j (resp. n h j ) those in the cooperative j, the segregation index writes:

• Revised segregation index that we built after running preliminary simulations. This index is the same as the one described above but each lonely agent is also considered as part of a cooperative (containing only one agent). We kept both indexes as explained in the result section.

Results

All results that are shown in figures are given for each value of risk aversion and as an average result for 30 simulations, each simulation running for 50 steps.


Zhou Xun

Michel Lubrano
email: michel.lubrano@univ-amu.fr

A Bayesian subjective poverty line, one dollar a day revisited

Keywords: Poverty line, Bayesian inference. JEL codes: C11, C21, I32

This paper provides a new estimation of an international poverty line based on a Bayesian approach. We found that the official poverty lines of the poorest countries are related to the countries' mean consumption level. This new philosophy is to be compared to the previous assumptions made by the World Bank in favour of an absolute poverty line. We propose a new international poverty line at $1.48 per day (2005 PPP) based on a reference group consumption level. This figure is much higher than that proposed by the World Bank ($1.25 in 2005 PPP), but still within a reasonable confidence interval. By this standard, there are more than 1.7 billion people living in poverty.

Introduction

In different countries and at different times, the definition of poverty changes according to people's living situations and to people's varying poverty perception. Even within a given society and at a given point of time, the critical level of income at which individuals are recognised as being poor is not perceived in the same ways by different income groups. The meaning of poverty differ between those groups as poverty can be, at least partly a social construction. This was illustrated for instance in van [START_REF] Van Praag | The welfare function of income in Belgium: An empirical investigation[END_REF] as a preference drift.

Between different countries, the minimum basket of goods that ensures physical and even mental health is not the same, just because living standards are different and social characteristics and habits are different. The common view is that in the poorest countries, poverty is anchored to basic human needs, such as enough food, clean water, sanitation, clothing, shelter, health care and basic education. A poverty line in those countries is usually defined as an "absolute poverty line" that focus only on how much humans need for living, independently of the national income distribution. For richer countries, once the basic needs are satisfied, individuals tend to desire more expensive basket of goods, e.g. more varied diets, suitable cloth, comfortable shelter, better health and higher education, just to be like the others and take a decent part in social life. The definition of "poverty" in this case becomes more complex and is influenced largely by the perception of "economic inequality". An individual who considers himself as being poor may not face a problem of survival, but he's suffering from an envy comparison for what others have in his living society. The latter definition of poverty line is called "relative poverty line" Where could we put the limit between these two definitions of a poverty line. What is the list of countries which are considered as being sufficiently rich in order to afford a relative poverty line and what is the list of the other countries? [START_REF] Ravallion | Quantifying absolute poverty in the developing world[END_REF] showed that official poverty lines varied little with mean consumption for poor countries while above a critical level of mean consumption, official poverty lines had a much stronger elasticity with respect to mean consumption. Based on that previous finding, [START_REF] Ravallion | How did the world's poorest fared in the 1990s[END_REF] and [START_REF] Ravallion | How did the world's poorest fared since the early 1980s[END_REF] provided an international poverty line (a worldwide absolute poverty line) to be "$ 1 per day" ($1.08 at 1993 PPP). In a more recent paper, [START_REF] Ravallion | Dollar a day revisited[END_REF] clearly identify two group of countries in a new data set covering 74 developing countries from 1988-2005. They estimate a non linear regression relating official poverty lines to mean consumption, allowing for a different coefficient between the two group of countries. From this model, the new international poverty line had risen to $1.25 per day at 2005 PPP. [START_REF] Greb | Dollar a day re-revisited[END_REF] then re-revisited this study, using different econometric techniques and a different specification and found a higher international poverty line at $ 1.45 per day. The difference between $1.25 and $1.45 does not seem important while it actually is. This adjustment means that 317.6 millions supplementary people would fall in poverty in 2005. This gives us enough reason to revisit the problem of setting an international poverty carefully.

We are going to use [START_REF] Ravallion | Dollar a day revisited[END_REF]'s new data set as given at the end of their paper and adopt a Bayesian approach in order first to take fully into account the uncertainty of the estimated parameters and second to provide a posterior density for the obtained poverty line. We re-estimated the same empirical model used in [START_REF] Greb | Dollar a day re-revisited[END_REF] and illustrate graphically where the difference lies between the two papers. We then show how to define a poverty line as a function of the mean level of consumption of a reference group of poor countries, the composition of that group being endogenously determined. We found in this latter analysis, that an international poverty line is eventually higher at $1.48 per day. The posterior density of our international poverty line is much more concentrated than those obtained using any of the previous studies.

The paper is organised as follows. In section 2, we show how a subjective approach to poverty perception can be introduced, using macro cross country data. With section 3, we develop the econometric techniques involved by a Bayesian approach to our problem. With section 4, we show how a Bayesian approach can illustrate some misspecification problems and provide a rational route to derive the posterior density of a world poverty line. Section 5 concludes.

Poverty line and preference drift

A poverty line can be defined on a subjective basis. We refer to the minimum income question (MIQ) that can be found for instance in [START_REF] Kapteyn | Some methodological issues in the implementation of subjective poverty definitions[END_REF] and which be phrased as follows: what is the minimum income that you would need in order to make the two ends meet?. If z i is the reported answer, y i the actual income of the household and x i a vector of characteristics of the household (such as its composition) then the following regression can be estimated z i = α + βy i + γx i + ǫ i using individual data. An estimated subjective poverty line corresponds to a fixed point for every type of composition x and is given by:

z * = α + γx 1 -β .
Individuals having an income below z * would be classified as being poor. A poverty line at a country level cannot be determined independently of any subjective perception of poverty. A European comparison was made using this approach in for instance [START_REF] Van Den Bosch | A comparison of poverty in seven European countries and regions using subjective and relative measures[END_REF]. This approach can have however some bias just because it assumes that z * is common to all individuals and does not depend on their income level. Using a fixed point for determining z * eliminate any preference drift. So that in [START_REF] Van Den Bosch | A comparison of poverty in seven European countries and regions using subjective and relative measures[END_REF] subjective poverty lines are much above official poverty lines.

A preference drift, at least in developed countries is clearly identified, see for instance [START_REF] Van Praag | The welfare function of income in Belgium: An empirical investigation[END_REF] for Belgium. That means, that individual do not have the same perception of poverty. The perceived minimum necessary income rises with the level of own income.

In order to determine a revised world poverty line, [START_REF] Ravallion | Dollar a day revisited[END_REF] start from a subjective definition of poverty, recognising that the perception of poverty can vary between countries because of different habits, different perceptions and social traditions. Absolute poverty lines are defined with respect to a consumption basket meant to provide the necessary calories to survive. But the composition of that basket is socially determined. [START_REF] Atkinson | The Economics of Inequality[END_REF] quotes the example of English workers who went on strike because tea was planned to be withdrawn from the official basket of goods for computing a poverty line in the nineteen century. Despite the fact that tea has no nutritional value, it had a social value. The preference drift transposed at the country level means that the minimum necessary income for an individual would depend on

z ij = α + βy i + γ Cj + ǫ ij
where Cj is the mean consumption level of country j. The same fixed point algorithm provides the level of the poverty line for country j:

z * j = α + γ Cj 1 -β .
But this time there is a country specific effect. [START_REF] Ravallion | Dollar a day revisited[END_REF] assumed that for very poor countries, the preference drift is zero so that γ = 0 for that group of countries. Working on a data set of 77 developing and developed countries, they identified a group of 15 very poor countries with an average level of private consumption per capita of less that $60 a month. They estimate the following model

z j = 1I(C j < $60)α 1 + 1I(C j > $60)(α 2 + γC j ) + ǫ j (1)
where z j is an official poverty line in PPP dollars and C j the average level of private consumption per capita in PPP dollars. The estimated poverty line is given by the regression coefficient α 1 . This coefficient represents an estimate of the empirical mean of the z j computed for the countries for which C j < $60. This group of countries is said to represent the reference group to compute the poverty line. [START_REF] Greb | Dollar a day re-revisited[END_REF] estimate a slightly different model

z j = 1I(C j < θ)α 1 + 1I(C j > θ)(α 2 + γ log C j ) + ǫ j (2)
where the reference group is endogenously determined by estimating θ.

Preference drifts among the poor

The main assumption made in [START_REF] Ravallion | Dollar a day revisited[END_REF] and all the related works is that a poverty line has to be an absolute line for the poorest countries. When looking at the figures reported in the data base of [START_REF] Ravallion | Dollar a day revisited[END_REF], there is a relation between z j and C j for the group of very poor countries, even if that relation is not of the same amplitude as for richer countries. For poorer countries (those with a mean C lower than $60 a month), the poverty line represents on average 0.92 of the mean consumption while that factor drops down to 0.45 for the richer group of countries. This last figure is much more in accordance with the usual definition of a relative poverty line which corresponds usually to half of the mean income. The first figure of 0.92 can be completed by computing the average poverty line for the poorer group which is $38 ($12). So there is a large standard deviation which has to be explained. In this group the minimum and maximum poverty lines are $19 and $59. These figures confirm that the usual relative poverty line concept can be applied only to richer countries. However, there is a relation, even if it is of a different nature, between the average level of consumption and a reasonable poverty rate, which is not necessary the official one. To illustrate that point, we would like to report the controversy that took place recently in India around the decision of the Indian Government to reduce the level of the official poverty line, following the recommendation of the Tendulkar commission Tendulkar (2009). The official rate was reduced to Rs 28.65 per capita daily consumption in cities and Rs 22.42 in rural areas. The objective was to reduce the poverty rate which went down to 29.8% with these new figures (the world Bank estimates the rate of poverty in India to be 32.7% in 2010 with $1.25 a day). The Indian press reported large protestation, which can be understood when we know that India is a very fast developing country. But certainly inequality is also very fast increasing. This is at odds with what was claimed by the commission who said that "Fundamentally, the concept of poverty is associated with socially perceived deprivation with respect to basic human needs". So there is a large gap between what society perceives and what the official agencies publish. This discrepancy between an official poverty line and what individuals perceive is not specific to developing countries. In [START_REF] Van Den Bosch | A comparison of poverty in seven European countries and regions using subjective and relative measures[END_REF], we see that in countries of southern Europe, a subjective poverty line is also much higher than the official line. Our model should take that into account in order to define a world poverty line.

A new poverty line definition

The poverty line which is proposed both in [START_REF] Ravallion | Dollar a day revisited[END_REF] and [START_REF] Greb | Dollar a day re-revisited[END_REF] consists in computing the mean poverty line of a reference group when that reference group is given in [START_REF] Ravallion | Dollar a day revisited[END_REF] or endogenously determined in [START_REF] Greb | Dollar a day re-revisited[END_REF]. The idea we would like to illustrate here is that a poverty line for the poorest countries has still to be determined as a function of the characteristics of a reference group, but that this poverty line should depend also on a reference income (or a reference consumption level) of the reference group. Consequently, the model that we shall estimate is

z j = s j (α 1 + γ 1 C j ) + (1 -s j )(α 2 + γ 2 C j ) + ǫ j (3)
s j = 1 if C j < θ 0 otherwise
where θ is an unknown threshold. The new poverty line will be determined as a conditional expectation

E(z j |s j = 1) = α 1 + γ 1 E(C j |s j = 1). ( 4
)
In words, the poverty line we propose for developing countries is a function of a reference group consumption level which is taken to be equal to the mean consumption of that reference group. It is different from a usual relative poverty line in the sense that it depends not on the national mean consumption but on the mean consumption of a more general group, called the reference group. We call this new poverty line a subjective poverty line not because it depends on subjective data, but because it relates to a common group where countries are supposed to identify. They judge their poverty line by reference to that group. The notion of a reference group appeared in the happiness economic literature as a possible explanation to the Easterlin paradox. Individual satisfaction is a function not mainly of the level of income, but of the difference between their income and a reference income which is taken as the mean income of the reference group. See for instance Ferreri-Carbonell (2005) for an empirical investigation. We try here to translate that concept to countries and poverty lines.

The convenient way to both determine an estimate of the threshold parameter θ and to take into account the uncertainty in the determination of the reference group is to adopt a Bayesian approach, as we shall see in the next section.

3 Bayesian inference for regression models with a break

The generic model we want to estimate is a two regime model with a break determined if a variable is lower or greater than an unknown threshold:

E(y i |x i ) = x ′ i β 1 if 1I(w i < θ) (5) E(y i |x i ) = x ′ i β 2 if 1I(w i > θ). ( 6
)
y i is the dependent variable, x i a set of exogenous variables and w i is the regime shift variable which is supposed to be exogenous or predetermined. θ is a threshold parameter and 1I(.) is the indicator function defined as

1I(w i < θ) = 1 if w i < θ 0 otherwise.
The corresponding regression model is

y i = 1I(w i < θ)x ′ i β 1 + (1 -1I(w i < θ))x ′ i β 2 ) + ǫ i
, where the error term ǫ i is supposed to be normal with zero mean and variance σ 2 . For inference purposes, it is useful to define the following matrices

x ′ i (θ) = [x ′ i 1I(w i < θ), x ′ it 1I(w i > θ)], (7) X(θ) = [x ′ i (θ)], (8) β ′ = [β ′ 1 , β ′ 2 ]. (9)
Thus we have the more compact form:

y = X(θ)β + ǫ.
where y is a vector containing the N observations of y i .

Bayesian inference

Considering N observations, the likelihood function of the single variance model is:

L(β, σ 2 , θ; y) ∝ σ -N exp - 1 2σ 2 N i=1 [y i -X ′ i (θ)β] 2 (10)
Conditionally on θ, this is the likelihood function of a usual regression model, so that natural conjugate prior densities for β and σ 2 belong the normal inverted gamma2 family:

π(β|σ 2 ) = f N (β 0 , σ 2 M -1 0 ), π(σ 2 ) = f Ig (σ 2 |ν 0 , s 0 ) (11)
The conditional posterior density of β and σ 2 are:

π(β|θ, y) = f t (β|β * (θ), s * (θ), M * (θ), ν * ), (12) π(σ 2 |θ, y) = f Ig (σ 2 |ν * , s * (θ)) ( 13
)
where

M * (θ) = M 0 + N i=1 X ′ (θ)X(θ), β * (θ) = M -1 * (θ)[X ′ (θ)y + M 0 β 0 ], s * (θ) = s 0 + β ′ o M 0 β 0 + y ′ y -β ′ * (θ)M * (θ)β * (θ), ν * = ν 0 + N
The posterior density of θ is proportional to the inverse of the integration constant of the above Student density times the prior density of θ:

π(θ|y) ∝ |s * (θ)| -(N -k)/2 |M * (θ)| -1/2 π(θ). ( 14
)
As there is no conjugate prior for θ, we are free to use any form of parametric density. A convenient choice is to use a uniform prior between bounds or a non-informative prior. The marginal posterior densities of β and σ 2 have to be found using numerical integration as:

π(β|y) = f t (β|β * (θ), s * (θ), M * (θ), ν * )π(θ|y)dθ, and π(σ 2 |y) = f Ig (σ 2 |ν * , s * (θ))π(θ|y)dθ.
As the dimension of θ is one, we can use a traditional deterministic integration rule, like the Simpson rule in order to evaluate these densities. However, as we can also be interested in transformations of the parameters, a simulation method is better.1 As ( 14) is a marginal density, we have simply to find a feasible grid over which to evaluate it, compute numerically the cumulative and then use the inverse transformation method to draw a value for θ. The grid over which to evaluate (14) has to be chosen carefully. It should cover most of the probability, but it should also avoid identification problems as detailed in Bauwens et al. (1999, p. 235). To speak quickly, the grid should be chosen in such a way that there are enough observations per regime. The domain of definition of θ is given by [Min(w i ), Max(w i )]. But its bounds cannot be reached, because otherwise, the model would not be identified.

We draw a value of θ from π(θ|y i ). Using this draw, we draw a β from the conditional posterior π(β|θ, y) which is a Student density.

The two variance case

For modeling purposes, it will be useful to consider the possibility of having different variances in the two regimes. We keep the same dichotomous variable s i as in the original model and assume this time that:

Var(ǫ i ) = s i σ 2 1 + (1 -s i )σ 2 2 = σ 2 2 (s i φ + 1 -s i ) = σ 2 h i (θ, φ), (15)
as detailed in Bauwens et al. (1999, p. 236). Let us now divide the observations by h i (θ, φ) in order to get a regression model with homoskedastic errors of variance σ 2 :

y i (θ, φ) = y i / h(θ, φ), (16)
x ′ i (θ, φ) = [x ′ i 1I(w i < θ), x ′ it 1I(w i > θ)]/ h(θ, φ), (17) X(θ, φ) = [x ′ i (θ, φ)], (18)
The final model is y(θ, φ) = X(θ, φ)β + ǫ, its likelihood function being

L(β, σ 2 , θ, φ; y) ∝ σ -N N i=1 h i (θ, φ) -1/2 (19) exp - 1 2σ 2 (y(θ, φ) -X(θ, φ)β) ′ (y(θ, φ) -X(θ, φ)β) .
In this expression, y(θ, φ) represents the N observations of y i (θ, φ). The conditional posterior densities of β and σ 2 are the same as before. We just have to replace y and X(θ) by y(θ, φ) and X(θ, φ) in the necessary expressions.

The joint posterior density of θ and φ has the form

π(θ, φ|y) ∝ N i=1 h i (θ, φ) -1/2 |s * (θ, φ)| -(N -k)/2 |M * (θ, φ)| -1/2 π(θ)π(φ). ( 20
)
It is slightly more difficult to draw jointly θ and φ from this bivariate density than to draw θ from the univariate density (14). It is always possible in theory to decompose a bivariate density into π(θ, φ|y) = π(φ|θ, y) × π(θ|y), so that we first draw in the marginal density π(θ|y) and then in the conditional π(φ|θ, y). To apply this method, we have first to determine a grid over θ and φ in order to fill up a matrix. From this matrix of points, we can determine numerically the marginal density π(θ|y). For a given draw of θ, we have to find the corresponding conditional π(φ|θ, y). Of course, we will not have a draw of θ that corresponds exactly to a line of the initial matrix of points. So we shall have to proceed by linear interpolation between two lines as explained in the appendix.

Data and estimation

The data come from [START_REF] Ravallion | Dollar a day revisited[END_REF] who have considered 74 developing countries. The data set includes national official poverty lines (PL) (or academic poverty lines in some cases) and Private Consumption Expenditures (PCE). These data report to different years from 1988 to 2005. They have been adjusted by the household consumption PPP's collected at the occasion of the international comparison program of 2005 (World bank, 2008).

The PCE and PL are reported on a monthly basis. This data set is an improvement over the old data set used in [START_REF] Ravallion | Quantifying absolute poverty in the developing world[END_REF] which covered only 33 countries and had a weaker price adjustment. Bayesian approach provides us the adequate tools to discuss and compare those two alternative specifications. When we estimate both formulations with an unknown threshold θ, we observe that the model in levels provides a rather unprecise estimation for θ as we have E(θ|y) = 98.77 (44.36) while the model in logs provides a much higher value for θ as E(θ|y) = 138.99 (34.78), but also a more precise standard deviation (as given between brackets). As a consequence, the poverty line is better estimated with the model in logs. This is again well apparent if we examine the posterior density of θ in both models. From Figure 1, we see that in order to deliver a reasonable message the model in levels has to be equipped with a strong prior on θ in order to limit its range to the first mode of the posterior density of θ, say [32,120].

Revisiting the initial model

The posterior corresponding to the model is logs is uni-modal, delivering thus a single message and does not need a correcting prior. We report in Table 1 estimation results for the model using the log of consumption:

z j = s j α 1 + (1 -s j )(α 2 + γ 2 log(C j )) + ǫ j , (21)
using a non-informative prior and 5 000 draws. An estimate for a World Poverty Line is obtained by re-scaling the posterior density of α 1 , considering α 1 /365 * 12. A graph is given in Figure 2. A 90% confidence interval is [1. 05, 1.79]. This result is the similar to that reported in [START_REF] Greb | Dollar a day re-revisited[END_REF] who found a 90% confidence interval of [1.10, 1.72]. Of course the Bayesian posterior interval is slightly larger, just because and as underlined in [START_REF] Hansen | Sample splitting and threshold estimation[END_REF], the distribution of the estimated threshold θ is not standard (see Figures 1 and2) and thus using an asymptotic approximation as in [START_REF] Greb | Dollar a day re-revisited[END_REF] is not the right method to report the empirical uncertainty.

A Bayesian approach provides the small sample distribution of θ and thus allows to take into account uncertainty in the determination of an empirical poverty line in a rational way. Whatever the estimation method, (21) leads to the determination of a much larger reference group than that obtained in [START_REF] Ravallion | Dollar a day revisited[END_REF]. We have on average 38 countries when there was only 15 in [START_REF] Ravallion | Dollar a day revisited[END_REF] where θ is fixed and equal to $60. This leads to a slightly larger value for the poverty line.

A graph of the predictive density, as reported in Figure 3, suggests that the variance of the error term is not the same in the two regimes. A model with two variances is even more coherent with the theoretical model of [START_REF] Ravallion | Dollar a day revisited[END_REF] as in the first regime, the poverty line is supposed to be constant and the level of consumption rather low. We thus consider the

z j = s j α 1 + (1 -s j )(α 2 + γ 2 log(C j )) + s j ǫ 1j + (1 -s j )ǫ 2j . (22)
Table 2 validates the existence of two variances as the ratio between the variances of the two regimes is much lower than 1. α 1 is estimated in a much .20, 1.62] for the poverty line. However, the posterior density of θ becomes bimodal which leads us to look for a better model.

Preference drift

In the approach of both [START_REF] Ravallion | Dollar a day revisited[END_REF] and [START_REF] Greb | Dollar a day re-revisited[END_REF], the assumption is that for low income countries, the poverty line should have an absolute definition, which means that it is independent of income or consumption. When we look at Figure 3, we see that this assumption is not fully coherent with the data. In the first regime, the official poverty line seems to depend on the level of consumption, however with a much lower slope than in the second regime. We shall now estimate our preferred model (3) but including two variances so as to obtain:

z j = s j [α 1 + γ 1 log(C j )] + (1 -s j ) [α 2 + γ 2 log(C j )] + s j ǫ 1j + (1 -s j )ǫ 2j .
In Table 3, we have reported two versions of this model. Apparently we the countries that look like the country under investigation. We have computed the posterior mean that a country belongs to this reference group. We have 26 countries for which this probability is equal to 1: Bangladesh, Benin, Burkina-Faso, Cambodia, Chad, Congo-Rep, Ethiopia, Gambia, Ghana, Guinea-Bissau, Malawi, Mali, Mongolia, Mozambique, Nepal, Niger, Nigeria, Rwanda, Senegal, Sierra-Leone, Tajikistan, Tanzania, Uganda, Vietnam, Yemen, Zambia. In this first group the maximum consumption is $81 for Vietnam (to be compared to the $60 of [START_REF] Ravallion | Dollar a day revisited[END_REF]. With a 95% bound, we get a group of 38 countries where those 12 countries are added: Cameroon, China, Cote d'Ivoire, Djibouti, India, Kenya, Kyrgyz, Lesotho, Mauritania, Moldova, Pakistan and the Philippines.

How to simulate the posterior density of IP L

We have to find a posterior density for IP L, based on the first regime characteristics. It is obtained as a transformation of the parameters. We define IP L as being

IP L = γ 1 E(log(C j )|C j < θ),
in accordance with (4). In order to simulate IP L, we must have draws of γ 1 and θ. For each draw of θ, we determine the corresponding reference group and compute a value for the sample mean of log(C j ). The algorithm is as follows. We have kept draws from γ 1 and θ.

1. Start a loop in j 2. Given θ j , determine a sample separation and n j the sample size in the first regime

3. Compute IP L j = γ 1 j n i=1 log(C i )1I(C i < θ j )/n j 4. End loop
We get an IPL of 1.48 dollars a day with a standard deviation of 0.096. A 90% confidence interval is [1.32-1.64]. We have given in the previous subsection a list of 26 countries which had a probability equal to 1 to be included in the group of reference. We can compute the posterior density of the IPL when the reference group is limited to this group. There is no longer any uncertainty in the composition of this group. So the mean poverty line is of IPLs 1.39 dollars a day (0.086). We give in Figure 5 We have also added the posterior density of the poverty line using the first model in logs as defined in [START_REF] Greb | Dollar a day re-revisited[END_REF] as well as the poverty line corresponding to the approach of [START_REF] Ravallion | Dollar a day revisited[END_REF]. For this last option, we had to specify a prior information θ which was compatible with range of θ representative of the approach of [START_REF] Ravallion | Dollar a day revisited[END_REF]. From Figure 1, we have used θ ∈ [32, 120], thus eliminating the secondary mode. We get a mean poverty line of $1.26 (0.33), compared to the $1.25 found in [START_REF] Ravallion | Dollar a day revisited[END_REF].2

The four poverty lines presented in Figure 5 illustrate four different possible approaches.

1. The first poverty line of the plot corresponds to a fixed reference group.

That group is used to compute a reference consumption level, common to that group of countries (however determined by the model). The common poverty line is defined as proportion of this reference consumption level. Uncertainty comes from that proportion γ 1 . It corresponds to a sample -based prior for θ.

2. The second poverty line is slightly higher as it corresponds to a larger reference group, but it takes into account the whole uncertainty of the model. This is our proposed poverty line.

3. The last two poverty lines obey a different philosophy. They measure simply the mean of different national poverty lines, using a reference group which is determined by the model. In one case, we use a model formulated in logs, which was shown to correspond to a better specification. In the other case, we use a model specified in levels which corresponds to the initial model of [START_REF] Ravallion | Dollar a day revisited[END_REF].

With a poverty line of $1.25, following the data published in Chen and Ravallion (2008), we have 1.4 billion poor people in the developing World. With a poverty line of $1.48 as we found, this figure goes up to more than 1.7 billion and the headcount index passes from 25.7 to 31.5.3

Gauging official national poverty lines

In Figure 6, we have listed a group of countries, ordered by increasing level of private consumption with the latter being represented by a continuous line. The top group of horizontal lines indicates a posterior lower probability bound for θ which determines the corresponding sample separation, and consequently the reference group used for computing an international poverty line. The vertical lines indicate the national official poverty line of each country while the bottom group of horizontal lines correspond to a posterior confidence interval of our new international poverty line. We have skipped the upper part of the sample for clarity. From Figure 6, we see that Figure 6: Official poverty lines in the group of very poor countries, there are some countries which report an official poverty line which is both greater than our proposed poverty line and greater than their own PCE. It is difficult to find a rationale behind this, except suspecting the way these data are elaborated, in particular concerning the 2005 PPP.

There are emerging countries (members of the BRICS group), India and China which report an official poverty line which is well below either the new poverty line of [START_REF] Ravallion | Dollar a day revisited[END_REF] and of course of our proposed poverty line. There is also one country, Indonesia which belongs to the edge of the reference group (countries with a probability between 0.90 and 0.95 of belong to that group) that is not so poor and which report a national poverty line lower than ours. Finally, there is Tunisia which is not in the reference group and which report a very low poverty line. We give the list in Table 4. It is not surprising to find India in this list, if we remember the reference we made to the [START_REF] Tendulkar | Report of the expert group to review the methodology for estimation of poverty[END_REF] report. And the official poverty of China is even lower than that of India, despite the fact that it has a higher PCE than India. Using the figures provided in [START_REF] Chen | The developing world is poorer than we thought, but no less successful in the fight against poverty[END_REF], we can compute by difference the evolution of the number of poor for various levels of the poverty line in China. This will provide an explanation for the behaviour of those countries. With Figure 7, we see all the interest there is for an official agency to monitor the official poverty line. Poverty in China had dropped a lot between 1981 and 2005, but this drop was not regular. Adopting a poverty line of $1.25 or $1.45 does make a difference. At the beginning of the period the decrease in the number of poor is much stronger with the $1.00 poverty line than with the two other options. The increase in 1987 is steeper with the $1.25 line. At the end of the period the number of poor is roughly the same with the $1.25 and $1.45 lines, but of course it remains much larger than with the $1.00 line.

Conclusion and comments

Defining an international poverty line is an important objective, because it leads to measure the number of poor countries and of poor people in the world. Knowing these numbers and the localisation of the poor, it is easier to devise an economic policy and to evaluate the results of these policies later on.

We have seen in this paper that it is not an easy task to devise a poverty level. The one dollar a day line had to be reformed and [START_REF] Ravallion | Dollar a day revisited[END_REF] was a major attempt to do this. Their newly proposed poverty line is the lowest of the different poverty lines we have reviewed. But we have shown that a large uncertainty is attached to them. In fact, the posterior density of our poverty line covers all the other point alternatives. They all rely a different definition of a reference group.

The final poverty line we obtain (1.48 dollar a day) is larger than the 1.25 of [START_REF] Ravallion | Dollar a day revisited[END_REF]. But it is well in a reasonable confidence interval. Due to the way it is computed, it compels to the logic of a subjective poverty line.

The final point we would like to make concerns the Bayesian approach. We have used quite standard tools, even if they could seem complex for a reader not familiar with the field. With these tools we have visualised the origin of some questions concerning the model to be used and its specification. The posterior density of the break point θ was particularly useful in this respect. And finally, we could compare various assumptions concerning the determination of an international poverty line.

Figure 1 :

 1 Figure 1: Why a model in logs is better

Figure 2 :

 2 Figure 2: Rescaled posterior density of α 1 and posterior density of θ

Figure 3 :

 3 Figure 3: Posterior fit of the initial model in logs

Figure 4 :

 4 Figure 4: Posterior density of θ and posterior fit of the last model

  a graph of the posterior density of

Figure 5 :

 5 Figure 5: Posterior density of four poverty lines

Figure 7 :

 7 Figure 7: Number of poor in China

Table 1 :

 1 Bayesian inference for initial model

		Estimate std.error
	α 1	43.44	7.02
	α 2	-467.95	109.44
	γ 2	104.45	10.56
	θ	138.99	34.78
	σ 2	1425.70	246.09
	n	74
	Estimated IP L	1.43	0.23

Table 2 :

 2 Bayesian inference the two variance case

		Estimate std.error
	α 1	43.16	3.93
	α 2	-460.00	135.92
	γ 2	103.16	9.99
	θ	144.48	32.37
	σ 2 1 σ 2 2	312.06 2518.2	100.4 647.4
	φ	0.13	0.050
	n	74
	Estimated IP L	1.42	0.13
	more precise way with a standard deviation that goes down from 7.02 to 3.93,

Table 3 :

 3 Model with preference drift and two variances Estimate std.error Estimate std.error parsimonious parametrisation in the first regime and suppress the constant term if we include the log consumption. So the final model is that corresponding to the second panel of this table. Compared to the initial two variance model (22), we have a slightly larger variance in the second regime, but the first regime which is of direct interest to us is by far more precisely determined. log(pce) does have an influence in determining the official national poverty line, but its impact is ten times lower than what it is in the second regime.With equation (4), we have proposed a new type of absolute poverty line. It can be qualified as a subjective poverty line because it is computed as a function of the mean consumption of a group of reference, formed by

	α 1	-26.62	26.87	-	-
	γ 1	16.92	2.40	10.63	0.67
	α 2	-497.8	151.9	-496.2	142.9
	γ 2	110.2	3.85	109.1	3.81
	θ	172.0	19.29	167.2	18.24
	σ 2 1 σ 2 2	284.6 2780.4	69.93 737.0	297.92 2668.0	74.59 692.8
	φ	0.11	0.038	0.12	0.041
	must have a

Table 4 :

 4 Richer countries with a low OPL in 2005

	country	Prob OPL	C	Gini
	China	0.9738 0.85 120.78 44.9
	India	0.9992 0.90 84.24 36.8
	Indonesia 0.9316 1.07 139.96 37.6
	Tunisia	0.0000 1.35 240.63 39.8

It is very easy to compute the posterior density of a transformation of a parameter when we have posterior draws from this parameter. We just have to take the transformation of each draw as draws from the posterior of the transformed parameter. This task would analytically very demanding.

If we had restricted the prior range of θ to [32-60] so as to follow more closely the options of that paper, we would have obtained a poverty line of $1.23 with a larger standard deviation (0.44) and a reference group of 13 countries instead of 22.

In fact the figures given in[START_REF] Chen | The developing world is poorer than we thought, but no less successful in the fight against poverty[END_REF] are for a range of poverty lines which are $1.00, $1.25,$1.45, $2.00 and $2.50.

Appendix

Simulating from a bivariate density Let us consider a bivariate posterior density: π(φ, θ|y) = π(φ|θ, y) × π(θ|y)

We know the analytical form of the joint density π(φ, θ|y), but neither its marginal π(θ|y) nor its conditional π(φ|θ, y). We want to draw random numbers for the joint posterior density. To do so, we are first going to evaluate this bivariate density on a grid, filling a matrix F where the rows will corresponds to θ and the columns to φ. From this matrix of points, we can determine numerically the marginal density π(θ|y) by summing over the columns. Using this marginal density and using the inverse transformation method, we can draw a value for θ. For a given draw of θ, we have to find the corresponding conditional density π(φ|θ, y) as a row of matrix F . Of course, the draw will not correspond exactly to one of the predetermined point of the grid in θ. So we shall have to proceed by linear interpolation between two lines. 1. Compte numerically the cumulative and then use the inverse transformation method to draw θ j from π(θ|y)

2. Find the two nearest points of θ j on the grid of θ, denoted as θ j-and θ j+ 3. Calculate the differences: a = θ j -θ j-, b = θ j+ -θ j and c = |θ j+ -θ j-| 4. Obtain the conditional posterior densities π(φ|θ j-, y) and π(φ|θ j+ , y) from the joint posterior matrix (20)

5. Compare each point of the two above conditional posterior densities in order to get π(φ|θ i , y) by line interpolation: