Luke Haywood
email: haywood@pse.ens.fr.

Marc Robin

Katherine Cuff

Fabrice Gilles

Watch your Workers Win Changing job demands and HRM responses

Keywords: job satisfaction, wealth, HRM, job mobility, turnover JEL codes: J28, J32, J63, M52, M54

This paper considers how the demand for non-material aspects of jobs evolves over changing wealth levels and how firms may want to react. We first consider the importance of non-material job aspects in general before turning to two specific human resource practices: flexible working hour arrangements and employer pension provision. In order to estimate the effect of wealth on job preferences without confounding it with the potential effect of job preferences on wealth due to earnings differentials, we focus on non-labour income (e.g. lottery winnings). We test how it affects workers' preferences using an approach based on duration data.

When firms decide on workers' wages, an important factor is the risk that too low wages may induce workers to leave. Many factors may determine the value of a particular job to an individual. A worker who has a higher wage, enjoys the work she does, has friendly colleagues and flexible working hours has been found to be less likely to quit. However, the relative weights of different aspects of employment may vary over time and circumstances. For example, material aspects may become more important when a worker's spouse loses their job and becomes dependent. Similarly, the number of children may have an influence on the preferences for certain job characteristics. With the importance of worker turnover in mind, we may thus be interested in estimating the effect of different influences on job leaving behaviour. Most changes in life circumstances are related in complicated ways to individual behaviour (couples may choose to have a child also as a function of their expectations of future labour market outcomes), making estimation of the impact of such changes on preferences for job characteristics difficult. This paper tests the impact of changes in wealth levels on preferences over job characteristics. The empirical strategy uses information on windfall gains (mainly from lottery winnings and inheritances) as an instrument for wealth changes. These windfalls are expected not to be caused by labour market behaviour and thus provide an original source of identification for the effect of wealth on job preferences of workers.

Section one places our research question within the framework of literature on turnover and job search.

Section two presents a measure of non-material aspects of a job based on subjective survey data. The single-item measure of job satisfaction for work "in itself" measures a particular dimension of job satisfaction different from monetary compensation.

Section three reviews the British household panel data that is used, covering job durations with information on job characteristics as well as workers' windfall gains.

Section four first reviews descriptive evidence on job leaving rates across different groups that appears suggestive of a role for wealth shocks in influencing job preferences. The empirical model to explain job leaving as a function of windfalls and job satisfaction is then presented and estimated. Results confirming an influence of wealth on job preferences are discussed.

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Luke Haywood

Section five considers which management practices may become more important as workers become more wealthy: what might managers do to avoid turnover by workers who become less dependent on the material payoffs of their job? The impact of working time arrangements and employer pension fund contributions on changing job leaving is tested.

Determinants of Worker Turnover

The current paper considers influences on voluntary turnover decisions by workers. There is a considerable literature on voluntary turnover, and in particular on individuals' motivations for turnover. The classic analysis in the management field by [START_REF] March | Organizations[END_REF] distinguishes two factors influencing workers' mobility decisions: perceived desirability of an individual's current employment and perceived ease of movement away from current employment. Our research question can be framed in these terms as follows: When ease of movement changes as a result of a wealth shock -e.g. because individuals have the resources to sustain a transitional search period without earnings -does perceived desirability become more important? Turnover research has focused on various dimensions of individual differences which influence turnover decisions. Personality, gender and industrial sector have for example been found to be of importance for job leaving [START_REF] Barrick | Effects of impression management and self-deception on the predictive validity of personality constructs[END_REF]). Wealth differences appear to constitute an important difference across individuals. In fact, whilst certain personality traits (such as the so-called Big Five) are arguably intrinsic, others may be affected by wealth shocks -e.g. we may expect rising self-confidence. To the extent that self-confidence is important for job leaving [START_REF] Barrick | Reducing voluntary turnover, avoidable turnover through selection[END_REF]), the impact of wealth may pass via self-confidence. This article is related to a research agenda instigated by [START_REF] Lee | An alternative approach: The unfolding model of voluntary employee turnover[END_REF]'s "Unfolding Model" which proposes five determinants of turnover decisions: shocks, scripts (pre-existing leaving plans), image violations (broken expectations), job satisfaction and job search. These are said to combine to generate four different turnover path types. In the first path, a shock triggers a pre-existing plan. If we take wealth as the shock, it may be hypothesized that workers with lower job satisfaction may be more likely to have preexisting plans to leave. In path two, shocks lead workers to reconsider their 2 Luke Haywood situation. They then leave only after a period of deliberation. The value of duration models is to not restrict the period over which we allow the shock to operate. Instead of looking at the impact of a wealth shock for turnover decisions in the following month or year, we consider the impact over the whole length of tenure. The hypothesis of this paper is that a re-evaluation of a worker's situation after a wealth shock will include, in particular, their level of job satisfaction. Compared with path two, path three concerns those individuals that leave for another job -i.e. after assessing alternative options.

To the extent that we cover voluntary turnover both to unemployment and to alternative employment, we cover this type also. Path four is defined as a path where lower job satisfaction independent of the shock is the "precipitating" factor of movement.

The idea of the current work is to test whether by changing the relative importance of monetary and non-monetary job components, wealth shocks lead to a re-evaluation of jobs' desirability in terms of job satisfaction. If this is the case, wealth and job satisfaction interact to determine turnover outcomes. It appears that this is consistent with paths one to three -which we will not attempt to distinguish in the empirical part. The approach resembles that of a study by Holtom et al. (2006) 1 who compare the degree of job embeddedness (a concept related to individuals' attitude towards their employer) compared across stayers and leavers, distinguishing between shock leavers and non-shock leavers. The duration data model that we use contrasts durations of individuals before and after receiving a shock and over different levels of job satisfaction.

In a recent paper, Lee et al. (2008) show that job satisfaction has a strong impact on voluntary job leaving for workers in a variety of situations: Workers leaving to search for a new job; workers leaving to accept a solicited or unsolicited job offer as well as workers leaving for family reasons. Testing the relative importance of "push" versus "pull" factors, they also find that unemployment is not a significant factor in voluntary job leaving decisions.

In economics, turnover has mainly been considered under the prism of job search theory, in which the informational frictions in the labour market are modelled using a stochastic job offer arrival rate which individuals can sometimes influence by adjusting their search effort. One of the results of simple job search models is that workers' best strategy for on-the-job search 1 Cited by [START_REF] Holtom | Turnover and Retention Research[END_REF] 3

Luke Haywood is simply to compare the utility of their current job to the level of utility in an alternative job. Thus a many-period game with potentially complex dynamics has a simple static solution, this is the insight famously used in [START_REF] Burdett | Wage Differentials Employer Size and Unemployment[END_REF].

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A requirement for this kind of enlightened job choice is that individuals are aware of the different levels of utility associated with a job (the salary, but also the non-monetary component) -for their current job and for any job offers. To test whether this is the case, [START_REF] Gielen | Repeated Job Quits: Stepping Stones or Learning about Quality?[END_REF] looks at workers' job-to-job transitions using the same data analysed here and finds that only a limited amount of movements can be explained by information-gathering, whilst many more moves are consistent with informed workers choosing jobs with higher levels of utility.

Informed transitions provide the basis for our treatment here. It is assumed that jobs consist of a monetary and a non-monetary componentand that the utility of the non-monetary component is adequately measured by answers relating to "job satisfaction for work in itself". We now turn to this two-dimensional measure of jobs and consider whether such a simple characterisation of individuals' appreciation of working conditions can be justified.

Job Satisfaction: domains and measurement

Job satisfaction is a multi-facetted concept for which psychologists have elaborated sophisticated models. This paper investigates how a wealth shock may change the relative importance of non-monetary aspects of jobs. Nonmonetary job aspects are not always easy to observe. As a measure for non-monetary aspects of the job we require something that workers evaluate independently of their wages. We might consider subjective job satisfaction, i.e. answers to the question "How satisfied are you with your job?". However, if we assume that evaluations of overall job satisfaction include both job satisfaction with pay as well as job satisfaction with non-monetary aspects of a job, it is clear that this single-item measure of overall job satisfaction cannot be used. We require a measure only of the non-monetary element of job satisfaction. However, it is clear that we cannot test the potential independence by testing the correlation between a measure of non-monetary job characteristics with wages: If it is the case that wealth influences the demand for non-monetary job characteristics, incomes will also influence this demand by changing wealth levels. Then we would suspect a correlation between non-monetary job satisfaction and wages in the absence of any problem in our measure of non-monetary job aspects.

It may be thought that our requirements for a measure of non-monetary job characteristics independent of wage are difficult to be met for another reason: Tenets of perfect markets may argue that wages should be expected to compensate for differences in the quality of non-monetary job aspects. This would lead to pay levels being set according to job disamenities as Adam Smith already predicted (the classic modern exposition is by [START_REF] Rosen | The Theory of Equalizing Differentials[END_REF]). However, in line with the vast literature on job search, we only need to acknowledge the lack of perfect information on the job market in order to avoid the prediction of a deterministic relationship of job amenities and wages (see in particular [START_REF] Hwang | Hedonic Wages and Labor Market Search[END_REF]). As long as there is some stochastic variation in the relationship between our measure of job characteristics and wages -which job search models would, for example, predictwe can use this to identify the effect we are interested in.

The measure that appears to fit rather well the requirements is "job satisfaction for work in itself": clearly directed at a subjective evaluation of intrinsic characteristics of the workplace whilst at the same time clearly distinct from financial rewards3 . Thus the present study uses this one dimension of job satisfaction (job satisfaction for work "in itself") as a measure for workers' appreciation of all non-monetary aspects of a job. The other element assumed to be important in the eyes of workers is the wage. These two components entirely describe a workplace4 .

In the large panel data used for this research, several subjective measures of domain job satisfaction are available: In particular, table (1) gives The precise question was: "I'm going to read out a list of various aspects of jobs, and after each one I'd like you to tell me from this card which number best describes how satisfied or dissatisfied you are with that particular aspect of your own present job." The scale (on the card) varies from "1-Not satisfied" over "4-Not Satisfied, not dissatisfied" to "7-Completely Satisfied".

the different measures of subjective job satisfaction for the period under consideration we have information on. We do not need to make assumptions about the links between objective job characteristics (other than wages) and the subjective evaluation of non-monetary job satisfaction. However, in this article, job quality, job characteristics, job satisfaction and working conditions are used interchangeably: our subjective measure is considered to be a sufficiently good instrument for some objective characteristics. We do not require that individuals have the same attitudes towards these characteristics -the same work may be viewed as in itself very satisfying by one person and not by another. In order to test whether wealth shocks lead to more importance given to these considerations we need attitudes to be stable across time however. Whilst the subjective nature of the instrument may mean it varies not just across individuals but over time, numerous studies have confirmed the strong correlations between this subjective measure and later objective outcomes: the fact that job satisfaction predicts future job turnover rates is an illustration of the possibilities afforded by this measure and ensures a degree of stability over time. Early papers to note this fact are [START_REF] Hamermesh | Economic Aspects of Job Satisfaction[END_REF] and [START_REF] Freeman | Job Satisfaction as an Economic Variable[END_REF]. Using subjective data on job satisfaction, [START_REF] Clark | What really matters in a job?[END_REF] shows that data on job satisfaction predict quitting behaviour in the UK, and that it can be used to establish "what matters most in a job". Using the same data used in the present study he finds that of the different domain satisfaction measures of In what follows we will use job satisfaction, job quality and working conditions interchangeably for only one aspect of job satisfaction -our measure of the non-material aspects of a job, i.e. job satisfaction for work "in itself".

The British Household Panel Survey

We use the British Household Panel Survey, a panel of around 10,000 persons broadly representative of the British population covered over 17 yearly waves from 1991 onwards.

The main variables of interest are windfalls, subjective job satisfaction, job duration and wages, as well as demographic and firm control variables. Job market data falls into two categories: In every wave, workers are first asked about the characteristics of their current job, including their subjective evaluation of working conditions and detailed information on earnings. Second, they are asked about transitions within their firm (e.g. promotions) and across firms (changes of employer or labour market status) in the preceding year. Our sample thus combines information on wages, working conditions and job durations.

The Sample

We focus on employed individuals for whom we have multiple observations who encounter at least one windfall during the panel observation period.

For this subsample we have information on 10386 completed job spells from 3488 workers. Given the specific factors involved in choosing retirement, the sample is restricted to those aged between 16 and 50, with a sample average age of 34 years. The sample is 49% female, 51% are married and the average level of education is around 12 years. 2% of the male and 30% of the female sample work part time.

Windfalls

Windfall earnings are recorded from lottery winnings, gambling gains, inheritances, life insurance and accident payouts and money received as a result of 7

Luke Haywood building society conversion 5 . The assumption made here is that these windfalls were not anticipated such that no behavioural changes can be made prior to the windfall. We have no information on the exact date of the windfall, and assume that the windfall occurred at the beginning of an observation period (between two waves). If a worker reports having received a windfall in the preceding period and also reports job mobility, we thus assume that the windfall occurred before the mobility decision. This assumption ensures that windfalls are not anticipated as a result of our recording scheme.

Consider the worker in figure (1): In September 2002 she reports having received a windfall in the previous year. Since the timing of this windfall is unclear, we do not know whether it occurred before or after January 2002 -i.e. before or after the job change. We assume that the windfall occurred prior to January 2002. To the extent that windfalls are received later, any effects of windfalls will occur in later waves -i.e. we can test for lagged effects in later waves.

We focus on respondents who record some windfall earnings over the period of the panel. The majority of windfalls involves small amounts, with a large spike of £100 6 and a majority of this results from lottery winnings, gambling and inheritances. Whereas lottery winnings tend to be small, with a recorded mean of £150, accident (£237) and life insurance (£584), but especially inheritances (£2294) are larger, as table (2) shows. Whilst most windfalls appear fairly small compared to earnings, table (4) shows that 5% of windfalls exceed annual earnings of workers.

Job duration data

The measure of job duration used here is firm tenure, such that within-firm mobility does not constitute job leaving. The advantage of considering total 5 This is somewhat of a British peculiarity: After the financial Big Bang of the early 1990s a number of building societies (basically consumer cooperatives) demutualised to become banks. In the process, considerable windfalls were distributed to customers. The deregulation of the banking sector was a sudden policy change making it very likely that these earnings qualify as windfalls. There was some speculative behaviour (joining building societies in the hope of demutualisation), but often clauses were enacted to avoid payouts to recently joined-up members.

6 All monetary values provided are deflated to their values in 2000. For building society conversion earnings we have no information on windfall size -payouts here varied considerably.

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Luke Haywood Note: For one year (wave 5) only aggregated data are available -included in total.

duration of a worker with a particular firm rather than in a particular job position is that within-firm mobility includes promotions which are most certainly endogenous to workers' behaviour7 . As a result, there may be some fluctuation in the wages and working conditions within a job (as workers change jobs within a firm). This appears acceptable given our estimation strategy (focussing on differential quitting rates) if we use information on the last available wage and working conditions before a worker left a job. In the rest of the paper we refer to movements of workers away from employment at a certain firm as job leaving.

As an example, consider the timeline of a worker depicted in figure (1). She is observed in the survey interview at wave 11 as being employed at a certain employer and provides information on job satisfaction, working conditions etc. In wave 12 she reports having changed employment in the previous year and provides information on working conditions and earnings at her new employer. Since the worker provides the exact end date of the job, we have precise information on job duration. For the job up to January 2002 we assume the covariates provided in September 2001 apply, for the job from January 2002 onwards we assume it is characterised by wage and working conditions reported in wave 12 (first interview reporting the new job). Thus we only assume that working conditions remain constant from the time of the last interview until movement away from the job. This also means that we cannot use spells for which we have no corresponding survey information -this is the case of the worker in figure (2) for whom we have no information on the job between January and April 2002.

Stock sampling

In our estimation we must control for the fact that we are more likely to have all the information we require for spells that last longer. In the example outlined in figure (2), whilst the job spell from April 2002 onwards is observed, the spell from January to April 2002 is not observed. We condition job duration (tenure) at date of entry into the sample to control for the likelihood of observing an observation -the likelihood of a job lasting until the interview date.

We exclude individuals for whom we only have one spell in order to increase the efficiency of estimation8 . We have on average 2.8 job observations per individual.

Censoring

As in most duration data sets, we do not observe the spell end date for all individuals. The typical assumption made is that of random censoring, i.e. that those individuals who are censored do not have different characteristics than those for whom we observe spell end dates.

Censoring may be a particular issue if we have no end dates for the longest spells which may be ongoing. In our data this concerns very few spells as there is considerable mobility in the British labour market over the period studied. Many more spells are censored as individuals drop out of the panel for unknown reasons. For these we assume random censoring.

Transitions

All labour market transitions may be affected by the change in wealth. In particular, we may wish to distinguish transitions within the labour market from transitions to non-participation. We observe voluntary transitions to another job, to university, retirements and jobs ending for family reasons as well as involuntary transitions -redundancies, sacking or ending of part-time jobs. In order to keep sample sizes reasonable, we do not differentiate between different transitions in a competing risk framework. Rather, we focus on all voluntary transitions -excluding all different types of dismissal, redundancy and retirement.

Job Satisfaction

The measure used for non-monetary job characteristics is the answer to the question "How satisfied are you with your job in itself ?", with potential answers ranging from "1 -not at all satisfied" to "7 -completely satisfied". The presence of other questions relating specifically to satisfaction with financial rewards should reassure us that the dimension of job quality measured here relates exclusively to factors other than remuneration, as the empirical strategy requires.

Empirical Analysis

Descriptive evidence of the impact of wealth shocks on transition behaviour is presented in figures (3) and ( 4) which give the smoothed hazard rate and the survival rates by receipt of a large windfall (defined as a windfall greater than £2000) and by job satisfaction (high job satisfaction defined as "nearly completely" or "completely satisfied" workers), focussing on the first 100 months of a job.

We find that quit rates are highest for the group of workers with low job satisfaction and windfalls (maybe a call centre lottery winner) and lowest (over most of the job duration) for workers with high job satisfaction and windfalls (e.g. an artist lottery winner who can continue her low-paid job). This variation in the impact of wealth shocks on mobility decisions over different levels of job satisfaction provides a starting point for a more rigorous causal analysis.

The Cox Proportional Hazard model

In order to map the factors influencing job leaving, we focus on the hazard rate, a measure of the duration distribution. The hazard at time t is the probability of job leaving conditional on having stayed in a job spell s up until t. Because it is thought that for many dynamic processes, the duration 12 Luke Haywood Large windfall here defined as windfall greater than £2000

t is itself a factor determining the rate of exit out of a state (here, an employment), researchers typically distinguish between a baseline hazard θ 0 (t) depending on duration and a systemic part of the hazard θ 1 (x) which gives the influence of the covariates. A simplifying assumption many researchers make is that the two parts are multiplicatively separable, generating the proportional hazard model. The implications of the resulting formulation ( 1) is that the covariates -amongst others here income and job satisfaction -have a proportionally constant influence on the hazard rate over different job durations.

In order to take into account potential differences across individuals that are not covered by observables x, an individual effect for unobserved heterogeneity η i(s) is added in (1).

θ s (t s |x s ) = θ 0 (t s ) exp(x s β + η i(s) ) (1)
Or, defining ξ i = exp(η i ):

θ s (t s |x s ) = θ 0 (t s ) θ 1 (x s ) ξ i(s) (2)
13 Luke Haywood Large windfall here defined as windfall greater than £2000

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The Cox Proportional hazard model (CPH) allows for many types of different functions for duration dependence θ 0 (t) -in fact any multiplicative time-dependent baseline hazard rates are acceptable. The procedure is semiparametric in the sense that the baseline (unobserved) hazard (θ 0 in (1)) is not estimated and the partial likelihood estimates of the coefficients (β) are nonetheless consistent. The intuition for the partial likelihood is -for a spell s -to use the conditional probability that spell s ends, given risk set R s defined as the set of spells ending at or after s. Due to the proportionality assumption, the baseline hazard -assumed to be the same for all individuals -drops out.

Thus we write the partial likelihood conditional on the individual effects as:

L P L s (β|t s=1...S , x s=1...S ) = s=S s=1 θ 1 (x s ) ξ i(s) r∈R s θ 1 (x r )ξ i(r) (3)
The CPH model buys semiparametric identification at the cost of efficiency: Only the ordering of job durations influences the likelihood, not the precise timing -so not all information is used for estimation.

For ξ we here follow a parametric route and assume ξ D → Gamma. The gamma distribution is not only a flexible distribution but it has also been shown that very general random processes can be well approximated by this distribution [START_REF] Abbring | The unobserved heterogeneity distribution in duration analysis[END_REF]). Multiple observations per individual somewhat weaken the strong assumptions on orthogonality between random effects and other explanatory variables (van den Berg ( 2001)). For the individual likelihood contribution this implies integrating out the individual effect:

L i (β|t s=1...S(i) , x s=1...S(i) ) = s=S(i) s=1 ∞ -∞ θ 1 (x s ) ξ i(s) r∈R s θ 1 (x r )ξ i(r) dG ξ (4)

Changing importance of job satisfaction

Our empirical analysis looks into the evolution of two key job characteristics hypothesized to influence job leaving decisions: the wage and job satisfaction for work in itself. In line with previous research, both are found to have a strongly significant impact on individuals' job mobility choices. The regressions control for individual characteristics (age (non-linearly), education 15 Luke Haywood (non-linearly), family situation (4 dummies)), objective working conditions (part-time (dummy), working hours), as well as 13 sector dummies thought to be potentially related to working conditions. The dependent variable in regressions is the hazard rate, i.e. the probability of job leaving at any point in time. Using the Cox limited information structure, the models allow for unspecified types of duration dependency (subject to the proportional hazard assumption). Furthermore, gamma-distributed shared random effects are included in the model and unobserved heterogeneity in job leaving is found to be significant.

In line with the initial hypothesis, it is found that the impact of windfalls on job leaving depends importantly on the level of job satisfaction, i.e. that the interaction effect is negative, such that the coefficient on job satisfaction becomes larger in absolute terms after a windfall. In table ( 5) we find that the interaction is in fact non-linear in the windfall with higher windfalls showing over-proportionally more impact on the size of the coefficient of job satisfaction.

To gain an understanding of size and significance of the effects over a range of windfalls and keeping over covariates constant we do a simulation9 the results of which are presented in figure (5). We investigate the difference in job leaving of a one-point increase in job satisfaction (from level 6 to 7) for a 35-year old never married person working full-time 35 hours in the manufacturing sector earning £12,000 yearly. We find that the marginal effect starts to become significantly different from the baseline for windfall values of around £1,000, changing quite significantly for larger windfalls. For windfalls above £100,000 the precision of the estimates is strongly reduced as we face small samples.

Human Resource Management responses

Employee turnover may carry considerable costs for firms. Whilst estimates of the precise costs vary a great deal10 , one aim of human resource practices such as flexible working hours or pension contributions may be employee retention [START_REF] Houseman | Why employers use flexible staffing arrangements: Evidence from an establishment survey[END_REF]). In the previous section, subjective job satis- faction for work "in itself" was used as a measure of working conditions in a firm. Can we go beyond that and find objective characteristics that might become more important to workers as a result of the wealth shock? First, model (2) in table (5) indicates that the impact of a wealth shock depends importantly on an employee's level of earnings. In fact the model achieves a better fit if we express windfalls as a percentage of annual earnings. Thus changing job preferences are clearly more relevant for less well-paid workers.

Put another way, firms can still retain workers who care more about the intrinsic job characteristics by paying higher wages.

What else might managers do? We explore two particular practices that have received some attention in the human resource literature: flexible working arrangements and whether the employer participates in a private pension scheme.

Latent preferences for flexible working hours

Most types of employment contract stipulate not only a given number of hours, but also the specific schedule when these hours must be worked. This second constraint might be particularly costly for individuals who have important family commitments, and in particular, childcare obligations. How might demand for flexible working hours be affected by changing wealth lev-

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Luke Haywood els?

On the one hand, increased levels of wealth should increase demand for leisure activities, some of which may be difficult to reschedule. More generally, to the extent that flexible working hours are an immaterial job characteristic we might expect the relative weight of flexibility to have increased after a wealth shock.

On the other hand, the impact of these "scheduling costs" [START_REF] Wilson | Scheduling Costs and the Value of Travel Time[END_REF]) may be reduced when childcare provision can be purchased by families. This might suggest a decrease in the relative importance of flexible working hours to workers after a wealth shock.

We use a series of questions related to specific working hour arrangements and generate a dummy for flexible working hours. Jobs for which individuals declare that employers run a scheme of "flexitime", "annualised hours", "job share", "nine day fortnight" or "zero hours contract" are all classified as flexible working arrangements. In practice, 85% of the flexible working hours arrangement concerned flexitime. Whilst flexitime arrangements vary, workers are in general required to work at "core" times and free to schedule their working hours around these -subject to a total number of hours being worked in a given period.

The results in table (6) provide strong evidence that the second effect dominates: the population is split three times: the two panels on the right give the estimates for childless individuals and for parents, the middle two panels for women and men whereas the left two panels compare mothers to men and childless women. Note that we here focus only on dependent children, such that parents with grown-up children are here categorised as childless. The change in preferences after a wealth shock can be assessed by the interaction term of flexible working hours and the windfall. For a large windfall the coefficient on flexible working hours is shown to change significantly -but for certain groups only: for mothers, less strongly for the group of all women and not significantly for parents (including men) and men. Given ample evidence that the majority of childcare is still carried out by mothers this supports the idea that the worker engaging in chlidcare is the key factor in determining the reduced importance of flexible working hours after a wealth shock. Luke Haywood

Latent preferences for employer pension contributions

Around two-thirds of the employment spells (72%) in our sample are at employers who run a private pension scheme. Why might preferences regarding this benefit change as a result of the wealth shock? If individuals who win a large windfall gain are less focused on the immediate material needs and more likely to consider long-term perspectives, we may expect an increased importance of this job characteristic in workers' transition decisions.

The results in table ( 7) indicate a significant increase in the importance of employers' pension contributions after a wealth shock. When we split the population by gender we find that this effect is driven by male workers. How can we interpret this finding? Employer pension contributions can be described as forced savings: A part of workers' earnings is saved for the period after retirement. As workers become more wealthy, this type of arrangement may become more interesting -in particular for those individuals who manage their spending habits less well. This might fit with evidence showing that women are more risk-averse. However, differential savings behaviour may also be the result of time-inconsistent preferences such that men would benefit from the comitment device of forced savings and may show a stronger revealed preference for employer pension contributions11 .

Concluding remarks

Using British panel data it has been found that the degree to which wealth influences mobility decisions depends on the degree of job satisfaction individuals have in a particular job. The way in which the labour market distributes utility may depend importantly not only on human capital and luck (as models of the labour market focussing on productivity and frictions imply) but also on non-human capital. The basic finding should be relevant to a number of researchers interested in the interactions between inequality, unearned income and subjective wellbeing in the labour market. It also holds lessons for managers who might be confronted with workers whose high motivation to work may be related to the monetary dimension of their job and thus susceptible to decrease over time as workers accumulate wealth. Whilst the current paper focuses on wealth shocks, further research is called for regarding the rôle of job quality in mobility decisions in a model allowing for wealth accumulation.

Different potential HRM responses have been outlined here. Behavioural changes in preferences over job characteristics were found to be a function not only of the absolute size of windfalls but of windfalls relative to current income. For a given windfall, individuals with lower incomes will thus be more strongly influenced by the arguments advanced here. Consistent with the efficiency wage literature this underlines the strong importance of earnings on job mobility decisions and provides a first potential managerial response.

Whilst subjective job satisfaction is known to be related to objective working conditions in numerous ways, the shift in preferences for job satisfaction was found to go hand in hand with a shift in preferences for two objective job characteristics: whether an employer offers flexible working arrangements and whether an employer participates in an employer pension scheme. We have seen how changing wealth levels impact job demands by workers and considered potential HRM responses: Wealthier workers are less

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Changing job demands interested in flexible working arrangements and more interested in employer pension contributions.

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Figure 1 :

 1 Figure 1: Example of timeline of survey interviews and events

Figure 2 :

 2 Figure 2: Example of timeline of survey interviews and events

Figure 3 :

 3 Figure 3: Moving to another job: by Windfall and Job Satisfaction

Figure 4 :

 4 Figure 4: Staying in a job: by Windfall and Job Satisfaction

  Figure 5:

Table 1 :

 1 Measures of Job satisfaction

	Mean Standard Deviation Min Max

  table (1), job security and pay are the most impor-

	tant, followed by work in itself and hours of work. Overall job satisfaction is
	found to be a good predictor of future separations. Clark et al. (1998) find
	that dissatisfied German workers, too, are more likely to quit.
	6	Luke Haywood

Table 2 :

 2 Size of windfalls received

		Inherit. Lottery Life ins. Accident ins. Total
	1-1000 pounds	98	2255	50	41	2695
	1000-5000 pounds	158	78	198	148	627
	5000-10000 pounds	79	8	32	23	149
	10000-50000 pounds 115	2	42	11	183
	50000+ pounds	35	1	5	1	49
	Note: For one year (wave 5) only aggregated data are available -included in total.

Table 3 :

 3 Size of Windfalls as fraction of annual earnings

		Inherit. Lottery Life ins. Accident ins. Total
	1-10% of annual income	141	2264	78	61	2,788
	10-50% of annual income	153	69	184	122	578
	50-100% of annual income 67	8	29	29	139
	100+% of annual income	124	3	36	12	198

Table (

 (

		Table 4: Destinations of job leavers
		number of spells percent
	to new job	1985	16.47
	to unemployment 1107	9.19
	to university	83	0.69
	retirement	24	0.20
	family-related	229	1.90
	health-related	170	1.41
	other	1512	12.55
	censored	6940	57.59
	total	12050	100
		1) provides information about different measures of job
	satisfaction.
		11	Luke Haywood

Table 5 :

 5 Job leaving as a function of wage and job satisfaction. Model 1 adds log windfalls and interactions in levels; model 2 adds windfalls as a percentage of annual income and interactions. See the footnote for the numerous controls included in the regressions. Dependent variable: hazard rate of voluntary job leaving

	Model 1	Model 2
	β (robust s.e) β (robust s.e)

Table 6 :

 6 Job leaving as a function of wage, windfalls, flexible working hours and an interaction term. The first two columns on the left compare mothers to childless women and men; the middle two columns compare coefficients for men and women and the two columns on the right compare parents to individuals without dependent children. See the footnote for the numerous controls included in the regressions. Dependent variable: hazard rate of voluntary job leaving

		Mothers	Childless or Men Women	Men	Childless	Parents
		β	β	β	β	β	β
	Log Wage	-0.561*** -0.691***	-0.711*** -0.616*** -0.835*** 0.549***
		(0.198)	(0.138)	(0.186)	(0.173)	(0.170)	(0.147)
	Log Windfall (LWF)	0.142	0.787**	0.118	1.101**	0.770**	0.165
		(0.407)	(0.343)	(0.382)	(0.492)	(0.389)	(0.302)
	Flexible Hours (Flex) -0.445*** -0.547	-0.386**	0.068	-0.123	-0.205
		(0.173)	(0.111)	(0.161)	(0.139)	(0.128)	(0.129)
	(Flex * LWF) Inter'n 0.138**	0.012	0.125**	-0.028	0.020	0.072
		(0.055)	(0.034)	(0.056)	(0.043)	(0.037)	0.045
	Log Pseudolik.	-930.523	-3513.079	-1359.321 -2307.101 -2473.244 -1957.831
	N*S (spells)	1886	6630	2865	4321	4743	3773
	N (individuals)	880	2790	1257	1722	2188	1764
	Controls: age,age 2 ,education,education 2 , 13 industry dummies, 4 family situation
	dummies, working hours, part-time dummy;
	Significance levels:10%(*),5%(**),1%(***); All standard errors robust to
	heteroskedasticity
			20		Luke Haywood

Table 7 :

 7 Job leaving as a function of wage, windfalls, employer pension contributions and an interaction term. The two columns compare coefficients for men and women. See the footnote for the numerous controls included in the regressions. Dependent variable: hazard rate of voluntary job leaving

	Sample	Women	Men
	β	β	β

Controls: age,age 2 ,education,education 2 , 13 industry dummies, 4 family situation dummies, working hours, part-time dummy; Significance levels:10%(*),5%(**),1%(***); All standard errors robust to heteroskedasticity

It is only when more sophisticated considerations such as using offers as bargaining chips (see e.g.[START_REF] Postel-Vinay | Equilibrium Wage Dispersion with Worker and Employer Heterogeneity[END_REF]) are taken into account that more complicated dynamic strategies are generated.4Luke Haywood

It may serve as a useful instrument even if this measure does not give a complete characterisation of relevant job characteristics -it may for example abstract from contextual or relational aspects (e.g. relationships with superiors etc.). Furthermore, we may be concerned about relying on a single-item measure. Considering job satisfaction overall -i.e. not the specific domain satisfaction we use -[START_REF] Wanous | Overall Job Satisfaction: How Good Are Single-Item Measures?[END_REF] discuss to what extent the summary single-item measure is correlated to more sophisticated measurements of job satisfaction and conclude on an optimistic note.

 4 Given that numerous sector and individual dummies are added in the regression analysis, the identifying assumptions in the empirical part are in fact less

restrictive.5Luke Haywood

In practice, results did not seem very sensitive to using alternative definitions of spells as job or firm tenure.9Luke Haywood

[START_REF] Van Den Berg | Duration Models: Specification, Identification, and Multiple Durations[END_REF] shows that shared frailty models (random effects models with several observations per individual) offer considerable advantages over models with only one observation per individual.10Luke Haywood

[START_REF] Ai | Interaction terms in logit and probit models[END_REF] note that the interaction effect in non-linear models is not the marginal effect we would like to use for interpretation.

[START_REF] Holtom | Turnover and Retention Research[END_REF] cite a range from less than one times a worker's monthly income to more than two yearly incomes.

See Sunden and Surette (1998) for a US perspective on gender differences in savings.


Damien Besancenot
email: besancenot.damien@univ-paris13.fr

Radu Vranceanu
email: vranceanu@essec.fr.

BANKS'RISK RACE: A SIGNALING EXPLANATION

Keywords: Banking sector, Risk strategy, Signaling, Imperfect information, the Great Recession. JEL Classi…cation : G21, G32, D82

published or not. The documents may come

Introduction

The 2007-2009 recession has been by all dimensions one of the most severe since the Great Depression [START_REF] Brunnermeier | Deciphering the liquidity and credit crunch 2007-2008[END_REF]. Social costs were huge, both in terms of output loss and rising unemployment [START_REF] Imf | World Economic Outlook[END_REF]. Fiscal positions of many countries were deeply undermined, with lasting consequences on growth prospects. While scholars will debate for many years about the deep causes of the crisis, a majority of experts claim that one essential determining factor was the abnormal accumulation of risk by banks (inter alia, [START_REF] Borio | The …nancial turmoil of the 2007-?: A preliminary assessment and some policy considerations[END_REF][START_REF] Trichet | Ré ‡exions sur les turbulences …nancières, Intervention auprès de l'Institut Montaigne[END_REF]Diamond & Rajan, 2009;[START_REF] Stiglitz | Freefall. Free Markets and the Sinking of the Global Economy[END_REF]Pomfert, 2010). This observation begs the question on why, in the …rst place, did banks engage in what can be described as a genuine race for risk?

The answer submitted in this paper brings into the picture the asymmetric information between banks'shareholders and managers with respect to the structural risk of these …nancial institutions.

One key feature of banks'activity is their opacity in functioning and management. As emphasized by Morgan (2002, p.874) "risks taken into the process of intermediation are hard to observe from outside the banks". Indeed, banks are traditionally very reluctant to disclose any information about their clients on both sides of their balance sheets. Furthermore, the composition of their asset portfolio is both a strategic decision and a key factor of success; no bank will eagerly disclose this information. Over the last twenty years, the complexity and opacity of banks' …nancial intermediation has increased dramatically, being driven essentially by a shift from the traditional "deposit and loan" model to the "orginate-to-securitize" model (Diamond & Rajan, 2009;[START_REF] Stiglitz | Freefall. Free Markets and the Sinking of the Global Economy[END_REF]. In theory, securitization should have allowed banks to transfer most of the credit risk to a myriad of investors; in practice, it turned out that during this crisis US banks kept large amounts of high risk securities on their books; furthermore, when di¢ culties arose, they had to cover the risks embedded in o¤-balance sheet entities they had created for securitization purposes. European banks have also aggressively invested in CDOs with a hidden content in US subprime loans, that had a true risk known only to a minority of insiders.

In general, given that higher risks command higher returns, banking technology allows the manager to choose the preferred risk/return combination. As the experience of the 2007-2009 recession has shown, banks are not equally equipped to face adversity. Newspapers extensively covered the dramatic situation of failing banks; other banks came out as unchallenged winners from the restructuring process.1 Actually, banks di¤er in their portfolio of activities, investment and loan opportunities, risk management systems and operating costs. Empirical literature has revealed that problem loans and cost e¢ ciency are related measures, with banks approaching failure featuring both high ratios of problem loans and low cost e¢ ciency [START_REF] Berger | Problem loans and cost e¢ ciency in commercial banks[END_REF][START_REF] Deyoung | X-e¢ ciency and management quality in national banks[END_REF]. In order to keep the model as simple as possible, in this paper we focus only on di¤erences in cost e¢ ciency, but the structure of the problem and the conclusions would not change much if we alternatively consider that less e¢ cient banks have also less investment opportunities than the more e¢ cient banks.

We assume that there are (only) two types of banks, the high cost-e¢ cient (or "good") ones and the low cost-e¢ cient (or "bad") ones.2 A caveat is needed here: if a banks spends more resources monitoring the quality of its portfolio and is quicker to write-down unperforming assets, then these higher costs would not necessarily be representative of a bad management [START_REF] Boot | Competition and entry in banking: Implications for stability and capital regulation[END_REF]. In our setup, cost e¢ ciency is a basic measure of wasted resources in a static perspective. Thus, a less e¢ cient bank can deliver the same returns as a highly-e¢ cient bank only if it takes more risks on its balance sheet. The bank-speci…c risk/return relationship is introduced in a very stylized way, inspired by the traditional model by Markovitz (1952). More precisely, the bank is represented as a portfolio manager, investing shareholders wealth in a convex combination between the riskless asset and a portfolio of risky assets.

Building on this simpli…ed bank model, we analyze banks' risk/return strategies within the framework of a classical signaling game [START_REF] Spence | Job market signaling[END_REF][START_REF] Spence | Signaling in retrospect and the informational structure of markets, Nobel Prize Lecture[END_REF][START_REF] Riley | Competitive signalling[END_REF] that opposes banks' managers to shareholders. In a perfect information world, all investors would ‡ee the less e¢ cient banks to join the most e¢ cient ones, and the former type of bank would be pulled out of the market.

However, if returns can be observed by private investors but risk cannot, then less e¢ cient banks would survive if they manage to conceal their type. They can do so by increasing the amount of risk in order to deliver the same returns as the high e¢ cient banks. In equilibrium, the less e¢ cient banks take too much risks.

Another set of strategies can be obtained if the regulator can set an upper limit on the amount of risk that banks can take, but, to protect banks'business model, does not publicly disclose the bank speci…c information. In general, regulators get information on bank risk exposure through on-site and o¤-site examinations or sta¤ members working within the banks. Their task is not an easy one given limitations on disclosure and the underlying exposure in banks'trading books, particularly when complex instruments are in question. After 2008, regulators carried out time-consuming stress tests that measure a bank's resilience to simulated additional macroeconomic shocks. In May 2009, the US Federal Reserve System made public the methodology and results of these tests for each of the nineteen large banks subject to these checks, while the European regulators decided to keep the bank-speci…c information secret and disclose only aggregate information. 3So in the European context, the regulator's information was private knowledge. Some prudential ratios might also be communicated only to the regulator, without being publicly disclosed. If such an upper bound on risk exists, and this limit is known by banks only (and the regulator), good banks can signal themselves by increasing returns (and risk) up to the point where bad banks cannot follow them. If not all good banks undertake the costly "super-signaling" strategy, then bad banks can further survive by taking excessive amounts of risk. This extended game presents several types of equilibria, depending on the proportion of good banks and the gap in operating costs. The main contribution of the paper is to emphasize that, under imperfect information, in any of these equilibria at least some banks have no other optimal choice than to hold on their balance sheet more risk than in a perfect information set-up. Furthermore, for some parameter values, we get a typical con…guration of multiple equilibria; which one of them actually materializes ultimately depends on investors'beliefs. In a multiple equilibria setting, the economic system is prone to extreme instability.

Our simpli…ed bank model does not allow to bring into the picture the capital-to-asset ratio, an indicator traditionally used to assess a bank …nancial health. 4 After the crisis, policymakers took signi…cant steps with the aim of capping the amount of risk that banks can take. One important decision adopted by the Basel Committee on Banking Supervision on September 12, 2010 was to increase the minimum common equity requirement ratio to 7%, as compared to 2% before the crisis (banks have time until 2019 to comply with this new target). We would notice that the extensive literature on the relationship between capital requirements and risk taking is somehow inconclusive; some authors have even suggested that higher capital bu¤ers may actually facilitate higher risk-taking [START_REF] Kahane | Capital adequacy and the regulation of …nancial intermediaries[END_REF][START_REF] Koehn | Regulation of bank capital and portfolio risk[END_REF]; see also the survey by [START_REF] Vanhoose | Theories of bank behavior under capital regulation[END_REF]. Since a high capital-to-assets ratio is both observable and involves an opportunity cost for the bank, this indicator could also be used for signaling purposes. If weak banks can set aside as much capital as strong banks (and take higher risks), then strong banks might be able to signal themselves by rising the capital ratio well above the legal minimum, to the point where weak banks cannot follow them. This alternative approach would provide an interesting path for further research.

The paper is organized as follows. The next section introduces our main assumptions. Section 3 analyzes the equilibria of the game. Section 4 presents our conclusions.

Main assumptions

The model is cast as a game between banks' managers and shareholders under imperfect information of the latter about the structure of the asset portfolio of the bank. The banking sector is made up of publicly listed banks, that take deposits and issue debt in order to grant loans and buy …nancial assets. If the bank has access to a risk-free asset and to a portfolio of risky assets and loans, the manager can pick any combination of risk/return along the capital market line, a tangent to the e¢ cient frontier [START_REF] Markowitz | Portfolio selection[END_REF]. A higher return can be obtained only if the bank takes more risks (by investing more in the e¢ cient portfolio of risky assets and loans).

Denoting the net return by R and the amount of risk by v; this typical trade-o¤ between risk and return for a bank of type i can be written:

R i (v) = R 0 + v c i ;
(1) where R 0 > 0 is the interest rate of the risk-free asset, > 0 is the slope of the capital market line and c i stands for the bank-speci…c operating cost. 5 The inverse function, indicating the risk needed to achieve a given return for a bank with operating cost c i is also of interest for further developments:

v i (R) = 1 R R 0 + c i : (2)
We assume that, depending on the quality of their management, banks di¤er in their operating costs. 6 To keep the model as simple as possible, we assume that banks can be of two types:

"good" banks (of type g) with a low operating cost c g that can be normalized to zero without loss of generality and "bad" banks (of type b) with a high operating cost c b = c > 0: In Figure 1, we represent the capital market lines of such a good and bad bank. It can be noticed that in order to provide shareholders with the same net return, bad banks must take riskier bets, that is

v b (R) > v g (R); 8R > R 0 .
Let q be the proportion of good banks in the total population of banks, 1 q being the proportion of bad banks. This distribution of banks is common knowledge.

In a general form, we represent the utility of the representative risk-averse shareholder's by a quasiconcave function U (R; v); with @U (; )=@R > 0; @U (; )=@v < 0. Resulting indi¤erence curves are convex.

Shareholders agree to pay the bank's manager a compensation that is proportional to their perceived utility, or

W = E[U (R; v)];
where E[ ] is the expectations operator;7 the compensation factor is not essential, so we normalize it to 1.

Under these assumptions, in a perfect information set-up, a manager running a good bank

R v R S R H R L v L U(R L ,v L ) U(R H ,v H ) 0 A B A' v b (R) v g (R) B' R 0 v H v v g (R S ) _ R 0 -c v b (R H ) U(R S ,v g (R S )) U(R H ,v b (R H ))
Figure 1: Return strategies, risk and utilities would simply choose the bundle (R H ; v H ) that maximizes his income given the bank's capital market line, such as indicated at point A in Figure 1.

Notice that the manager of a bad bank would prefer the bundle (R L ; v L ); at point B in the same …gure. Yet, given that shareholders'satisfaction is higher for good banks than for bad banks, no investor would hold the bad bank stocks: therefore less e¢ cient banks cannot survive in this perfect information world.

However, the assumption of perfect information is not very realistic given that a bank's risk exposure is a very complex commodity. As noticed in the Introduction, it is very di¢ cult for outsiders, even for expert ones, to evaluate the full amount of risk taken by a bank. Building on these basic fact, we further assume that the level of risk exposure of a given bank is private information to its manager, while the stock return is public information. In this context the set of strategies of the banks is more sophisticated:

For bad banks, like in the perfect information set-up, the strategy of playing R L is never optimal since it reveals the type of the bank and all the shareholders would leave it. At di¤erence with the perfect information case, in the imperfect information environment a bad bank can survive if it manages to conceal its type. It can reach this outcome by increasing the riskiness of its portfolio such as to deliver the return R H ; i.e. the perfect information return of the good bank (at point B'in Figure 1).

Good banks can play their perfect information optimal strategy R H as well. However, if good banks want to make sure that no bad bank has an incentive to imitate them, they should pay a su¢ ciently high return that a bad bank cannot deliver it. In order to analyze the broadest set of equilibria, we consider that regulators can set an upper limit on the total risk allowed to be taken by any bank, a limit denoted by v (the vertical line on Figure 1).

This upper limit on risk is known by banks. We assume that shareholders no not have the same ability to measure risk as the regulator.8 Then, if a bank pays a return slightly above R S , de…ned by:

R S = R 0 + v c; (3)
it unambiguously signals itself as a good bank, given that bad banks cannot further increase risk such as to copy them. R S is thus the second return strategy of the good bank.

Turning now to the manager's payo¤, we argued that if investors perceive that a bank is of the bad type, this bank leaves the market and the compensation of the manager becomes zero. So, a positive compensation for the manager can be de…ned only for banks that stay on the market (i.e., those that are not been perceived as being bad banks). Denoting by Pr[ijR j ] the probability shareholders assign to the event that a bank is of the type i; with i 2 fb; gg; if the return strategy is R j , with j 2 fL; H; Sg, the manager's payo¤ can be written:

W (R j ) = 8 > > < > > : 0; if Pr[bjR j ] = 1 Pr[gjR j ]U (R j ; v g (R j )) + Pr[bjR j ]U (R j ; v b (R j )); if Pr[bjR j ] 2 [0; 1[ (4)
Notice that return strategies R S and R L reveal perfectly a bank's type. Thus W (R S ) = U (R S ; v g (R S ))

and W (R L ) = 0.

Figure 2 presents the decision tree.

The typical sequence of decisions goes like this:

R S R H R H q 1-q Type g Type b α 1-α R L W(R S )=U(R S ,v(R S )) W(R L )=0 W(R H )=E[U(R H ,v i (R H ))]
Manager's payoff At step 1, depending on their type, banks chose their return strategy.

At step 2, shareholders make their opinion about the type of bank given the observed returns and pay the manager a compensation proportional to their own expected utility; the game ends.

Finally, we notice that the gap between U (R S ; v g (R S )) and U (R H ; v b (R H )) depends on the operating cost c: Figure 3 shows that there is a critical c such that

U (R S ; v g (R S )) = U (R H ; v b (R H )):
Taking R H as given and using Eq. ( 2), we can show that the utility of the manipulating bad bank is decreasing in c:

dU (R H ; v b (R H ; c)) dc = @U (R H ; v b (R H ; c)) @v b (R H ; c) dv b (R H ; c) dc = 1 U v < 0: (5)
On the other hand, using Eq. ( 3) to determine dR S =dc = 1 and Eq. ( 2), to get

dv g (R S )=dR S = R v R S R H U(R H ,v H ) 0 A A' v b (R) v g (R) B' R 0 v H v v g (R S ) _ R 0 -c v b (R H ) U(R H ,v b (R H ))=U(R S ,v g (R S )) ~Figure 3: The critical operating cost c increasing in c : dU (R S (c); v g (R S (c))) dc = @U () @R S dR S dc + @U () @v g dv g (R S (c)) dR S (c) dR S dc = = U R U v 1 = 1 U R [ M RS(R S ; v(R S ))]: (6)
Outside the optimum of the good bank, for R S > R H ; the marginal rate of substitution M RS = U v =U R > ; thus the derivative has a positive sign.

Thus, for any c < c we have

U (R S ; v g (R S )) < U (R H ; v b (R H ))
and for c > c; we have

U (R S ; v g (R S (c))) > U (R H ; v b (R H ; c)
). This latter case was depicted in Figure 1.

Equilibria

An equilibrium of this game is de…ned as a situation where banks' strategies are optimal (i.e., allow to their CEO to earn the highest compensation) given shareholders'beliefs about the type of bank, and shareholders'beliefs are correct given banks'optimal strategies. We may distinguish between a separating equilibrium (where the return strategy of a bank perfectly reveals its type), a pooling equilibrium (where all banks implement the same strategy and thus no information about the type of bank can be inferred from the observed return strategy), and a hybrid equilibrium (where banks play a Nash mixed strategy and this strategy carries some but not full information about their type).

In order to rule out a trivial situation, we admit that, by increasing risk enough, bad banks can deliver the perfect information optimal return of the good bank, or, in an equivalent way, that R S > R H :9 If bad banks cannot copy the strategy of the good banks, only the latter do survive and implement the perfect information strategy R H :

3.1 Signaling equilibrium: good banks do R S

We can show that an elementary separating equilibrium where all good banks deliver their signaling return R S and bad banks have left the market is always possible.

In such an equilibrium, good banks' strategy is s(g) = R S : The equilibrium beliefs are:

Pr[gjR S ] = 1; and, given that any bank that pays less than R S should be a bad bank, Pr[gjR L ] = 0

and Pr[gjR H ] = 0:

Bad banks can play either R L or R H ; but given the system of beliefs, the manager's payo¤ is:

W R L = W R H = 0.
There is no incentive for a bad bank to stay in the market. Furthermore, R S is the optimal strategy for the good bank: indeed, the condition W R S > W R H = 0 is always true.

In this equilibrium, the risk exposure of good banks exceeds the perfect information level, H ): Good banks resort to excessive risk taking as a barrier to entry.

v g (R S ) > v g (R

Pooling equilibrium: all banks do R H

We can now put forward the existence of a pooling equilibrium where all banks play R H : bad banks play the perfect information optimal strategy of good banks, and good banks decide not to signal themselves by doing R S : Banks'single strategy is s(i) = R H ; 8i 2 fb; gg:

Shareholders'equilibrium beliefs can be written: Pr[gjR S ] = 1; Pr[gjR H ] = q and Pr[gjR L ] = 0:

Necessary conditions for this equilibrium are: (1) W (R H ) > W (R S ) for the good bank and

(2) W (R H ) > W (R L ) = 0 for the bad bank. Since W (R S ) > 0; the equilibrium exists under the single condition W (R H ) > W (R S ): Given the de…nition of the manager's payo¤ (Eq. 4), this condition becomes:

qU (R H ; v g (R H )) + (1 q) U (R H ; v b (R H )) > U (R S ; v g (R S )) (7)
or:

q > q 1 U (R S ; v g (R S )) U (R H ; v b (R H )) U (R H ; v g (R H )) U (R H ; v b (R H )) : (8)
In the small operating cost case (c < c), we have

U (R S ; v g (R S )) < U (R H ; v b (R H )) that implies
q 1 < 0: the previous condition is always true. The pooling equilibrium always exists if the loss of utility of shareholders who support a bad bank is not too large; this can happen if the operating cost gap c is small.

In the large operating cost case (c > c), we have

U (R S ; v g (R S )) > U (R H ; v b (R H )); thus q 1 > 0: Furthermore, q 1 < 1 as U (R S ; v g (R S )) < U (R H ; v g (R H )):
We can conclude that in the large cost case, the pooling equilibrium exists only if the frequency of good banks is large enough. If there are not too many bad banks who imitate the good banks, the manager of the latter has not too much to lose, and it does not worth for him to implement an expensive signaling strategy.

In the pooling equilibrium, bad banks take too much risks as compared to the perfect information set-up, but they all survive in this environment.

3.3 Hybrid equilibrium: some good banks signal themselves, all others play R H

In this equilibrium, a proportion of the good banks decide to signal themselves by playing R S ;

and (1 ) play their perfect information strategy R H . All bad banks copy the latter and play

R H :
The mixed equilibrium strategy of the good banks is s(g) = f R S + (1 )R H j 2 [0; 1]g and the bad banks'strategy is s(b) = R H : Using Bayes rule, and denoting by Pr[R j ji] the probability that a bank of type i plays strategy R j , equilibrium beliefs can be written: Pr[gjR L ] = 0;

Pr[gjR S ] = 1 and:

Pr[gjR H ] = Pr[R H jg] Pr[g] Pr[R H ] = (1 )q (1 )q + (1 q) : (9)
In equilibrium, a good bank must be indi¤erent between strategies R H and R S :

W (R S ) = W (R H ) , U (R S ; v g (R S )) = Pr[gjR H ]U R H ; v g (R H ) + Pr[bjR H ]U R H ; v b (R H ) : (10)
This equation allows us to determine ( 1) with respect to predetermined variables:

(

1 ) = (1 q) q U (R S ; v g (R S )) U R H ; v b (R H ) [U (R H ; v g (R H )) U (R S ; v g (R S ))] : (11)
We …rst notice that U R H ; v g (R H ) U (R S ; v g (R S )) > 0. Hence, in the small cost case (c < c),

since U (R S ; v g (R S )) < U R H ; v b (R H )
we have (1 ) < 0 : the hybrid equilibrium is impossible.

In the large cost case (c > c); we have

U (R S ; v g (R S )) > U (R H ; v b (R H )); so 1 > 0:
The equilibrium exists if 1 < 1; which is equivalent to:

q > q 1 U (R S ; v g (R S )) U R H ; v b (R H ) U (R H ; v g (R H )) U (R H ; v b (R H )) : (12)
This is the same existence condition as that of the pooling equilibrium in the same large cost case (Eq 8).

In the hybrid equilibrium, both bad banks and a fraction of the good banks are taking an excessive risk. The maximum amount of risk in the economy appears for ! 1; we infer that if the hybrid equilibrium is in place, the amount of risk in this economy reaches its highest level for c & c.

A summary of possible equilibria

Table 1 presents a summary of the possible equilibria. Except for the case of large costs and a small proportion of bad banks (q < q 1 ), the game features a typical con…guration of multiple equilibria, where the equilibrium that will actually materialize depends on investors'beliefs.

"Small" cost (c < c) "Large" cost (c > c)

"A few" bad banks: q < q 1 Separating, Pooling Separating "Many" bad banks: q > q 1 Separating, Pooling Separating, Pooling, Hybrid Table 1. Summary of possible equilibria This analysis was performed using general forms for shareholder utility functions. There is no need to argue much about the modelling bene…t of general forms as compared with speci…c forms. The reverse side of the coin is that in absence of explicit forms for our critical thresholds q 1 and c; it is not easy identify what type of equilibrium would describe the best the banking sector situation in the eve of the crisis. If we consider the overall number of banks in the US and Europe (more than 8000 in the US and about 7500 in Europe), the proportion of "bad" banks can be seen as being small. However, if we limit our analysis to the "exclusive" segment of large, multi-product banks with a global reach, then the proportion of banks that were deeply a¤ected by the crisis turned out to be substantial. In general, di¤erences between these large banks in terms of investment opportunities and operating costs are small. Our analysis does not allow to say "how small is small" as compared to the critical cost c (for which we provide only an implicit de…nition). Thus, on a pure theoretical ground it is impossible to rule out any of the equilibria, including the hybrid one. In a con…guration of multiple equilibria, which one actually materialize depends on shareholders'"sandy" expectations. In any case, the most important result from our general analysis is that in any of the possible equilibria banks take too much risks as compared to the perfect information case.

Conclusion

According to a broad expert consensus, an abnormal accumulation of risk by banks throughout the world was at the origin of the 2007-2009 recession. The analysis in this paper connects this race for risk to the imperfect information in the banking sector. In a world where returns can be observed but risk cannot, banks running with high operating costs would take more risk only to deliver higher returns and be perceived as highly e¢ cient banks. The latter can signal themselves by further increasing risks well above their perfect information optimal level.

Our analysis was built on a very simple model of bank. The representative bank has been represented as a portfolio manager able to arbitrate between risk and return. Such a simpli…ed approach does not allow to address the important role played by bank capital as a bu¤er for adverse shocks or for signaling purposes. This was the price to pay for a transparent analysis of the various signaling strategies.

The game presents several equilibria, all being characterized by excessive accumulation of risk by banks compared to the ideal, perfect information case. Furthermore, depending on the proportion of less e¢ cient banks and the di¤erential in operating costs, the model presents several multiple equilibria con…gurations, an economic environment prone to instability.

Policy implications are straightforward. Any reform able to reduce the asymmetry of information between banks'managers and shareholders should eliminate a key reason of the race for risk.

Yet there is no miracle solution able to achieve this result spontaneously. If banks'true exposure to risk is hard to assess by outsiders and at least some good banks implement their high-risk signaling strategy, then stronger regulation is needed to cap the maximum amount of risk that banks can take. Yet, from a theoretical perspective, what regulation would allow to reach this target is not a settled issue. As already mentioned, the Basel Committee on Banking Supervision decided on September 12, 2010 to gradually rise the minimum requirement for common equity, from 2% before the crisis, to 7% by 2019. It still has to be seen whether such a measures goes in the right direction.

Figure 2 :

 2 Figure 2: Decision Tree

The dramatic fall of Lehman Brothers, or the massive support of the respective governements to rescue Citigroup, UBS, Royal Bank of Scotland, Northern Rock, Hypo Real Estate or Dexia, etc. can be weighted against rather successful stories of Barclays, HSBC, Nomura, Santander, Goldman Sachs, JP Morgan Chase, and so on.

Empirical studies show that over a long period, the average bank incurs costs 20 to 25% higher than costs at the most e¢ cient banks (DeYoung, 1997).[START_REF] Haslem | DEA e¢ ciency pro…les of US banks operating internationally[END_REF] use data envelopment analysis to show that about 20% of the large US bank were ine¢ cient in 1992.

In June 2010, Spain decided to make public the results of stress tests performed on national banks. Despite some protest on behalf of Germany, the other European leaders agreed on making public the stress tests carried out by the Committee of European Bank Supervisors.

Implicitly the model assumes that good and bad banks have the same capital and assets, and di¤er only in operating costs.

Financial literature focuses on return variance (denoted often by 2 ) as a proxy for risk. The risk considered in this text is of the nature of an extreme event. The impact of such event on the …rm cannot be directly inferred from the observed return variance.

The structure of the problem and the solution would not change if, instead of di¤erent operating costs, we assume that banks di¤er in their investment opportunities (thus, the Markovitz frontier would be broader for the stronger bank, and the slope of its capital market line would be steeper than for the weaker bank).

These expectations will be determined over the set of beliefs about the type of bank given the return strategy.

As mentioned in the Introduction, this situation is characteristic for instance of the European banking landscape between June 2009 and June 2010.

we can show that the utility of the good bank that implements the signaling strategy is

In turn, this condition is met only if the operating cost is not too big, i.e. if c < R 0 + v R H :