Ulrich Horst

Alan Kirman

Member, IAS Richard B Fisher

Princeton

Miriam Teschl

Changing Identity: The Emergence of Social Groups

Keywords: Economic agent, social preference, social identity, personal identity, concept of change, social equilibrium JEL Classification: D01, D63, Z13 1

Homo Economicus has progressed from an atomistic and self-interested individual in standard economics to a socially embedded agent in modern economics who is endowed with a particular social identity or with specific preferences for the latter. While this vision makes the economic agent more realistic, its representation by adding variables in an agent's utility function poses problems. The distinction between the agent's own preferences and those that society imposes on the agent becomes both blurred, and difficult to make. In order to separate these two facets of preferences, we explore the idea of an agent's personal identity of which his social identity is one aspect. To consider personal identity means endowing the economic agent with the capacity to change, in particular, in the social space. This greatly complicates the problem of making welfare judgements since it is difficult to define a welfare improvement in the future. We next present a model in which the evolution of peoples' identities is stochastic. This can explain the appearance of individuals who seem to defy social convention. Our model shows how personal identity changes. We can find a social equilibrium only if personal identity matters little. But this suggests, paradoxically, that for individuals to have well-established identities, society must be one which is is not in equilibrium, but one which will continue to change and in which both the social groups and the individuals are constantly being modified.

Introduction

The typical Homo Economicus of modern economic theory has, for a long time, been thought of as an atomistic individual, unconnected to and unconcerned about other people and their behaviour and who is represented by a strictly private, i.e. personal and purely self-concerned preference ordering. More recently, however, the idea that Homo Economicus is a social being and thus not independent of other people's behaviour or of his cultural and social background, has become a serious consideration in economics. Different sorts of social preferences, i.e. preferences that have in one sense or the other a relation to the agent's membership in a social community have been attributed to him. He is now considered as caring about other people or about his social status in a given society. This status is seen as depending on his class-membership or social background, and he is now seen as having developed a preference for norm and rule following which lead him to take into account notions of fairness and to adopt pro-social behaviour. Yet, while this development has been a necessary step towards a more realistic depiction of the economic agent, it suffers from the drawback that in many models the individual has been fitted with an exogenously given social identity that simply shapes and influences the agent's behaviour without his being directly involved in any way. Thus, endowing the economic agent with social preferences has the disadvantage that it induces the agent to act -under certain circumstances -against his " personal preferences" without leaving him any choice. Our aim in this paper is to present a model that accounts for endogenous identity formation and which allows for the individual behaving as a member of a social community.but also gives him a choice as to which community to belong to.

Underlying our model is the observation that as the economic agent has become socially delineated in economic models, he has lost his sovereignty -yet the latter is a fundamental assumption of modern economics. That is, he became social but did not simultaneously develop the capacity to reflect upon and evaluate how acceptable to him were social influences on his behaviour. Thus, he is not sovereign with respect to the evaluations that he can make. However, he is also not sovereign in the sense of preferring whatever he thinks proper for himself as an individual. The social preferences his choices reflect might be in direct conflict with what he prefers as an individual. This is conceptually inconvenient and troublesome for at least two reasons.

First, in reality even if an individual has a social identity, which influences his behaviour, he is not totally defined by it. Indeed, the individual has the capacity to change and thus to shape and to influence his social identity. That is, an individual is a specific human being with a personal identity that is represented by his desire to be and to become who he wants to be.

Social identity is just one aspect of this individual's personal identity. Having a social identity does not necessarily preclude an individual from choosing another social identity. To do this he must be able to think and to reflect on whether his social and cultural background or his position within society is the most adapted to his personal endeavours. Thus, in order to restore the economic agent's sovereignty, we must develop a model of behaviour that depicts an individual who is embedded in society but is still capable of choosing to change, as the result of his own critical reflection.

Second, a given social identity that influences and shapes an individual's behaviour will make welfare evaluations more problematical if not impossible. Once again, if we do not have a way of distinguishing between an individual's personal preference and a preference that his social identity has led him to have, we can never be sure if the individual's preferences or the society's preferences as implanted in the individual will be satisfied. Thus, making sensible welfare judgements would seem to involve making a distinction between an individual's chosen or premeditated social preference and an individual's unquestioned social preference that he assumed for his social identity as a result of his place in society. Without this it seems difficult to say whether an individual is really better off or not and this undermines our capacity to make any judgements about social welfare. This, of course, raises a whole series of questions as to the nature of utility functions and the well-being that they are intended to representThis critique is reminiscent of Amartya's Sen's analysis of the "utility" maximising approach and which finally has led him to develop his "Capability Approach" [START_REF] Sen | Well-Being, Agency and Freedom. The Dewey Lectures 1984[END_REF]..

In this paper, we present a model that attempts to give a direct response to the concern about the agents' role in shaping his social identity. Indeed, instead of presenting an individual as having an exogenously given social identity or some presupposed social preferences, our model incorporates endogenous social identity formation and an evolution of the latter over time both at an individual as well as a social level. Individuals, represented by choice probabilities that depend on personal characteristics, will choose to belong to specific social groups ("types"), which have certain characteristics. The choice of the social group will modify the characteristics of the individuals, for example an individual may try to adapt his characteristics to those of his group.

However, when several individuals choose a particular type this will in turn, modify the type's characteristics over time, and this change may motivate individuals to revise their ideas and to move on and choose yet again, different types. The model shows that this society that consists of individuals and types will, under certain conditions, reach a "steady-state equilibrium" with a fixed number of types with established characteristics. However, although the characteristics of the types settle down in the limit the proportion of individuals associated with each type becomes fixed, the individuals continues to change types but choose each type with a given probability which no longer changes over time. This is true, if the individuals' choice of types depends in a "continuous" way on their current personal and type-characteristics and the overall availability of types in the society. This means that a society will be composed of a given set of social identities if when the personal identity of individuals changes a little,, the probabilities of choosing different types over time does not change too much, Put alternatively, individuals with similar characteristics will have similar probabilities of moving to different types. This condition indicates that social groups crystalise more strongly and are more stable the less sensitive an individual is to a change in his personal identity. Our model attempts to solve a problem of social identities, which by the logic of the individual's choice behaviour are likely to be chosen if the characteristics of these identities match with the individual's preferences. However, the latter are influenced by that selfsame social identity. Our model breaks this circularity by explicitly introducing a dynamic of change of individual as well as of social group characteristics. This is reasonable since there is no reason to assume that individuals and groups always remain the same However, once we allow social identity to become a matter of choice, the problem of evaluating welfare becomes mrore delicate.,How is it possible to say whether the individual reached his highest possible welfare level or not. if the social identity which governs his preferences was a matter of choice?

In the next section (2.), we will present the conceptual foundations of our model. In order to do this, we proceed by introducing three different concepts of identity, namely the what-, where-, and who-identity of the economic agent (Kirman andTeschl 2004, 2006) and these will provide the basis of our model together with a measure of continuity of an agent over time. Whereas the what-identity refers to the representation of the standard economic agent,the where-identity models of the economic agent have been developed over recent years and incorporate either directly or indirectly the idea that the economic agent is a social being who occupies a place in a social space and that this conditions his choices. We will review some of the models that account for a variety of individuals' social preferences and point out that while these models paint a more realistic picture of the economic agent, they do not, consider the individual's and society's capacity for change and thus, do not allow for endogenous identity formation. Finally we will introduce the idea of the who-identity by considering psychological and philosophical accounts of personal identity. This information, provides the groundwork for the model that will follow in the subsequent section (3.). The last section (4.) will conclude and outlines some of the possible avenues for future research on the basis of endogenous identity formation.

The What, Where and Who of Personal Identity

The standard economic agent, in particular as he is formally represented in general equilibrium models, is typically characterised as exclusively concerned about her personal and self-interested well-being. In addition to this hypothesis about his motivations, this Homo Economicus was considered to be an atomistic and socially isolated being, and this is derived from the implicit assumption that the agent's action did not have any influence on other people's behaviour and vice versa. If he did have social interactions they were beyond the control of the individual and taken as given. Homo Economicus is represented by a stable preference ordering over consumption bundles or consumption paths, and exogenously given budget-constraints 1 . The agent then simply chooses the best attainable bundle. This characterisation explained what the economic agent is, namely a rational utility maximiser, whose choices are automatically determined by her preferences which are unchanging over time and her budget constraint. We call this representation of the individual and the characterisation of behaviour the what-identity of the economic agent. It is a very simplified description of the economic agent, especially insofar as it relies on the rather audacious claim that an individual has a preference ordering over goods and actions for the present as well as for all future periods. Of course, underlying the idea of attributing a preference ordering to an individual is the idea that the individual is capable of choosing a best element of that ordering. Where this ordering comes from is not specified but many hold that its existence implies a consciously reflecting individual, who is able to evaluate different choices and to order them. This parsimonious model has been constantly amended and adapted to different choice situations that go far beyond pure market behaviour. Indeed, in the last few decades, more and more economists have come to accept that "man is a social animal and that his choices are influenced by the preferences and choices of others. An act of choice for this social animal is, in a fundamental sense, always a social act" (Sen 1973, pp. 252-3). This was a general recognition that economics could not abstract from the social environment in which production, exchange and distribution would take place. The social background was recognised to have specific influences on the individual and the individual was thought to affect, even involuntarily, other people's behaviour and beliefs through his choices. Individual action was no longer externality-free. The economic agent thus left his conceptual isolation and his preferences were no longer solely and exclusively self-concerned. The economic agent was thought of as caring about other people, about social and behavioural norms and rules, about fairness, his status, his personal characteristics and his distance to others in terms of these characteristics in a social space. Thus, information about where the economic agent is and in which social context he lives was introduced into economic reasoning. We call this the where-identity of the economic agent.

In most cases, economic agent's where-identity has been incorporated into a what-identity model. That is, the mechanism of choice, namely constrained utility maximisation, has been maintained and different aspects of an individual's social preferences and his dependency on other people's actions have been introduced as arguments into the utility function. A social interaction model such as Akerlof's (1997) Social Distance and Social Decision for example, presupposes that individuals occupy different locations in a social space that they partly inherited and partly acquired through their choices. They thus have, from the outset a given where-identity.

Individuals maximise their utility by choosing an action that affects their intrinsic preferences as well as their social position, i.e. where they want to be, by "interacting" with other individuals who equally occupy different social locations. Assuming that the benefits of interaction increase when interacting with people who are close to the individuals' position and that people form static expectations about the future position of their partners, the model shows that choices to interact with those who are socially close might outweigh choices according to individuals' "intrinsic" preferences. This eventually leads, according to standard criteria, to non-optimal economic outcomes, as individuals do not attain the welfare maximising equilibrium they would achieve if they focused solely on their individual preferences.However, the old problem as to what precisely what should be the welfare criterion arises here again. "Interacting" with those who are socially close might also have negative consequences. However, this does not preclude social interaction with them. Indeed, as Akerlof states: "The jealousy and envy of friends, relatives, and neighbors This also explains class stability or the existence of subcultures.

The conceptual idea of social distance is also kept in Akerlof and Kranton's series of paper on Economics and Identity (2000[START_REF] Olson | Personal Identity[END_REF][START_REF] George | Identity and the Economics of Organisation[END_REF]. They introduce an individual's identity-function into a standard utility function. This identity function represents an individual's social identity and epitomises the psychological idea that people act in order to guard or to achieve a sense of self. Akerlof and Kranton characterise identity as the social categories to which an individual belongs. This means that the individual is already provided with an explicit where-identity. These categories have behavioural prescriptions and each individual has a particular distance from these prescriptions depending on her own characteristics. The individual will choose effort levels or actions that fit and enhance the salience of her identity and thus sense of self. This means that the individual attempts to minimise the distance between her own characteristics and a socially determined self-image to which she adheres, in order to attain her highest possible utility.

Akerlof and Kranton indicate that the choice of identity might be one of the most important decisions, but note that "identity 'choice ', however, is very often limited" (2000, p. 726). That is, social categories are given and it is this, unexplained, feature that, provides society with a stable structure. Indeed, the choice of a category to which one does not naturally belong might violate other people's sense of self and incite these people to undertake punitive actions that are aimed at restoring their social self-image derived from their internalised identity-rules. The expectancy of punitive actions might thus prevent individuals from choosing categories other than their own.

Other models, such as [START_REF] Gary | Social Economics: Market Behavior in a Social Environment[END_REF] for example, consider how change is possible if a person has a particular place within the social structure. Their models shows that it is very difficult for individuals to increase the distance from their social group. Becker and Murphy introduce, as well as different goods social capital into an individual's utility function. Social capital represents social influences on utility and is supposed to be complementary to certain goods in the utility function. This complementarity is a key element in explaining "that social forces have a tyrannical power over individual behaviour" (p. 9). Assuming that the stock of social capital is the average of a specific good chosen by all members of the same group, the complementarity of the social capital with this specific good makes it clear that a change in an individual's idiosyncratic variable will not particularly modify this individual's behaviour concerning this good as long as the complementary social capital did not change either. Equally, when a variable that is common to all members of a given group changes, a single individual action will not alter social capital. Only the sum of different individual choices will affect social capital through the mechanism of social conformity and social interaction.

Conformity is, to a large extent, built into the above models, often because the individual has from the outset a specified where-identity. Other models, however, show that individuals end up with a specific where-identity influenced by their specific personal and social preference structure and thus without presupposing any particular place in society yet. [START_REF] Douglas | A Theory of Conformity[END_REF] for example introduces the idea that individuals care about social status or esteem in addition to their intrinsic preferences for standard consumption. This specific preference will, under certain conditions, give rise to conformity. Status should be conferred by a person's motives or virtues.

However, because individual proclivities cannot be observed directly, individuals signal to other members of a society their motives by choosing those actions that they think will convey a positive image. Conformity emerges if individuals put a sufficiently high weight on social status. They are then willing to suppress their individual preferences in order to adhere to a single homogenous norm of behaviour because they know that even small departures of the social norm will damage their status.

Where they are has thus been influenced by their individual preferences for social esteem.

Whereas this result only considers the case where all individuals value status and esteem equally, the model can also be adapted to the case where individuals care about the opinion of different population subgroups. In that case, several norms arise to which individuals adhere and thus lay the basis for subcultures. Norm following in general will be more manifest in those societies, where esteem and social status are more important. Bernheim notes that there is therefore scope for public policies: "If we assume that the government can, through publicity, make one value [...] more focal than another, then policy makers may be able to affect real decisions without traditional forms of economic intervention" (p. 865). The where-identity thus turns economists into social engineers even without economic tools. This, however, can also have unexpected consequences as Oxoby (2004) shows in a model that explains Cognitive Dissonance, Status and Growth of the Underclass.

The underlying idea is that individuals experience some psychological uneasiness (or cognitive dissonance) if they are unable to achieve what a social norm or status imposes (e.g. poor people cannot achieve the status associated with high consumption). That is, status-seeking individuals are concerned as to how they stand in relation to the average population regarding a specific norm.

If they fare much worse than others and if the relative value of their status in overall utility is high, they suffer psychological pain in terms of envy or disappointment. In order to reduce it, they engage in (costly) dissonance reduction by changing the social norm to which they adhere and thus modifying what they think to be status worthy. The underclass is thus created: poor people abandon the norm of attaining high consumption by reducing their effort and favouring the status derived from leading a leisurely and relaxed life (which brings them to the edge of criminality and welfare dependency). This model thus shows that making some norms more focal might indeed have negative consequences for the social welfare.

Where-identity models thus recognise the importance of individual as well as social characteristics and consider the interaction between people as a major ingredient of behaviour. It also becomes clear that including social aspects in the standard what-identity model complicates welfare assessment on an individual as well as social level. More specifically, it opens the perspective that not only might individual behaviour impair social welfare (as traditional economic theory would readily accept), but also that social behaviour may harm individual welfare. The latter aspect is, however, a controversial claim, since where-identity models still start from the assumption that it is the individual who has these social preferences. And if he does not reach the social optimal position, it is the externality of other people's behaviour that constrains in most cases the economic agent's welfare. The very existence of well defined preferences, has, however, not been questioned. Indeed, [START_REF] George | Economics and Identity[END_REF] do not presume, for example, that people are aware of their motivations and argue that "Whether or not [the individual] consciously realizes she is maximizing a utility function [that contains an identity function], she does so nevertheless" (p. 719, FN 6). Thus, one cannot even claim that reflection about the nature and origin of a preferences happened at the moment of forming one's utility function. This is certainly contrary to the idea of Homo Economicus being a sovereign individual. To restore the idea of sovereignty, it seems to be important to know if an agent's preferences are explicitly this individual's preferences given his social identity or if these are preferences that this individual is made to have because of his social identity. Assuming that the individual is not aware of motivations, as Akerlof and Kranton do, excludes any such distinction.

This calls for what Amartya Sen describes as Reason Before Identity (1999). His critique of the unthinking use of social identity can readily be extended to the adherence to any social norm and rule or "myopic" status-seeking that influences human behaviour. "In any explanatory exercise," he writes, "note has to be taken of local knowledge, regional norms, and particular perceptions and values that are common in a specific community " (2004, p. 91). However, this should not preclude any choice and reasoning about identity. Indeed, even if cultural beliefs and attitudes influence a person's behaviour and the nature of reasoning, it will never determine them fully. In fact, culture and respectively communities or social groups do not have one single set of defined attitudes, beliefs and norms. There is enough internal variation and difference that the individual has not to adhere to a unique set of norms but can choose between different ones, even within the same group. A person, furthermore, can always be seen as belonging to several groups and social categories and it is up to the individual to recognise her multiple membership and to decide if she wants to adhere to them, how much weight to put on them, or even if she wants to leave particular groups. A person's identity is thus, as Sen would say, a matter of choice.

Yet, as we will argue, this identity does not only depend on choice of the person's membership in different groups. Identity is also defined through those characteristics that make this choice possible. That is, choice depends on the individual's own characteristics, preferences and desires and the reflective relationship she engages with her surrounding according to who she wants to be. We will call this aspect the who-identity of the economic agent. This who-identity is inspired by the idea that an individual has not only a social identity, but first and foremost a personal identity that the individual shapes and creates throughout her life. This means that a person acknowledges the fact of her own change over time. It also means that there is a reciprocal relationship between the circumstances that influence individual decisions and individual choices that will change those circumstances in order to achieve who one wants to be.

Personal identity and a person's self are fundamental concepts in psychology and widely discussed in philosophy. Indeed, there is a long tradition in psychology assuming that a person has different facets of the self or different self-images. Psychologists such as James (1890), Cooley [START_REF] Charles | Human Nature and the Social Order[END_REF][START_REF] Sigmund | The Ego and the Id[END_REF]), Mead (1934), etc. famously distinguished between aspects such as the "spiritual self" and the "social self", or the "ego-id-superego" indicating that a person with personal aims and goals occupies not only a place in a social space, but also in moral and spiritual "spaces", which affect a person's aspirations as well as images of herself. Of course, occupying these different spaces not only guides a person's action but also involves the risk that the respective aspirations enter in conflict with each other, i.e. that the various images of the self are at a certain distance from each other and are not coherent. This causes psychological discomfort (or cognitive dissonance), which again reflect back on people's actions.

Modern versions of these "multiple selves" visions of the human being recall the idea of selfdiscrepancy (Higgins 1987) and indicate that a person possesses different domains of the self, namely the actual self, the ideal self and the ought self. These selves represent respectively the attributes someone possesses, would like to possess or should possess, either seen and evaluated from the point of view of the person herself or from that of another person. Inconsistencies or discrepancies between these self-concepts, for instance between the actual self of the person and her ideal self, or between the actual self of the person and the ideal self but such as another person ideally wants the person in question to be, will cause different amounts of emotional discomfort and uneasiness. The person will thus attempt to reduce these distances by changing the standpoint of evaluation, the domain of the self or some actions that restore coherence between these selves.

Hence, different selves or self-images serve as regulatory references, with the intention to reduce certain distances or discrepancies by moving the current state as close to a desired end state as possible [START_REF] Charles | Principles of Self-Regulation: Action and Emotion[END_REF] or by distancing the current state as far as possible from an undesired state or self-image [START_REF] Tory | Self-Regulation and Quality of Life: Emotional and Non-Emotional Life Experiences[END_REF]. This implies that people have a rather extensive knowledge about themselves and have the possibility as well as the intention to change over time [START_REF] Nancy | On Motivation and the Self-Concept[END_REF]. They work on certain life-tasks based on the knowledge of their preferences, abilities, acknowledgement of social roles and other personal characteristics. These life-tasks change over time and so will their goals and achievements. By pursuing these different life-tasks, people develop different images of themselves in the future. That is, they are guided by their imagination of their own future possible selves, which are cognitive representations giving a personalised vision of their overall motives. These possible selves thus represent the potential of change of a person and visualises who they could be and would like to be. However, while psychologists explore the cognitive representations and interactions of different selves that contribute to a person's sense of identity and thus motivate this person to change his actions or beliefs, philosophers have concentrated on the nature and essence of personal identity especially on the problem that a person changes over time, but his identity is claimed to be the same. The philosophical question of personal identity can be split in different, partially overlapping questions2 . It considers, for example, what it is to be a person. It furthermore asks what it takes for this person to persist. Philosophers were thus looking for an identity-criterion according to which a person at one time can be considered to be the same person at another time. Personal identity is also seen as representing and characterising what makes a person who he is. There is always something unique about a person. This it is often claimed, stems, at least in part, from the fact that he has a capacity of choice and can, at least marginally, influence the course of his life.

Thus, the philosophical account of personal identity attempts to provide models which reconcile persistence and change of a person over time, including the possibility of the person changing himself 3 There are several difficulties to overcome. First, if the person's identity were defined in terms of an identity-criterion that remains qualitatively the same throughout time, the person would, strictly speaking, lose his identity as soon as this criterion changes. We thus have to drop this strict identity relation and allow the criterion to change. However, if the criterion changes, we need at least some more or less enduring or suitably connected successive representations of that criterion in order to be able to talk of the same individual through time [START_REF] Derek | Reasons and Persons[END_REF]). Yet, we need more than simple connectedness if we wish to account for the individual's possibility to change himself. One way of doing this is to decouple personal identity into at least two identity criteria or functions one of which remains the same but is nevertheless responsible for the change of the other. [START_REF] Pierre | La pluralité cohérente des notions de l'identité personnelle[END_REF].

Applying the psychology and philosophy of personal identity to the economic agent by specthen undergoes an operation that successively replaces his memories by Greta Garbo's memories, when does Parfit cease to be himself and be Garbo instead? In this case it seems that personal identity is a question of degree and there will never be a precise answer. The line where Parfit ceases to exist can be drawn arbitrarily and may be a matter of convention. This is far from a new preoccupation in economics, since when Sidgwick (1907 pp. 418-419) was discussing utiltarianism he observed, "If the Utilitarian has to answer the question, "why should I sacrifice my own happiness for the greater happiness of another?" it must surely be admissible to ask the Egoist, "Why should I sacrifice a present pleasure for one in the future? Why should I concern myself about my own future feelings any more than about the feelings of other persons?" It undoubtedly seems to Common Sense paradoxical to ask for a reason why one should seek one's own happiness on the whole; but I do not see how the demand can be repudiated as absurd by those who adopt views of the extreme empirical school of psychologists, although those views are commonly supposed to have a close affinity with Egoistic Hedonism. Grant that the Ego is merely a system of coherent phenomena, that the permanent identical "I" is not a fact but a fiction, as Hume and his followers maintain; why, then, should one part of the series of feelings into which the Ego is resolved be concerned with another part of the same series, any more than with other series? [START_REF] Derek | Reasons and Persons[END_REF] was led to the conclusion that the question about personal identity is not so much what identity is, but what matters in identity. The idea is that there must be enough psychological connectedness between two successive moments of time in order to say that A is identical with B (or survives in B if B's body is different from that of A). An overlapping chain of psychologically connected time-slices of some human carrier guarantees continuity of a person even if the psychological features of that person at the beginning of the chain are not identical anymore with those at the end of the chain. A different solution to the above questions is to say that one criterion of identity cannot be responsible for both aspects of personal identity, i.e. persistence and change [START_REF] Pierre | La pluralité cohérente des notions de l'identité personnelle[END_REF]. Instead, the interplay of four different identity-functions (a person's body, her memories, preferences and social status), each in itself too weak to assure identity of the person, contribute together to the "functioning" and persistence of personal identity over time. Moreover, whereas the body and memory assure that the individual is, in some concrete sense, the same over time, preferences and social status guarantee the possibility of change, including of the person changing herself. If the consequences of some actions that were carried out on the basis of the person's social status enters in conflict with her personal preferences, the person will revise the priorities of her social status from the point of view of her personal preferences and vice versa.

While one identity-function is in the process of being changed, the other remains constant and it is this that guarantees persistence. Personal identity over time then is the overlapping chain of changing and persisting identityfunctions.

ifying who he is thus means not only explaining say what the economic agent is and where he is situated in a social and moral space, but also endowing the economic agent with a capacity for change. Thus introducing the who-identity of an economic agent means accounting for continuity and change of this agent through time.

The pure what-identity of the economic agent specifies that the individual is a rational utility maximiser with a given preference ordering. If we thus take this unchanging preference ordering as the criterion of personal identity, we trivially assure this individual's continuity but exclude any change other than by external constraints. However, if we think that an individual is not only determined by what he is, we have to introduce the possibility for the individual to choose to change. Thus, to account for an economic agent's identity, we need more than an unchanging preference ordering. If this ordering alone would represent his identity, then as soon as it changes, he would lose the basis of his identity.

The where-identity of the economic agent accounted for the fact that the individual has not only self-interested preferences, but also social preferences. On the one hand, these preferences are influenced by the position in the social space that he occupies, on the other hand the individual has preferences about the place in the social space that he wants to occupy. We will consider that place as being an identification with one of a number of possible social groups each with its own characteristics. A standard argument is that, although the individual is guided by two different kinds of preferences, they can both be included in the same utility function. But this simply suggests that there is a trade-off between the two and a more critical reflection about social influences in terms of reason before identity is excluded. Nor, is the possibility of a considered change of identity envisaged. The who-identity of the economic agent is influenced by the idea that underlying the individual is a dynamic of change. In the who-identity model of the economic agent that we will present in the next section, preferences are grounded in the agent's characteristics and because his characteristics change over time, so will his preferences. In a general sense, we can think of an indivdual's preferences as being conveyed by his choice probabilities (p xt ), which thus stand for what the economic agent is at each moment of time. Hence, we call choice probabilities the what-criterion of the economic agent's identity. Furthermore, the economic agent is living and acting in a social world, thus occupying a place in a social space. However, he will choose where this place should be by selecting his participation in certain social groups according to his preferences. Social groups themselves have specific characteristics and it is in fact, not the group that the economic agent is choosing, but the characteristics of the group.

One vision which is discussed by Akerlof and Kranton, is that an individual can use the characteristics of the group as a means to reduce the distance between the agent's current characteristics and those of some self-image that the agent wants to realise. The characteristics of the social groups (or types as we call them) are the where-criterion of the economic agent's identity (x t ).

The choice of social groups will thus change the economic agent since he will acquire new characteristics. However, the relationship is reciprocal: the agent's characteristics that he brings to the social group will affect the characteristics of the social group. Moreover, supposing that there are several individuals with different characteristics who choose the same social group, this social group will be changed by the choices of these individuals. This change in characteristics though, will motivate the agent, once again, to revaluate his previous choice of that social group. Hence the individual will make a new choice whereby he decides on the basis of his preferences either to stay in the group or "to move on" and to choose to participate at a different group. Thus, the idea of a dynamic change of the individual through time is matched with the idea of a dynamic change in the social structure. The probability of having certain characteristics and types and ending up with new characteristics in new types is the who-criterion of the economic agent's identity and it represents the idea of change through time, based on both, the individual' choices and the social change (π xt ). It symbolizes the possibility to change oneself through one's own choices. Evidently, given the stochastic nature of the agent's preferences, the who-criterion gives a picture of the possible selves that the agent may become given his current characteristics and group-memberships. However, while the who-criterion indicates the possible change of the agent from one time-period to the next, or, stated differently, his preference for the future, it does not yet tell us anything about the different connections between the agent's changes over time. For this, we need to introduce a measure of continuity (β) of the agent's personal identity. This measure consists of the sum of two different parameters (β 1 and β 2 ), one of which concerns the agent's connectedness to his past, the other his responsiveness to varying social opportunities. They can also be seen as a measure of personal connectedness and of social connectedness of the agent over time respectively. The former (β 1 )limits the change in the agent's new group-memberships (or types) and characteristics that can result from a small change in his current characteristics and group-memberships (or types). The latter (β 2 )analogously limits the change in the agent's future characteristics and types that can occur as a result of a change in the currently available groups in a given society. This indicates that where the agent is coming from and who he was in the past influences his future changes. The agent can thus be traced through time because one state of characteristics and types leads continuously to the other. A high responsiveness to varying social opportunities means that the agent's new characteristics and types depend strongly on which social groups are currently available. In this case, the agent can be followed through time according to the changes in his social environment to which he reacts. Thus, to sum up, personal identity is represented by the who-criterion that can be decoupled into two criteria, the what and the where-criterion that in turn constitute the economic agent's change over time, and by his measure of continuity that consists of his personal and social connectedness over time. In this sense, continuity and change contribute together to give an account of who the agent is.

What are the consequences and results of these individual and social dynamics? In what sense and under what conditions does the structure of the society stabilise? Our model shows that society will reach a stationary state if two conditions are satisfied. Society stabilises if modifications in the agent's state today do not have a radical effect on his future state (small β 1 ) and if the agent's choice of a future state does not depend too much on changes in the types available (small β 2 ). The latter is reminiscent of the "weak interaction" condition in [START_REF] Ulrich | Equilibria in systems of social interactions[END_REF]. In the long run the characteristics of the social groups and the proportions of individuals choosing them converge. This means that on a macroscopic level, the society no longer evolves. On a microscopic level though, individuals will continue moving from one group to the other and thus changing their characteristics. In the limit, individuals will move from one group to the other with the same probabilities which are reflected by the proportions of individuals in each group. In a sense, in the limit the agents lose their individual identity.

The result is surprising and stands in contrast to many models developed previously whereidentity models. Indeed, in those models, society stabilise because individuals have a well defined identity in terms of their preferences over personal and social outcomes, which are unchanging.

Furthermore, society stabilises even when individuals are very sensitive to social influences. In our model however society stabilises because individuals are relatively flexible, that is they are not too sensitive to who they are not to changes in the available groups. It can happen, as we will see in the next section, that a situation of multiple long run outcomes can occur, even if a certain degree of connectedness, or continuity, of a person is allowed. Such outcomes are dependent on initial conditions. Such difficulties arise, insofar as individuals have the capacity to choose and to influence the course of their life and in so doing modify the social structure that allows them to become who they want to be. In that sense, the most interesting society to live in is the one which is not stable.

An example

Why does our model involve stochastic rather than deterministic choices? It is clear that this plays an important role in the model for the evolution of the society or economy in which the agents live.. In the first place, individual choices will be conditioned by many factors which cannot all be incorporated into the model and the idea of using probabilities reflects the idea that there is some "noise" in the system which represents the heterogeneity of the agents and which is not directly observable by the modeller.

Another advantage of this approach is that it allows for some remarkable personal changes to take place. One finds occasionally individuals who despite the highly constraining nature of the social or economic environment in which they live, manage to develop a life which is not at all consistent with their background. In our framework these constitute low probability events but which, inevitably, happen from time to time. Consider the case of Cornelia Sorabji, a Parsee woman converted to christianity who was the first woman to obtain a law degree at Oxford and went on to become a distinguished barrister and a militant human rights advocate. Her career would be thought of as a choice in our context albeit an unlikely one. Indeed Amartaya Sen (2001) says of her: "She chose her plural identities influenced by her background, but through her own decisions and priorities. In the last respect, she was not unique, despite the uniqueness of her chosen combination of identities." This sort of switch is possible in our model and it will have an important effect on the evolution of the individual but, also, on the evolution of the social groups that make up society.

Searching for Identity

We now present the basic characteristics of our model. Firstly we specify the set of possible characteristics that an individual agent can have. We consider the characteristics of any individual as being represented by a vector in R n , though we might want, in general, to consider some more elaborate space. In particular, we think of the possible vectors of characteristics as being finite in number. Furthermore social types are also represented by vectors in the same characteristics space. One could think of each "social type" as having a "representative" whose characteristics correspond to the image of that type. At each point in time, the agent will have certain characteristics and will have chosen a type. The choice of the type constitutes the answer to the where criterion for the individual. The types themselves will, in our model, change depending on who has chosen them. Just as the image of the Republican party is modified as the people who declare themselves to be members or to vote Republican change. Furthermore the agents own characteristics will be modified over time, in part, for example, as the result of his choice of social type.

More formally: Let C = {c 1 , . . . , c l } be a finite set of characteristics and X ⊂ R n be a compact convex set of types, C ⊂ X. We denote by

x t := (x 1 t , . . . , x m t ) ∈ X and c t = (c a t ) ∞ a=1
the vector of possible types and the configuration of characteristics at date t ∈ N, respectively.

As we have said, this vector represents the agent's where-criterion and stands for the different where-identities from which the agent can choose.

Given his own characteristics c a t and the current vector of types x t , the agent a ∈ N selects a type x j t with a certain probability. We assume that, for the purposes of our model, information about the agents current characteristics and group completely defines them. This means that agents in exactly the same state today will have identical choice probabilites. This does, of course, not mean that they will make the same choice. The choice probabilities reflect the "preference" for a certain type. One might, for example, think of individuals as being more likely to choose a type which is close to their own characteristics or alternatively agents might have ambitions to change their social image. It suffices that agents have different choice probabilities. More precisely, the conditional distribution of agent a's type in period t takes the form

p xt (c a t ; •)
for a transition probability p from C × X to X where p xt (c a t ; •) is concentrated on the set X t := {x 1 t , . . . , x m t }. This choice probability depends on who the agent is now, that is, the what-criterion of the agent's identity and represents the "preferences" of the individual over social types. Now, at each point in time the social types evolve as a result of the choices of the individuals.

For technical reasons this is inconvenient since the set of types at each point is not the same.

To avoid this we can think of individuals as choosing from m "labels", even if what those labels represent is changing. Thus, the Republican party's characteristics change but agents can still choose to vote Republican.

Hence, we will label the types by 1, 2, . . . , m, and rather than choosing types, we assume that the agents choose a label y a t ∈ Y := {1, 2, . . . , m} with probability pxt c a t ; y a t+1 = p xt c a t ; x y a t+1 t .

After all the agents have selected their respective types or labels, their new characteristics are then determined stochastically. Thus the evolution of the agents is stochastic and not deterministic.

The choice of characteristics is made according to the probability measure pxt c a t , y It represents the idea of how an agent changes and the agent's preference for change. An agent changes both because his own characteristics, his what-criterion and -as we will see shortlythe characteristics of his group, his where-criterion, will be modified over time. Because of this, an individual always has a probability of revising his choices and may consequently either stay with the same type or "move on" and identify with a different type. Notice that, if we want to obtain specific results concerning the evolution of types we will have to put a lot of structure on the transition probabilities and then to justify that structure in economic terms. However, our overall aim here is to see under what general conditions the system has some sort of stability at the aggregate and individual level.

For the moment we assume that the agents act conditionally independently of each other,

given the individual characteristics and the current vector of types. This means that for any given sequence of types {x t } t∈N the dynamics of the process {z t } t∈N is specified in terms of the product kernel

Π xt (z t ; •) := ∞ a=1 π xt (z a t ; •).
In order to study the asymptotics of both individual and aggregate behavior, we denote by M the class of all probability measures on C × Y and by

t := lim n→∞ 1 n n a=1 δ z a t (•)
the empirical distribution associated to the configuration z t provided the limit exists in the weak sense, i.e., for any bounded function

f : C × Y → R, it exists f d t := lim n→∞ 1 n n a=1 f (z a t ).
Remark 4.1 Since C and Y are finite sets, t may be viewed as a vector t = ( i,j t ) (i = 1, 2, . . . , l, j = 1, 2, . . . , m) in R lm where i,j denotes the fraction of agents that choose type x j t and characteristic c i .

The long run dynamics of aggregate behavior is described by the asymptotics of the sequence { t }. What we are interested in is whether the types settle down and also whether the proportions of individuals choosing those types stabilizes over time.

By the law of large numbers of independent random variables,

t+1 = π xt (z a t ; z a ) t (dz a ) := H( t , x t ) Π xt (z t ; •)-a.s. (1)
Thus, for any given sequence {x t } t∈N the evolution of the process { t } t∈N can almost surely a described by a deterministic recursive dynamics.

Now, what we want to look at is the evolution of the types over time and how this is influenced by the movement of individuals between the types. For this we have to make assumptions about the way in which the characteristics of types are modified by the changes in the choices of the individuals. To obtain a first simple characterization of the dynamics of the sequence {x t } t∈N , we assume that types are modified as a function of average behavior. In the first place we restrict ourselves to a simple linear updating rule for the dynamics of types. This leads us to make the following assumption.

Assumption 4.2 There exist smooth functions F : U → X, F = (f 1 , . . . , f m ), defined on the compact convex U of all probability distributions on states and a constant α ∈ (0, 1) such that

x j t+1 = αx j t + (1 -α)f j ( t+1 ) and so x t+1 = αx t + (1 -α)F ( t+1 ). ( 2
)
We illustrate Assumption 4.2 by means of the following example where types are simply points on the unit interval and the way in which they change is determined by the fractions of individuals choosing them.

Example 4.3 (i) Suppose that x j t ∈ [0, 1] and that new types are convex combinations of the old type and the proportion of individuals choosing them. In this case

f j ( t+1 ) = 1 j (y) t+1 (dc, dy) = lim n→∞ 1 n n a=1 1 j (y a t+1 )
In view of Remark 4.1 we may view t+1 as a vector ( i,j t+1 ), and so

f i ( t+1 ) = n j=1 i,j t+1 .
(ii) The new types could also reflect the characteristics of those who selected them. In such a situation new types could be given as a convex combination of old types and the average characteristics of the members:

f j ( t+1 ) = c1 j (y) t+1 (dc, dy) 1 j (y) t+1 (dc, dy) .
In view of Remark 4.1 we have the following representation:

f j ( t+1 ) = l i=1 c i i,j t+1 l i=1 i,j t+1
.

(3)

In this case Assumption 4.2 requires that an agent chooses each state with strictly positive probability.

In view of (1) the joint dynamics of types and empirical distributions follows almost surely the deterministic dynamics To this end, we denote by | • | 1 the l 1 -norm and recall that

t+1 x t+1 = H(x t , t ) αx t + (1 -α)F • H(x t , t ) =: G(x t , t ) (4
1 2 |π x (z; •) -π x (ẑ; •)| 1 = 1 2 z |π x (z; z) -π x (ẑ; z)|
defines the total variation norm of the signed measure π x (z; •) -π x (ẑ; •). In terms of the total variation norm we can now introduce our weak interaction condition.

Assumption 4.4 There exist constants β 1 and β 2 such that β := β 1 + β 2 < 1 and

1 2 |π x (z; •) -π x (ẑ; •)| 1 ≤ β 1
uniformly in all the possible types and states, and

1 2 |π x (z; •) -π x(z; •)| 1 ≤ β 2 |x -x| 1 .
The coefficient β 1 measures the dependence of an agents' new type and characteristics on his current state z a t = (c a t , y a t ). In the special case where where agents choose their new states independently of their previous states,

π x (z; •) = π x (ẑ; •),
and we can choose β 1 = 0. This means that the agents have no personal connectedness and that they are non-continuous with themselves. That is, they do not care where they came from and what they choose is independent of their own past history. If, on the other hand, the agents "remain who they are" in the sense that π x (z; z) = 1 and π x (ẑ; z) = 0 for some ẑ = z we have that β 1 = 1 and our weak interaction condition is violated. This means that agents have a strong personal connectedness and that they stick to their previous decision and remain where they are. In the case of a high β 1 value, agents are continuous in terms of their personal connectedness and can be traced through time because one state of characteristics and types (can) depend on their previous one 4 . However, even in the case where people choose to remain the same, this does not mean that agents do not, in fact, change. Indeed, the type to which they belong has been changed, but this has not motivated the agent to move on and to select a different type.

The constant β 2 , on the other hand, places a quantitative bound on the dependence of z a t+1 on the current vector of types x t . A low β 2 implies that the agents' preferences for future states given their current states do not change much if the available types change a little. This means that agents do not react strongly to changes in available types, or, stated differently, that their preferences depend continuously on types. In that sense, the agents' social connectedness is very low. A high β 2 implies that agents can react very strongly to changes in available types and underlines the fact that the agents' social connectedness is strong. In this case, agents are continous in terms of their social connectedness and can be traced through time because their states in terms of characteristics and types (can) depend on previously available types 5 . Thus, the measure of continuity tells us that if β 1 is high and β 2 is low, personal identity is guaranteed through the agent's personal connectedness between his different states, and if β 1 is small and β 2 is high, personal identity is guaranteed through the agent's social connectedness to different states. That is, whereas in the first case we have a strong path-dependency between the different states of an agent over time, in the latter case, the individual's path over time depends 4 Note that β1 is a bound placed on the difference between two different transition probabilities πx. This means that in the case β1 has a low value, the two transition probabilities πx cannot be too different. In the case where β1 has a higher value, the two transition probabilities can be different, but do not need to be. Thus, in the latter case, there is a higher chance that an agent will end in a particular state, given that he was in a particular previous state.

5 As in the case of β1, β2 places a bound on the difference between the two transition probabilities πx and π x if the distance between the two vectors of types x and x is not too large. If the latter distance would be large, then even a very small β2 is unable to impose any restrictions on the transition probabilities and we cannot say much about this situation. However, again, whereas a small β2 implies that the two transition probabilities must be almost similar, a higher value of β2 tells us that the two transition probabilities can be different, but do not need to be. Thus, in the latter case, there is a higher chance that agents will end in a particular state, given that previously there were some particular types available. Most evidently, if the distance between the two vectors of types is very small, but β2 rather high, the difference between the two transition probabilities needs also to be small.

on the changes in his social environment. In that sense, we might say that the sense of personal identity stems from the idea of personal connectedness. Social connectedness bridges the different time-periods of a person only through changes in available types.

However, if both values are as small as the weak interaction condition requires, the agent is continuous neither in terms of a personal connectedness nor in terms of a social connectedness.

This means that an agent's preference for a future state is not only independent of his current state, but also of the choice between different available types.6 Thus, loosely speaking Assumption 4.4 says that agents do not put too much weight on changes in current states and types when revising their choices, and that preferences depend continuously on types.

Our weak interaction condition on the agent's individual behavior also constrains the macroscopic behavior of agents. The future distribution of agents over types is continuously dependent on the current distribution of agents and the current vector of types. That is, the weak interaction condition constrains the dependence of the future distribution of agents on the current distribution and the available types. Stated differently, if we had two almost similar but still different current distributions of agents and available types, the future distribution of agents in both "worlds" would be almost the same.

We are now ready to state our convergence result. Its proof is given in the appendix.

Theorem 4.5 Suppose that Assumption 4.4 is satisfied and that Lβ < 1.

a) The sequence {(x t , t )} t∈N of types and empirical actions converges almost surely to the unique fixed point (x * , * ) of the map G.

b) The stochastic kernel π x * has a unique stationary measure µ and the sequence of individual states converges in probability, i.e., for all states z,

lim t→∞ P[z a t = z] = µ(z)
The first part of the theorem states that in the long run types settle down to a unique limit

x * . Asymptotically, the agents new characteristics is thus chosen according to the transition probability

π x * (z a t ; •),
and in the long run, the full dynamics of the microscopic process {z t } t∈N is described by the stochastic kernel

Π x * (z t : •) = ∞ a=1 π x * (z a t ; •).
In view of (1), the transition law π x * has a unique stationary distribution µ and, for any initial condition z 0 , the distribution of individual states converges weakly to µ. In this sense the dynamics on the macroscopic level of types and average actions settle down to a unique deterministic limit (x * , * ) whereas the dynamics on the microscopic level of individual behavior settles down to the probabilistic limit µ. In equilibrium therefore, the characteristics of types' will not change anymore but each type will continue to be constantly chosen by a given fractions of agents.

Agents themselves will always move from one group to the other according to given probabilities.

Depending on the higher or lower entry-values of the probabilistic limit µ, agents will move more slowly or hop rather quickly from one type to the other. At each time, however, the individuals' probabilities of being in one of the future states do not depend much on changes in their current state or in available types. During each step, the agent assimilates to a greater or lesser degree the characteristics of the respective group. Hence, in equilibrium, agents are left with a social identity but one that changes according to when the agents will be where.

What would happen if our weak interaction condition of an agent's individual behaviour that led to the equilibrium is violated? Let us assume, for example, that an agent's preference does not continuously depend on available types, i.e. that his preference change radically in response to a small change in the types available. In that case, the individual's personal identity is largely determined by his social connectedness. In this situation we would be able to find multiple long run outcomes, and which of these will occur will, however, depend on the initial conditions.

Example 4.6 Suppose that there are two characteristics,

C = {-1, +1}, two types, Y = {-1, +1} and that x -1 t ∈ [-1, 0] and x +1 t ∈ [0, 1].
Let us assume that agents choose their new characteristics in reaction to current characteristics and types and let t = ( -1 t , +1 t ) be the empirical distribution of characteristics at time t. Types are updated in reaction to the proportion of people choosing the corresponding characteristics:

x ±1 t+1 = αx ±1 t ± (1 -α) ±1 t+1 .
Loosely speaking, the evolution of types reflect the dynamics of average characteristics. Let us further assume that an agents a sticks to his characteristics c a t if the majority of agents choose c a t . Otherwise he selects a characteristic with equal probability. More precisely,

π xt (1; 1) = 1 if x -1 t + x +1 t > 0 0.5 otherwise and π xt (-1; -1) = 1 if x -1 t + x +1 t < 0 0.5 otherwise.
Thus, the individual transition probabilities do not depend continuously on the current types. If

x -1 t + x +1 t > 0, the law of large numbers for independent random variables yields

+1 t+1 = +1 t + 1 2 -1 t while -1 t+1 = 1 2 -1 t and x -1 t+1 + x +1 t+1 > 0.
As a result, "initial conditions matter." The long run evolution of empirical characteristics depends on the starting point (x +1 0 , x -1 0 ) of the sequence of types. For instance, if x -1 0 + x +1 0 > 0, then

+1 1 = +1 0 + 1 2 -1 0 while -1 1 = 1 2 -1 0 and x -1 1 + x +1 1 > 0, so that +1 1 = +1 0 + 1 2 -1 0 + 1 4 -1 0 while -1 2 = 1 4 -1 0 and x -1 2 + x +1 2 > 0.
Thus, an induction argument shows that -1 t → 0 and +1 t → 1 as t → ∞. By analogy,

-1 t → 1 and +1 t → 0 as t → ∞ if x -1 0 + x +1 0 < 0. Overall, lim t→∞ +1 t =        1 if x -1 0 + x +1 0 < 0 0 if x -1 0 + x +1 0 > 0 0.5 if x -1 0 + x +1 0 = 0.
In this case, we can interpret the economic agent as being endowed with a specific set of preferences, whereby he prefers to stick to his previous decision if a majority of people choose in the same way as he does. The agent thus feels comfortable with himself as long as his decision is consistent with a majority of other people. If his choice is not consistent with that of the majority, the individual starts doubting his previous decision and becomes indifferent or undecided in terms of his preference in relation to his future behavior. This corresponds to a preference for conformism. The latter implies a social connectedness of the individual through time in terms of current available types. Thus, depending on which current types exist, the individual will behave differently. To put it in another way, his future state depends on current available types. This situation indeed resembles to some extent to the previously considered "social identity models".

However, the model at hand shows that a society can structure itself differently, i.e. have different equilibria, depending on the initial conditions. These initial conditions not only refer to the initally available groups, but also to the specific preference the individuals have. Of course, this is a special case and modifying it slightly would mean that an equilibrium cannot be guaranteed.

Again recall that the choice probabilities here depend only on the current state. In fact, we might imagine individuals' preferences to change if their connectedness through time not only depends on available groups, as in this case, but also on their past states and thus personal history.

Finally, we give an example to illustrate the fact that individuals do not necessarily settle down if β 1 = 1 even if the empirical distribution of characteristics is constant. This is a very simple example that shows that if a person's evolution is path dependent, they will have different preferences for, or probabilities of choosing different characteristics at different times.

Example 4.7 Let us assume that there are only two characteristics, C = {-1, +1} and that the agents do not react to types. If the individual transition laws take the form π(±1; ∓1) = 1, then β 1 = 1. Let us also assume that -1 0 = -1 1 = 1 2 . In this case, the empirical distribution of agents' characteristic does not change through time, but the probability of an agent choosing a certain characteristic does not converge because, for any t ∈ N any all agents,

P[c a 2t+1 = -1|c a 0 = +1] = 1 while P[c a 2t = +1|c a 0 = +1] = 0.

Discussion

Our model differs from previous models of identity insofar as it considers not only the aspects of people's social identity and thus of adding where they are to what they are, but goes further and tries to explain who they are in successive time periods. the idea is to add a further dimension to personal identity of which social identity is one particular aspect. Accounting for a person's who-identity focuses specifically on considerations about the change and the continuity of a person and of society over time.

There are two major differences between our model and social-identity-models. Here society stabilises neither because people are already situated within certain social groups which are fixed from the outset and thus act according to their norms and rules, nor because people develop a preference for specific norms and rules that then become embedded in specific social groups.

What happens here is that groups settle down to well defined characteristics because the members are not too sensitive either to slight modifications in their own identity nor to modifications in the social types available. An individual who was extremely sensitive to changes in his own characteristics and thus reacted strongly to such a change induced by some modification in the social groups would be very unstable. Thus, the dependence on the social structure and on own characteristics must not be too strong for society's structure to stabilise.

As, we have seen that if agents react strongly to variations in the various types available, then,

given their preference for social conformism, several long run outcomes of the society are possible, depending on the initial conditions of that society. This case resembles a number of social-identitymodels considered previously in the literature. However, the example shows clearly that it is the social connectedness of agents over time which becomes central in this case. Yet, this is a rather weak ingredient of the overall vision of personal identity that we wish to present, because it reflects the simple idea that people essentially change as a result of external modifications. Yet, given the overall dynamics of the general model, we have no reason to assume that individuals will not themselves, modify their preferences over time.

More importantly, if personal identity matters in terms of people's preferences that express a dependence on where they are coming from and who they want to be, i.e. if we take account of the personal connectedness of individuals, society might not become stable at all. This might sound counterintuitive, but it is not if we look at the dynamics of the model that relies on the simple idea that social groups affect people's characteristics, but people's characteristics also affect the outlook of social groups. If there is indeed an element of independence between social and individual preferences and people develop a particular preference for a social group because they want to adopt its characteristics and not because people want to adapt their characteristics to that of the social group, then we can conceive of a situation in which there is continuous change and movement of individuals. Each time they have the impression that they would need to start adapting to a social group instead of freely adopting certain characteristics of that group in order to fit to the group, they may decide to move on.

That is, people, by joining groups, are bringing their different characteristics to them and thus modify the outlook of those groups over time. This change in characteristics of the group might make people reluctant to continue staying in a particular group and motivates them to look out for different groups whose characteristics correspond more to those with which they feel less dissonance. Thus, the model shows why there can be situations in which a person's embeddedness in a social context or a person's holding of a preference for a particular place in society does not imply that she loses her capacity for change and that society's dynamics will settle to a steady state. In fact, it is not intuitively obvious that society's structure will stabilize in this way and individuals have to live with only a limited number of choices of who they can be. In the case where it does happen, we might say that society's preferences come to be more important than those of individuals.

It is clear that for the moment , the model so far is only a first rudimentary approach to the problem. Several issues are not yet taken enough into account in First of all, individuals do not have a well specified rationality. They are simply endowed with choice probabilities whose motivational origins we did not specify. However, even in this very general framework we were able to obtain some idea of what conditions might lead to individual choices and society's structure stabilising. if we wish to obtain more specific results, it is clear that people's transition probabilities, or what we called an individual's who-criterion, need more detailed specification.

Thus, the question is why people with particular characteristics and group-memberships end up with those characteristics and memberships.

The extension of the model that we intend to work on is the problem of "self image". We might imagine that people have particular self-images to which they would like to correspond and such images, could equally be described in terms of particular characteristics. Then people would choose those groups whose characteristics are most similar to those of their self-image. Given that group-characteristics change over time, this creates a distance between their self-image and the perceived characteristics of the group to which they have chosen to belong. To reduce this distance (or internally felt discrepancy or cognitive dissonance), individuals continue to move on and choose a different group that is closer to their self-images. However, it is also clear that, in certain circumstances, it might also be the self-image that changes. Indeed, one might conceive that a change in the perceived characteristics of the social group introduces new characteristics (or information) that induce the individual to reconsider her choices and aspirations. Thus, additional information might contribute to a different perspective of who one is and who one wants to be. The person than might either continue to change her characteristics in order to achieve her self-image or change her self-image and continue to be who (and where) she is. In that case, conditions must be found that differentiate between people adapting to a given situation by reducing their aspirations or adopting a new situation and thereby invigorating their desires. Whereas adaptation means to conform to given possibilities, adoption would imply creative involvement with new opportunities.

That this difference matters has long since been pointed out by Amartya Sen, who consistently raised awareness about those unfavourable social and economic circumstances to which people adapt and live through with "cheerful endurance" (Sen 1984, p. 309).

The same observation was made by George Bernard Shaw when he said, "The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself.

Therefore all progress depends on the unreasonable man". Man and Superman (1903) Maxims:

Reason

Another more realistic feature that we might wish to include in the model is to remove the assumption that individuals move freely from one group to the other. It is obvious that in the real world, people might be barred from entry to and obstructed from exit of social groups and this fact indeed acts as a brake on any change. However, while this is true regarding particular social groups, classes and access to different social facilities and is certainly interesting to analyse, our model does not need to be thought in terms of specific existing groups only. Indeed, the implicit set of what we call "types" is larger than that of existing social groups. "Types" can also be seen as objects, actions and even norms that are describable in terms of characteristics and whose feature it is that they are shared by different people. There are obvious if rather trivial cases such as for example that of specific sport-shoes which have been used first for the purpose for which they were intended, but because pop-stars and other people started using them, their characteristics changed over time and thus also the people who wear them. In that sense, our model is really intended as a way to think about continuity and change of a person choosing particular groups, objects and actions to create her personal identity over time.

Another aspect that is worth considering is that the model does not differentiate between the different ways in which the characteristics of social groups can change. Indeed, it might be claimed that certain social groups attract from the outset quite conformist individuals whose characteristics do not differ much from that of the group. In such cases (e.g. religious groups or political parties), the potential for, and rapidity of ,change of this group is much less compared to others, which attract a quite diverse range of people with varying characteristics (e. While these are certainly limitations, we think that the substantive contribution of the model is to provide a different view of the social embeddedness of an individual . Indeed, whereas social-identity models assume strong individual preferences for conformism and social status that contribute to the stability of society, we envisage an endogenous social identity formation on the basis of personal identity considerations, i.e. a continuous and connected change of personal and social characteristics over time with a strong feed-back between the two.. Under these assumptions, we see that society only settles down if individuals are sufficiently malleable and rather than pursuing an individualistic course, adapt, at least to some extent, at each point in time to the characteristics society imposes. This, on the one hand, is indeed in sharp contrast to socialidentity models, which assume that individuals who have strong individual preferences depending on their social group. On the other hand, in many cases, social-identity models do not consider any dynamics and are thus silent about the development of a person's preference and identity over time. In that sense, those models might be considered as special cases of our model once equilibrium has been attained. However, by taking this route one could argue that one is not making models of sovereign individuals, but of social fools.

Conclusion

The main argument of this paper is that we need a more complete model of economic identity.

Recently, the characterisation of the economic agent has been extended from an atomistic and strictly self-concerned individual to a socially embedded individual. However, while this is a necessary step to increase the realism of the concept of an economic agent, it suffered from the drawback that the individual has become to some extent over-socialised and this in two ways.

Either the individual is being endowed right from the start with a given social identity that specifies and influences the agent's preferences. Or the individual expresses a preference for a particular social position, status or norm with the effect that this preference might become so important that it overrules her intrinsic or private preferences. It is certainly an empirical fact that a person's social environment has influences on preferences or that a person adheres to specific rules despite of underlying and maybe even opposite personal preferences. However, these situations should not be considered as universal. The person, even if socially embedded, still maintains the capacity to evaluate social influences and to decide about the observance of specific norms and rules or about the importance to achieve a specific social position in a given society.

This capacity does not simply imply that the person trades her social preferences (for norms, status etc.) for her private ones and will choose one or the other according to its lower cost or higher utility. A person might rather feel a discrepancy between her own personal preferences that she is attempting to satisfy and the social necessities to which she thinks she must adhere.

This discrepancy may not be resolved by simply acting either according to her preferences or to the social expectations, but only by changing either her personal preferences or by attempting to change her social environment. Indeed, if the person decided to act according to her personal preferences, she would be constantly faced with a violation of social norms or with contradictions with her social environment. If she adhered to the standards of her social environment, she would overlook her private preferences. In that sense, either one or the other will be put on one side to some extent and her internal discrepancy cannot be claimed to be resolved. Standard welfare economics does not make any difference between these two situations. If the person acted such as to satisfy her personal preferences or her social preferences, in any case it is assumed that her choices reflected her overall preferences and her new welfare level expresses exactly this satisfaction. However,it is not obvious that the two aspects can be incorporated into one consistent utility function. Indeed, psychologists might claim that the satisfaction of personal preferences "feels" different from the satisfaction of or adherence to social rules. If, on the other hand, the person is trying to diminish the discrepancy between her personal and social preferences, she might look out for possibilities of change -either of herself or of her environment, that would rid her of her internal conflict and transmit a level of satisfaction that is compatible with both, her A Proof of Theorem 4.5

Our weak interaction condition guarantees that the maps → H(x, ) and x → H(x, ) are contractions with respect to | • | 1 with constants β 1 and β 2 uniformly in x ∈ X and ∈ U , This means that the future distribution of agents over types is continuously dependent on the current distribution of agents and the current vector of types. That is, the weak interaction condition restraints the dependence of the future distribution of agents on the current distribution and the available types. Stated differently, if we had two almost similar but still different current distributions of agents and available types, the future distribution of agents in both "worlds" would be almost the same.

Proof of Theorem 4.5: In particular, G is a contraction if βL < 1. Since G maps the compact convex set X × U continuously into itself, and so the assertion follows from standard fixed-point arguments.

  [...] result in the same incentives to keep close to one's origins as [...] positive benefits" (p. 1011).

)

  So far we have introduced the concepts of individual and social change in the immediate future, i.e from one time-period to the next. To analyse what happens over time, we also need to make assumptions about the continuity of the changes in the agent's choices. In effect we will put constraints on the connectedness between the different time-periods of an agent. These constraints consist of two parts, one that refers to the personal connectedness, i.e. the dependence of the current state in terms of characteristics and types of an individual on previous states; the other refers to the social connectedness of an agent over time, i.e. the dependence of the current state on previously available types. The imposition of a continuity condition on an agent's changes as the types available change over time can be seen as an interaction condition that puts restrictions on the agent's individual characteristics as a function of the change in society. More specifically, in order to guarantee an almost sure convergence of types and average actions, we need to impose a weak interaction condition on the agents' individual behavior.

  g. tennis club or university). That is, the model, by presuming one specific mechanism of change (e.g. the new type is a convex combination of old type and average characteristics of members) is silent about what determines the speed of change and thus eventually admits any degree of change.However, an interesting aspect of group change (and consequently also of personal change) over time is not only how characteristics change, but also what sort of characteristics contributes to change. Indeed, the idea that different people bring different characteristics to the group and thus participate inits change is quite convincing. However, depending on the characteristics that are brought to the group, the speed of change can be radically modified. If David Beckham joins Real Madrid, he definitely brings characteristics to the group that contribute to its continued fame and success as a football club. When, on the other hand, Michael Gorbatshov came to power in the communist regime in the Soviet Union, his particular characteristics contributed to its complete disintegration. Thus, whereas in some cases, many different characteristics of many different people are needed to change the overall outlook of particular groups, sometimes a few characteristics of a single person suffice to trigger an almost complete change of specific groups.

respectively 7 .

 7 More precisely, under Assumption 4.4,|H(x, ) -H(x, ˆ )| 1 ≤ β 1 | -ˆ | 1 and |H(x, ) -H(x, )| 1 ≤ β 2 |x -x| 1 .

a)

  In order to guarantee a convergence of the empirical distributions, we need to impose a growth condition on F. To this end, we equip X × U with the norm |(x, ) t | := max{|x| 1 , | | 1 }. and denote by |DF ( )| 1 the column-sum-norm of the derivative DF ( ) of the function F introduced in (2). With L := sup |DF ( )| 1 , the map → F ( ) is Lipschitz continuous with constant L:|F ( ) -F (ˆ )| 1 ≤ L| -ˆ | 1 .In particular, the mapsx → αx + (1 -α)F • H(x, ) and → αx + (1 -α)F • H(x, ) are Lipschitz with constant α + (1 -α)Lβ 2 and (1 -α)Lβ 1 , respectively. As a result, the map (x, ) → αx + (1 -α)F • H(x, )and the map G defined in (4) are Lipschitz continuous with respective constants α + (1 -α)Lβ and max{α + (1 -α)Lβ, β}.

  Thus we can write down the probabilities of the new characteristics given the old characteristics and vector of types. In particular, the conditional The transition probability π from C × Y × X to C × Y governs the dynamics of individual characteristics and types. This captures what we call the who-criterion of an agent's identity.

	distribution of the random variables c a t+1 given c a t and also the current vector of types x t is given
	by
	m
	πxt (c a t ; c a t+1 ) =	p xt (c a t ; y)p xt c a t , y; c a t+1 .
	y=1
	The distribution of an agent's new characteristics and type depends on his current one. Specif-
	ically, the joint conditional distribution of the random pair z a t+1 = (c a t+1 , y a t+1 ) takes the form
	π xt (z a t ; z a t+1 ) = p xt (z a t ; y a t+1 )p xt z a t , y a t+1 ; c a t+1 .

a t+1 ; • where p is a stochastic kernel from C × Y to C.

Although in equilibrium, constraints are in fact, endogenous, they are considered by the agent as exogenous, (price-taker assumption).

See e.g.[START_REF] Olson | Personal Identity[END_REF], Oksenberg Rorty 1976. For overviews on different philosophical accounts of personal identity see[START_REF] Stéphane | L'idéntité[END_REF][START_REF] Glover | I: The Philosophy and Psychology of Personal Identity[END_REF][START_REF] Harold | [END_REF]

The questions of what it is for something to be a person, as well as of her persistence through time have sometimes been matched together. However, whereas fingerprints or a genetical code can be considered as criteria that establish if a person at one moment of time is identical with a person at a later moment of time, it can hardly be claimed that fingerprints or the genetical code are what makes a person a person. Thus, in order to be able to say something specific about a person over time, there must be some way of recognising this person as the selfsame person over time.Evidently, a person has a body and some specific psychological or mental states (such as self-images) and memories that will change over time. But it is because they change that it seems to be difficult to consider them to be the criteria that help to recognise a person as the same person. One might thus say that a person is recognisable as the same person because she has a specific personality or character. But while saying that personality is an intrinsic attribute of what it is to be a person, it can hardly be considered as a criterion that stands for the persistence of a person through time, other than by saying that personality is some unchanging substance that remains the same throughout time and changes of that person. This, however, would imply that as soon as the personality of a person even marginally changes, this person ceases to exist and another person comes into existence. It would also contradict the idea that a person is able to change herself: for, as soon as the person (defined by her personality) changes her personality, she will become discontinuous with herself. One solution to these problems is to say that personal identity is only about numerical identity and not about qualitative identity. This means that a person is numerically the same if she remains one and only one individual over time. A person would be qualitatively the same if she would share at each moment the exactly same properties (except location in space and time). While in the first case that person remains the same even though she qualitatively changes, this is not assured in the second case. However, the numerical identity of a person still has to be defined in terms of some criterion. If A's memories are, say, transferred onto B's brain and vice versa, the question is if A is numerically identical with her memory contents or with her body. Furthermore, if this criterion can be changed, the question is how much change is allowed to say that a person is still the same person. If Derek Parfit is Derek Parfit because of his memories, but

To visualise what this means imagine that in your current state, you can be either Kate Winslet or Juliette Binoche. In your future state you join a football club. A low value of β1 thus means that you join the football club almost independently of whether you have been Kate Winslet or Juliette Binoche. Now imagine that you are Kate Winslet and that there exists a replica of Earth called Mars on which almost the same types are available as on Earth. A low value of β2 thus means that you as Kate Winslet will choose to participate in the football club, almost independently of whether you live on Earth or on Mars. Thus, your preference to participate in the football club is almost independent of whether you are Kate Winslet or Juliette Binoche and of whether you live on Earth or on Mars. Surely, in this situation, personal identity does not matter much.

For details we refer the reader to Horst(2000).

personal and her social preferences. Of course, in this case one must have to differentiate between a situation in which a person changes and thus adapts her personal preferences to her social environment because particular social circumstances did not leave her enough opportunities to do otherwise and a situation in which a person freely adopts a given set of characteristics that allow her to fully deploy her personal identity and who she wants to be. These are indeed important aspects, which should be included in any consideration of a socially integrated economic agent but which so far have not received much attention in the economic literature. b) Under Assumption 1, the stochastic kernel π x * has a unique stationary measures µ. Since the sequence {x n } n∈N converges almost surely to x * , we can apply similar arguments as in [START_REF] Ulrich | Asymptotics of locally and globally interacting Markov chains arising in microstructure models of financial markets[END_REF] to prove convergence is distribution of individual states.

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