Andrew E Clark
email: andrew.clark@ens.fr

David Masclet
email: david.masclet@univ-rennes1.fr

Marie-Claire Villeval
email: villeval@gate.cnrs.fr.

Alan Krueger

George Loewenstein

Andrew Oswald

Effort and comparison income Survey and experimental evidence

Keywords: M54, J33, A13, C92, D63 Effort, comparison income, rank, income distribution, peak-end, experiments M54, J33, A13, C92, D63 Effort, Comparison Income, Rank, Income distribution, Peak-End, Experiments

Introduction

One of the perhaps rare subjects that has inspired research across a variety of social science disciplines is that of status or social comparisons. A growing literature in economics has been devoted to the role of comparisons in explaining a number of behaviors, including financial market behavior [START_REF] Campbell | By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior[END_REF], criminal activity [START_REF] Glaeser | Crime and Social Interactions[END_REF], or subjective well-being [START_REF] Clark | Satisfaction and Comparison Income[END_REF][START_REF] Brown | Does Wage Rank Affect Employees' Wellbeing?[END_REF][START_REF] Ferrer-I-Carbonell | Income and Well-Being: An Empirical Analysis of the Comparison Income Effect[END_REF][START_REF] Luttmer | Neighbors as Negatives: Relative Earnings and Well-Being[END_REF]).

One part of this literature has focused on the relationship between relative income and labor market behavior. Quits have been shown to be negatively correlated with the average wage in the firm for similar workers [START_REF] Galizzi | Relative Wages, Wage Growth and Quit Behavior[END_REF]); women's labor force participation is influenced by relative income (specifically, whether the woman's husband earns less than the woman's sister's husband: [START_REF] Neumark | Relative Income Concerns and the Rise in Married Women's Employment[END_REF]; and rank in the local income distribution is a good predictor of migration [START_REF] Stark | Migration Incentives, Migration Types: The Role of Relative Deprivation[END_REF]. These behaviors mostly concern job choice. However, little is known about the impact of relative income on employee effort within the job, although efficiency wage theories are built on the concept of income comparisons between co-workers and justify wage compression via equity concerns [START_REF] Akerlof | The Fair Wage -Effort Hypothesis and Unemployment[END_REF][START_REF] Frank | Are Workers Paid Their Marginal Products?[END_REF].

In this paper we try to fill this gap. We analyze the influence of income comparisons on effort using both survey and experimental data. We suggest that such income comparisons may explain why some of the empirical evidence on the wage-effort relationship is mixed. It is commonplace to assume that wages have both incentive and selection effects [START_REF] Prendergast | The Provision of Incentives in Firms[END_REF][START_REF] Lazear | Performance Pay and Productivity[END_REF] and that reciprocity motivates employees to respond to high wages by greater effort [START_REF] Akerlof | The Fair Wage -Effort Hypothesis and Unemployment[END_REF][START_REF] Fehr | Does Fairness Prevent Market Clearing? An Experimental Investigation[END_REF]. Empirically, however, higher wages are not always associated with higher effort [START_REF] Gneezy | Pay Enough or Don't Pay at All[END_REF]. This is explained by a crowding-out effect of monetary rewards on intrinsic motivation [START_REF] Frey | The Cost of Price Incentives: An Empirical Analysis of Motivation Crowding-Out[END_REF], supra-optimal motivation generating choking under pressure [START_REF] Ariely | Large Stakes and Big Mistakes[END_REF][START_REF] Baumeister | Choking under Pressure: Self-Consciousness and Paradoxical Effects of Incentives on Skillful Performance[END_REF]), or an earnings target which bounds effort at some threshold [START_REF] Camerer | Labor Supply of New-York City Cab Drivers[END_REF]. Here, we suggest that under social comparisons, individual effort level will depend on both income and position in the income distribution. In the context of worker effort, this reference income produces classic omitted variable bias.

We introduce an explicit measure of reference group income into both survey and experimental data to isolate the effect of income comparisons on worker effort. There is a strong correlation between work effort and own income in all of our specifications. We also identify robust effects of income comparisons on effort supplied at work: those who are relatively well-paid work harder.

Given own income, we test which of relative income (i.e. the ratio of own income to comparison income) and income rank in the reference group is most important in determining effort: in other words, are social comparisons cardinal or ordinal? We further ask whether individuals compare up (to those who earn more), or down (to those who earn less), or both. Last, we consider whether income comparisons are not only horizontal (i.e.

to other individuals at the same point in time) but also intertemporal, so that changes over time in the individuals' own income or rank matter in determining their effort at work. Many find the idea of social comparisons seductive, but conclusive empirical proof of their existence has been elusive. This is partly because it is difficult to know to whom individuals compare, and because individuals' behavior may be correlated within a group, not because they compare to each other, but because they are exposed to common unobserved environmental factors [START_REF] Manski | Identification of Endogenous Social Effects : The Reflection Problem[END_REF]. Our empirical strategy for analyzing social comparisons in effort decisions allows both of these criticisms to be side-stepped.

Evidence is based on both experimental and survey data.

The experimental approach has the advantage of defining a priori the reference group, rather than having to infer it from survey data 1 , and of avoiding any possible role for contextual effects. In addition, it relies on actual and costly decisions instead of subjective reported behavior. Survey data, on the other hand, has the clear advantage of larger sample sizes, and avoids the criticism that laboratory experiments are to an extent unrealistic, either because participants are unrepresentative, or because behavior in an experiment is not typical of that in real life. The joint use of both survey and experimental data is still very recent [START_REF] Fehr | A Nation-Wide Laboratory. Examining Trust and Trustworthiness by Integrating Behavioral Experiments into Representative Surveys[END_REF][START_REF] Brown | Does Wage Rank Affect Employees' Wellbeing?[END_REF][START_REF] Carpenter | Do Social Preferences Increase Productivity? Field Experimental Evidence from Fishermen in Toyama Bay[END_REF][START_REF] Cummings | Effects of Tax Morale on Tax Compliance: Experimental and Survey Evidence[END_REF] and can be seen as a joint test of robustness. If there are consistent patterns in both types of data, we can have greater confidence in the external validity of laboratory experiments.

Our survey data come from the 1997 wave of the ISSP (International Social Survey Program). This survey includes data on both earnings and self-reported discretionary effort. We examine the correlation between effort and both reference group average earnings, and income rank within the reference group.

Our laboratory experiment extends the gift-exchange game between an employer and an employee, as introduced by [START_REF] Fehr | Does Fairness Prevent Market Clearing? An Experimental Investigation[END_REF], by allowing income comparisons between employees from various firms. The reference group for employees 1 More problematically, even if we are sure about the reference group, we have to suppose that individuals know what others earn. Alan Krueger (personal communication) has carried out work asking people in two companies to identify another employee at the company that they were close to and then to estimate that person's wage. The correlation between this estimated wage and what the other employee actually earned (using the company's payroll records) was low.

in this experiment consists of other employees participating in the same experimental session and placed under similar experimental conditions. All employees have the same cost of effort function. In contrast with recent attempts to show how co-workers' wages within firms affect effort [START_REF] Güth | Fairness within Firms: The Case of One Principal and Multiple Agents[END_REF][START_REF] Charness | Do Co-Workers' Wages Matter? Theory and Evidence on Wage Secrecy, Wage Compression and Effort[END_REF], we set up firms with only one employee, and a reference group consisting of other identical employees in different firms. In the first stage of this game, the employer offers a wage contract. In the second stage, employees who accept the contract decide on their effort level. In one of the treatments (the Information Treatment) we can identify income comparisons, as we inform employees, before they choose their effort level, about the wages offered by a sub-set of other employers in the labor market. In the Benchmark Treatment no such information is given. Both treatments are conducted according to a perfect stranger matching protocol. This game is well-suited to study how income comparisons affect the extent of reciprocity between employers and employees. 2We have four key findings. First, both the survey and experimental data show that individual effort depends not only on own incomes but also on relative incomes. This stands in sharp contrast to standard economic analysis of the income-effort relationship.

Second, rank in the income distribution matters more than the level of relative income. This suggests that comparisons are more ordinal than cardinal. Third, both survey and experimental data provide some evidence that individuals are more sensitive to downward than upward comparisons. The greater the distance of own income to the minimum income in the reference group, the higher the level of effort, at a given income level and income rank. Last, income is compared over time. For a given current income and income rank, effort is lower the higher the maximum income the individual received in previous periods; an analogous result is found for past income rank.

The paper is organized as follows. Section 2 surveys the literature on social comparisons, utility and behavior. Section 3 presents the empirical strategy with respect to both the ISSP survey data and the laboratory experiment. Section 4 reports the results from both data sources, and Section 5 discusses these results and concludes.

Literature

The existing literature on social interactions or comparisons can be broadly divided up into two strands: that on behavior and that on utility. This division can be illustrated by a direct utility function:

U i =U(a i , a j , ...) for j≠i (1)
which (most often) gives rise to a decision rule for i's utility-maximizing choice of a as:

a* i = f(a j , ....) (2)
The behavior and utility approaches to social interactions attempt to find empirical counterparts to (2) and (1) respectively.

There are a number of drawbacks to the behavioral approach. First, data on behavior is not always particularly accurate. Second, behavior often reflects the intersection of supply and demand, whereas we are interested here in individuals' preferences. Lastly, under separability conditions3 , others' behavior can affect my own utility, but not my behavior. There are equally problems with the utility approach, via equation (1): in particular, we do not necessarily know how to measure individual utility, U i .

Interactions in behavior have been widely modeled econometrically despite the identification problems emphasized by [START_REF] Manski | Identification of Endogenous Social Effects : The Reflection Problem[END_REF]. 4 Many of these studies have concluded that social interactions do indeed influence behavior, in the sense that if you do more of something, then I am likely to do more as well. One interpretation is that this reflects a concern for status or relative standing. Another is the possibility that individuals might be learning from each other about how pleasant or dangerous goods or activities are (so that their behaviors are correlated), rather than caring about their status. Rival explanations emphasize the perhaps key role of common omitted variables such as contextual effects, although much care is typically exercised in the empirical literature to defuse this interpretation.

An alternative approach appeals to proxy measures of utility, such as life satisfaction, job satisfaction, and happiness. Perhaps because of a scarcity of surveys which measure both proxy utility and behavior adequately, most attention has been concentrated on the role of income comparisons in the utility function. Empirical estimation has thus mostly been based on the indirect utility function, testing specifications such as V i =V(y i , y j , ...) for j≠i (3) rather than its direct counterpart (1) above.

Both the behavior and utility approaches require that the reference group be identified:

to whom does the individual compare? There are a number of potential candidates, including the individual's peer group (those who share the same characteristics), others in the same household, spouse/partner, friends, neighbors, work colleagues, and the individual herself in the past. 4 Recent contributions in this vein have analyzed saving [START_REF] Duflo | Participation and Investment Decision in a Retirement Plan: The Influence of Colleagues' Choices[END_REF], tax evasion [START_REF] Fortin | Tax Evasion and Social Interactions[END_REF], labor supply [START_REF] Woittiez | Social Interactions and Habit Formation in a Model of Female Labour Supply[END_REF][START_REF] Aronsson | Identifying Interdependent Behavior in an Empirical Model of Labour Supply[END_REF], and students' success at school [START_REF] Arcidiacono | Peer Effects in Medical School[END_REF][START_REF] Sacerdote | Peer Effects with Random Assignment: Results for Dartmouth Roommates[END_REF]).

An approach to modeling social comparisons which combines both of the above would be to consider behavior as a function of both absolute and relative income a i =a(y i , y j , ...) (4)

which can be operationalized empirically as

a i = A 0 + βy i + φy j + γ'X i + ε i (5)
This is the approach that we take in this paper. If only own income matters in explaining i's behavior, then the estimated value of φ will be insignificant. On the other hand, if relative income is important in explaining behavior, then both β and φ will be significant. If action a is normal then we expect β > 0 and φ < 0.

The behavior we consider here is effort expanded at work: we ask whether workers' effort depend on how much others earn, modeling

e i =e(y i , y*, ...) (6)
where we expect e 1 > 0 but e 2 < 0. Here y* is considered to be some transformation of the income vector of other people who are in individual i's reference group. Equation ( 6)

follows naturally from [START_REF] Adams | Toward an Understanding of Inequity[END_REF] equity theory, which states that, for person i and person j, individuals endeavor to ensure that the following equality holds:

Outcome j / Input j = Outcome i / Input i ( 7
)
The idea is that individual i has a comparison or reference person or group, j, and reduces his or her own input or effort until the ratio of their own income to effort is in line with that of the reference person.

Much of the efficiency wage literature is also based on the idea of the comparison of one's own wage to those of co-workers [START_REF] Akerlof | The Fair Wage -Effort Hypothesis and Unemployment[END_REF] or of workers in other firms [START_REF] Summers | Relative Wages, Efficiency Wages, and Keynesian Unemployment[END_REF][START_REF] Johansen | Efficiency Wages, Interfirm Comparison and Unemployment[END_REF]. However, empirical evidence that workers' effort does in fact depend on relative income remains slight.5

A recent literature in experimental economics examines the effects of social comparisons among co-workers. In the first stage of the game proposed by [START_REF] Charness | Do Co-Workers' Wages Matter? Theory and Evidence on Wage Secrecy, Wage Compression and Effort[END_REF], a principal can propose different wages to her two employees. These employees may have either homogenous or heterogeneous ability levels, but the direction and magnitude of these differences are unknown to employees. In the second stage of the game, employees choose their level of effort. According to the treatment, wages are either public or private. Income comparisons are shown to affect only weakly employees' behavior, whereas firms reduce any income difference between co-workers for fear of retaliation from the lower-paid employee in terms of effort. In other words, firms anticipate a negative effect from income comparisons on effort that is not actually observed in workers' behavior. This wage compression effect was also observed by [START_REF] Güth | Fairness within Firms: The Case of One Principal and Multiple Agents[END_REF] in a principal-agent game in which information about the contracts offered to each employee is manipulated. They show that principals tend to reduce the income differential between employees when contract information is made public.

In these experiments, productivity differences are introduced between co-workers in order to motivate firms to vary their wage offers. The weak reaction by employees to subsequent income comparisons may show that productivity differences are considered as a fair source of income differentials. In our experiment, on the contrary, all employees have the same productivity, each firm only employs one worker, and income differences result from firms' various choices (and not from any skill differences between workers). [START_REF] Gächter | Wage Comparisons and Performance: An Experimental Approach[END_REF] provide another experimental test using the strategy method:

subjects are asked to report their effort decision in reaction to various hypothetical income distributions. They identify a large subset of individuals who reduce their effort when faced with income inequality. In our experiment, incomes are actually chosen by real firm-subjects, and we infer the influence of income comparisons from individuals' observed effort decisions.

Empirical strategy

Our empirical strategy is based on the joint use of survey data and experimental data produced in the laboratory.

Survey Data on Work Effort

Our survey data, multi-country and cross-section, come from the 1997 Work Orientations module of the International Social Survey Programme, the ISSP (http://www.issp.org).

The key variables in our empirical analysis are effort, earnings and hours of work.

Individual yearly labor market earnings are converted to U.S. dollars using Purchasing

Power Parities from the OECD. Hours of work are measured at the weekly level.

Our dependent variable is effort at work. In the ISSP, this is crafted to measure discretionary effort, and is thus arguably well-suited to our analysis. All those in employment are asked to indicate the extent to which they agree with a number of statements. One of these is: "I am willing to work harder than I have to in order to help the firm or organization I work for to succeed".

The weighted distribution of answers to this question in the 1997 ISSP is shown below in percentages.

2004). We are interested in describing differences in the response to the effort question between individuals. A first pass is to look at the cross-country pattern in discretionary effort. To do so, we allocate a value of 5 to "strongly agree" through to a value of 1 for "strongly disagree" to the answers to the first effort question. Table 1 shows the number of observations and mean effort, ranked by country from the lowest to the highest effort. The main thrust of our paper is, conditional on country, to see how workers' discretionary effort is related to individual demographic and job characteristics. We are especially interested in the role of income. We control for individual income and hours of work, but also for reference group income. This latter is defined in a similar way to that in the Leyden school: by calculating average values of income in fairly broad demographic groups, 7 here country, gender, education and age. There are three education groups (10 or fewer years of education, 11 to 13 years education, and over 13 years education), and three age groups (16 to 29, 30 to 44, and 45 to 65). There are thus 17 (country) * 2 (sex) * 3 (education) * 3 (age) = 306 reference groups. These average income measures are called comparison income in the regression tables, and correspond to 6 There is a strong correlation between unemployment and the mean of this effort variable. The average OECD standardised unemployment rate in 1997 of the lowest seven countries in the ranking was 12.3%, as against 5.9% for the ten highest-ranked. Both the Pearson and Spearman correlations between mean effort and the unemployment rate are significant at better than the 2% level. One interpretation is that social reciprocity allows firms and employees to attain Pareto-superior employment outcomes. y* in equation ( 4) above. 8 Comparison income for individual i in cell j is calculated excluding i's own income, which obviates the need to cluster the regressions at the reference group level.

There are a certain number of limitations to this survey-based approach. One salient point is that income is measured with a certain amount of error in surveys. More specific to this paper's concerns, the effort measure is self-reported rather than being observed directly. Last, we have posited a certain reference group (by country, gender, education, and age), but we cannot be sure that this is the most relevant one. For these reasons, we complement our survey analysis with experimental evidence on income and effort.

Experimental Design

The game We identify the impact of income comparisons on effort using a version of the standard gift-exchange game [START_REF] Fehr | Does Fairness Prevent Market Clearing? An Experimental Investigation[END_REF]. Each session involves twenty subjects who are divided into two groups, ten in the role of firms and ten in the role of employees. Roles are attributed at random and are kept constant throughout the session. All employees have the same characteristics, in contrast to [START_REF] Güth | Fairness within Firms: The Case of One Principal and Multiple Agents[END_REF] and [START_REF] Charness | Do Co-Workers' Wages Matter? Theory and Evidence on Wage Secrecy, Wage Compression and Effort[END_REF]. Workers do not differ in ability and they do not have to form beliefs about the relationship between incomes and other employees' productivity. A Benchmark Treatment and an Information Treatment have been designed.

The Benchmark Treatment consists of the standard gift-exchange game. In each of the ten periods of the game, each firm is matched randomly with an employee. Each period consists of two stages.

In the first stage, the firm offers a contract consisting of a wage [ ] 20,120 w∈ to its employee. In the second stage, the employee decides whether to accept or reject the contract. If the contract is rejected, both the firm and the employee receive nothing.

Upon acceptance, the employee has to choose his level of effort,

[ ] 0.1,1 e ∈
. The higher the level of effort, the higher the firm's profits but the greater the effort cost c(e) borne by the employee. This effort cost is convex, as shown in Table 2. In this first-generation gift-exchange game, the employer's payoff is:

( ) P v w e π = -
where v is an exogenous redemption value; in our experiment, v=120. The employee's payoff is:

( ) A w c e π = - -20
with a fixed labor market participation cost of 20 (corresponding to travel costs, say).

These payoff functions are common knowledge. At the end of the period, the firm is informed about the level of effort chosen by the employee, and both the firm and the employee are informed about their respective payoffs. In each new period, the pairs of firms and employees are randomly reshuffled. We implement a perfect stranger matching protocol so that no subject is with another subject more than once, and this is made common knowledge in the instructions. This allows us to rule out any reputation-building behavior [START_REF] Gächter | Reputation or Reciprocity?[END_REF].

The Information Treatment has the same structure. The difference lies in the fact that at the end of the first stage, after the firm's income offer is revealed, the employee is told about the income offered by four other firms to their employee in the same period.

Employees can thus compare their own income to the income offered to other a priori similar employees on the labor market (but not co-workers) before rejecting or accepting the contract and thus before choosing a level of effort. In contrast, the firm is not informed about the other firms' income policies. We choose to display only partial information about other income offers (in each period, four other randomly chosen income offers, instead of the whole distribution) to produce a greater variety of income distributions within the reference group. However it might be possible that, in both treatments, income and effort be above the theoretically predicted levels. Indeed the literature has shown that an employee typically reciprocates a high (low) income offer by choosing a high (low) effort level that increases (decreases) her firm's payoff [START_REF] Fehr | Reciprocity as a Contract Enforcement Device: Experimental Evidence[END_REF]Fehr, Kirchsteiger, and Riedl 1998) 9

Equilibrium of the game

. In addition, it is also possible that information about the income distribution may affect the effort level in the Information Treatment if individuals are sufficiently sensitive to income comparisons. If subjects make horizontal comparisons (i.e. among employees), we may expect that effort will be positively correlated with both 9 One might argue that individuals may also reciprocate higher income rank and higher relative income with higher effort in the Information Treatment. However, in our experiment, firms were never informed about relative income and income rank. On the contrary, as firms are never informed about the income distribution, there is no reason why their behavior should differ across treatments. 10

Procedures The experiment was conducted in the experimental laboratory of GATE, Lyon, France, using the Regate software [START_REF] Zeiliger | A Presentation of Regate, Internet Based Software for Experimental Economics[END_REF]. A total of 120 undergraduate t-test, p<.0001) but there is no significant difference in average income between the two treatments. Firms do anticipate reciprocity from their employees, but they do not expect comparisons between them. Each employee made 10 contract acceptance decisions and, if the income distribution. As a consequence, ranking could not be considered as intentional on the part of the employer. 10 A firm cannot know whether a lack of reciprocity is due to comparisons or to the employee's selfishness.

the contract was accepted, chose an effort level. 11 The next section concentrates on the analysis of these effort decisions.

Each session lasted one hour on average, including the payment that was carried out in a separate room. Each subject earned on average €14 from the experiment.

Results

Effort may be influenced by relative income or income rank, by the width of the income distribution, or by the income the individual received in the past if there are intertemporal comparisons.

Effort and comparison income

We estimate discretionary effort equations on both datasets to determine whether income 12 Tobit models are used in the experimental data to account for both the right and left-censoring of the effort variable. It is also noticeable that social motivations lead employees to reciprocate higher incomes with greater effort as usually observed in the gift exchange game [START_REF] Fehr | Reciprocity as a Contract Enforcement Device: Experimental Evidence[END_REF]Fehr, Kirchsteiger, and Riedl 1998). We find a positive relationship between income and effort in both treatments, as shown by the mean income per effort level in Table 3. While the income-effort relationship looks steeper in the Information Treatment, the joint presence of income and comparison income makes such bivariate conclusions untrustworthy. The effort regression results are reported in Table 4 below. The first two panels correspond to the two different ISSP effort measures, and the third to the effort decisions in the Information Treatment of our experiment. Each panel consists of three regressions.

The first regression controls for own income or income and hours of work. The second regression adds the normalized income rank: a higher value of this rank variable thus corresponds to a higher position in the reference group income distribution. The third regression replaces income rank by average reference group earnings.

In the ISSP data we also control for age, gender, education and marital status, and country dummies. In the experimental data we only control for gender since there is little variance in age, marital status and education. In the latter, we also include a "period 1" dummy variable, which takes the value 1 for the first period of the game and 0 otherwise, to account for a possible learning effect, and a "period 10" variable, which takes the value 1 for the last period of the experiment and 0 otherwise, to pick up end-game effects.

Lastly, we control for a possible experience effect, with a dummy variable for having already participated in an economics experiment.

the two treatments or if it is attributable to the dissemination of income information. If the latter, some fraction of minimum effort decisions are motivated by social comparisons rather than selfishness. giving a rank/income elasticity of 1.6. This higher elasticity may reflect the wider distribution of income in the survey data, the fact that rank matters more "in real life", or that rank is more important when reputation-building is possible.

The evidence from both survey and experimental data thus points to income position within the reference group as being an important determinant of how much discretionary effort workers provide, over and above the actual income they receive, which latter has been the focus of the literature to date.14 This, to our knowledge, is one of only a small number of empirical findings pointing to relative income and status as a determinant of employees' behavior.

In the third column in each panel, average reference group income attracts a negative coefficient in two out of three regressions, but is not significant except in the experimental data. If we include both the normalized rank and the reference earnings in the third regression (not reported here), this marginally significant effect disappears, whereas the coefficient associated with rank remains significant. Our second key result is therefore that ordinal comparisons, as measured by normalized rank in the income distribution, are a more powerful predictor of employee behavior than cardinal comparisons, i.e. the differences in earnings expressed in currency units. 15Other results in Table 4 show that effort increases with hours of work in the ISSP data.

Also effort is higher for the married and the higher-educated. The estimates on the country dummies (not shown) largely reproduce the effort ranking in Table 1. Table 4 also shows that effort in experimental data is higher in the first round, and lower, but not significantly so, in the last round. There seems to be learning at the beginning of the game but no end-game effect. Last, those who have already participated in an experiment in the past exert lower levels of effort. 16The estimated coefficients on the male variable are of interest. These are positive and significant for the first ISSP effort variable (willingness to work hard). However, men are far less likely to agree with the statement that they work so hard that it interferes with their life. Rather than seeing this as a contradiction, we believe that this is a manifestation of the sharp differences in the time that men and women devote to domestic work. In the experimental effort regression, the male dummies are negative but not significant.

Upwards and Downwards Comparisons

Table 4 showed that individuals are motivated by both absolute and relative income. The specification in Table 4 treated all individuals as being equally important -they all contribute to mean income and to rank in the same way. However, it is also useful to ask if comparisons are symmetric upwards and downwards: Are some people more salient for comparisons than others?

This is easiest to see in the experimental data, where each individual is confronted with four other income offers that have been made to workers like herself before making her own effort decision. We estimate random effects Tobit models with own income and income rank, and look for up/down asymmetries by separately introducing the value of the minimum income in the distribution and the maximum income in the distribution.

We proceed in the same spirit in the survey data. Here, within each reference group, we calculate the income values corresponding to the 25 th and 75 th percentiles.17 These are added separately to the effort regression which controls for income and income rank within the reference group. We also include controls for gender, age, education level, marital status, and country dummies.

If individuals are only sensitive to the first moment of the income distribution, then we expect these new variables to be insignificant. If, however, they respond to the spread of the distribution then we may expect some effect of these new variables on effort. In addition, if individuals are more sensitive to the bottom of the distribution than to the top (or vice versa), then the estimated coefficients on the two new variables will not be equal in absolute size. The results are in Table 5. With the introduction of income dispersion, own income and income rank remain positive and significant in both the survey and experimental effort equations. The dispersion measures themselves are only significant in one direction: downwards. In the ISSP data, individuals report working harder the greater is the distance between their own income and that of the 25 th percentile. In the experiment, effort reciprocity is greater the larger the distance between the individual's own income and the minimum income revealed in the information treatment. It is worth emphasizing that these effects occur at given levels of both income and income rank.

These results emphasize the role of the income distribution, given the individual's position in it, in determining effort. If rank matters, then every point in the income distribution potentially affects workers' effort. Table 5 suggests that there is an additional asymmetric distributional effect from incomes towards the bottom of the distribution.

Comparisons seem more important downward than upward: individuals work harder when their income or rank is high, and when those beneath them are some distance away. This is consistent with loss-aversion in income rank.

Effort and Past Income

The results so far have discussed the relationship between comparison income and individual effort. Here we turn to comparisons to the income that the individual herself received in the past. The broad idea, as in the addiction literature, is that past exposure to higher incomes may reduce the utility associated with current incomes and thus decrease the current level of effort. This hypothesis has been tested with measures of satisfaction in panel data (see [START_REF] Clark | Are Wages Habit-Forming? Evidence from Micro Data[END_REF][START_REF] Weinzierl | Estimating a Relative Utility Function[END_REF]), but not with measures of behavior such as effort. In parallel, a separate literature on time-inseparability in behaviors such as consumption and labor supply has developed.

One difficulty in these literatures has been to ensure that ceteris paribus holds over the long time-periods between waves. Experimental data are ideally-suited to testing models of habituation since we impose the same environment over time, especially in the perfectstranger framework where there is no role for reputation building. We therefore investigate the role of previous income in determining current levels of effort, by estimating random effects Tobit models with the experimental data only. Our a priori is that a higher past income will reduce current effort, with past income acting as a benchmark.

We pick up the effect of past income by including the running maximum income and the running minimum income as additional explanatory variables. That is, does effort at time t depend on the highest (lowest) income the individual had been offered up to and including time t? We carry out an analogous analysis with respect to rank to determine whether effort is influenced more by past income or by past income rank. This running maximum/minimum specification is inspired by the peak-end transformation, which has been used to model how a flow of pain is converted into a final global evaluation [START_REF] Redelmeier | Patients' Memories of Painful Medical Treatments: Real-Time and Retrospective Evaluations of Two Minimally Invasive Procedures[END_REF]. 18 The regression results are shown in Table 6. Table 6 shows that the past matters: for a given income and a given income rank, effort is significantly lower the higher is the most generous income offer received in the past (column 1), and the higher is best income rank achieved in the past (column 2). The running minima of income and rank do not seem to matter (although the positive coefficient on running minimum is almost significant at the ten per cent level). High income and high income rank seem to be particularly salient, however, and this evidence suggests that they are used as benchmarks with which to evaluate the current offer's generosity, and thus the degree of reciprocity. The last regression in column 3 considers running maximum/minimum income and rank together and indicates that the past best position in the distribution matters more than the past best absolute income. Including the reference income in the regressions (not shown) does not change the level of significance of these key variables.

Discussion and Conclusion

Evidence for the role of status or comparisons in determining behavior remains elusive.

In this paper we have looked for effects of income comparisons on discretionary work effort in both ISSP survey and experimental data. We have four key findings.

First, effort at work depends both on the individual's own income, and on what others earn, both in survey and experimental data. Our results thus contribute to the still small literature showing that comparisons affect behavior via actual costly decisions and not only utility via self-reported well-being. We further believe this to be one of the first papers to combine survey and experimental data to do so.

Second, income rank (i.e. first, second, third in the relevant distribution) is a better predictor of effort decisions than is average reference group income. As such, comparisons are ordinal rather than cardinal.

Third, we suggest that the width of the income distribution matters independently of the individual's own position in it. Holding absolute income and income rank constant, individual effort is higher the greater the distance to the bottom of the distribution;

distance to the top of the income distribution did not matter. To this extent, income comparisons are stronger downwards than upwards.

Last, the income profile over time matters in itself. Those who received higher income or higher income rank in the past supply less effort today, at a given income and income rank. This result is potentially important for understanding for example the frequent failure of mergers. While the literature has concentrated on the role of income, mergers may involve substantial changes in rank as well; we have shown the latter to be a strong determinant of motivation.

There are a number of explanations of the rank-sensitivity of effort. We have presented our results in terms of income comparisons and concern for status. Alternatively, effort choice may derive from inequality-aversion (see for example [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF][START_REF] Bolton | Erc: A Theory of Equity, Reciprocity, and Competition[END_REF]: those who receive a high income increase their effort so as to reduce the difference between their own earnings (i.e. income minus effort cost) and those of lower (and particularly the lowest) income workers. While it is difficult to distinguish cleanly between theories, we note that inequality-aversion would predict a stronger effort role for others' incomes than for income rank, whereas in both experimental and survey data we find the opposite. Also inequality-aversion does not explain the role of past income and income rank in explaining current effort, whereas income comparisons do.

Another alternative explanation of the experimental results is that workers learn what the "fair income" is in the group: their effort does not depend on within-period comparisons as such, but by the search for the norm. However, if this were the case, they should reject more offers over time as they learn what the fair income is, and should reject more contracts in the Information Treatment than in the Benchmark Treatment. Neither of these predictions holds. Searching for the fair income also implies that individuals pay attention to both the top and the bottom of the income distribution, whereas for given income and income rank, individuals only care about the gap between their income and lower earnings.

A very general implication of our work is that combining survey analysis, based on subjective data, and experiments in a controlled environment serves as a validation exercise. While both approaches have been criticized for separate reasons, here they produce remarkably similar and consistent results.

Over 20 years ago, Bob Frank (1985) suggested that firms can trade off status and wages. This paper has shown that these two are indeed substitutes in terms of inciting worker effort. Worker effort is lower in face of both absolutely and relatively low incomes, where this relativity concerns both others in the same period and oneself in previous periods. This may explain why firms favor income secrecy, and also why the same income at a point in time might produce different levels of effort. They also demonstrate the concrete advantage accruing to firms paying rising income profiles. More generally, income comparisons, both to others and to oneself in the past, seem to be a pervasive element of economic life.

How are payoffs in each period determined?

The employee's payoff 1. If the employee has rejected his firm's offer, his payoff is zero for the period.

2. If the employee has accepted his firm's offer, the employee receives his income. He must subtract from this income both a transportation cost of 20 points and the cost associated with the quantity of work he has chosen.

3. The employee determines his quantity of work in choosing a number in between .1 and 1, as indicated in the Table below. The smallest quantity of work is .1 and the largest is 1. The higher the number chosen, the greater the quantity of work, and the higher the firm's payoff.

4. The greater the quantity of work chosen, the higher is the associated cost to the employee. The Table below shows how costs vary with the quantity of work.

5. In the case that the income offer is accepted, the employee's payoff in points is determined as follows:

Employee's payoff in points in each period = The employer's payoff 1. At the beginning of each period, the firm receives 120 coupons from the experimenter that can be used to pay the income of the current period. If the firm offers a income of 120 points to his employee and if this offer is accepted, then the firm has no coupons left. If the firm offers a income of 20 points to his employee and if this employee accepts this offer, then he has 100 coupons left. More generally, the firm keeps:

Income -cost of the quantity of work -transportation cost

120 coupons -the income paid to the employee 2. How are the remaining coupons converted into points? The number of coupons kept by the firm is multiplied by the quantity of work chosen by the employee. The result indicates the firm's payoff in points for the current period. Then, Firm's payoff in points in each period = (number of coupons -income) * quantity of work 3. If the employee does not accept his offer, the firm loses its coupons and its payoff is zero for the current period.

At the end of the period, the firm and his employee are informed about their respective payoffs.

At the end of each period, the next starts automatically. The firms and the employees are re-matched randomly to form new pairs.

Throughout the entire session, you are not allowed to talk if not invited to do so. Any violation of this rule will result in being excluded from the session and not receiving payment. If you have any questions regarding these instructions, please raise your hand. Your questions will be answered in private.

--------------------------

  The equilibrium of this game is a minimum wage -minimum effort pair of decisions, [w=20, e=0.1], with selfish and rational players. The minimum wage contract should be accepted since the employee has no better alternative. Equally, the employee should accept the contract and choose the same (minimum) effort level in both treatments since the incomes offered by other firms do not enter into the standard individual utility function. The firms should thus offer the same (minimum) income in both treatments.

  students, from three local Engineering and Business schools, participated in one of the six sessions we organized. Two of these concerned the Benchmark Treatment and four the Information Treatment. No-one participated in more than one session. Upon arrival, the subjects drew a label from an envelope, indicating the name of their computer. The instructions (see the Appendix) were distributed and read aloud. The subjects then filled out a questionnaire which allowed us to check their understanding of the rules of the game. Questions were answered in private. The subjects subsequently discovered their role (firm or employee). The program paired firms and employees randomly and anonymously. As the game was repeated 10 times under a perfect stranger matching protocol, each firm made an income offer to each of the employees. This leaves us with a total of 200 wage offers in the Benchmark Treatment and 400 wage offers in the Information Treatment. The average income offered is 53.51 (Standard Deviation 19.7) in the Benchmark Treatment and 53.09 (S.D. 20.0) in the Information Treatment. Both are clearly above the equilibrium wage of 20 (one-tailed

  comparisons affect individual behavior in both survey and experimental data. Two different specifications of comparison income are used. The first is in normalized rank form, defined as: rank in cell or group / number of observations in cell or group, with a correction for ties. This measure is bounded between just over zero for the bottom-ranked income in the cell to one for the top-ranked income. The second is average reference group income, excluding the individual's own income. Average and individual earnings levels are expressed in thousands of U.S. dollars per month in the ISSP data, and in experimental currency units in the experimental data. Estimation of effort in the ISSP survey data is via ordered Probit. Estimation of the influence of comparison income on effort in the experimental data is via random effects Tobit. 12 11 In the Benchmark Treatment, 20 contracts were rejected (respectively 22 contracts in the Information Treatment). We thus obtained 180 effort decisions in the Benchmark Treatment and 378 in the Information Treatment.

Table 1 .

 1 Mean Discretionary Effort by Country: ISSP 1997. There is something of a country pattern in the degree of social reciprocity at work. Mediterranean countries are broadly towards the bottom of this ranking, while workers in Anglo-Saxon countries are on average more willing to work hard to help their firm or organization. Portugal is an exception to this general rule, appearing towards the top of the ranking.6

		Employees
	Country	interviewed	Mean Effort
		No.	%
	USA	775	6.47	3.93
	Canada	546	4.55	3.75
	Portugal	843	7.03	3.71
	Switzerland	1 727	14.41	3.65
	Denmark	600	5.01	3.64

Table 2 .

 2 The cost of effort in the experiment

	Effort e	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
	Cost c(e)	0	1	2	4	6	8	10	12	15	18

Table 3

 3 shows that minimum effort was chosen 98 times out of 180 accepted contracts (54.4%) in the Benchmark Treatment, and 214 times out of 378 accepted contracts (56.6%) in the Information Treatment.13

Table 3 .

 3 Average income and effort levels in accepted contracts

	Effort level	.1	.2	.3	.4	.5	.6	.7	.8	.9	1
	Benchmark
	Number obs.	98	22	16	12	11	9	4	5	2	1
	Mean income	50.9	50.1	61.5	64.1	69.7	71.1	71.3	80.0	95.0	60.0
	Information
	Number obs.	214	45	32	29	13	18	13	8	0	6
	Mean income	44.4	59.2	65.4	64.0	69.6	75.6	80.8	79.4	0	93.3

Table 4 .

 4 Effort, Rank and Reference Group Earnings Robust standard errors in parentheses. *** significant at the .01 level; ** at the .05 level; * at the .1 level.

	Variables

Table 4

 4 shows that effort is almost always strongly correlated with own absolute income. This result also holds in the Benchmark Treatment. The same regression of effort in this base treatment, not reported here, has own absolute income positive and significant at the 1% level. Moreover, in the second regression in each panel, normalized

	rank attracts a positive and significant coefficient.	For the same number of
	dollars/experimental units earned, individuals are willing to work harder the higher their
	position in the reference group income distribution. In the experiment, a rise in rank of
	one position increases effort by 0.457 (=0.20*2.285), which is equivalent to a wage
	increase of 4.57 for given rank. Compared to average income per period (53.09), this
	represents a 8.6% rise. The rank/income elasticity is thus 0.43 (=8.6/20). With the ISSP
	data, a 20% rank increase is worth $606 per month, which is 32% of average income,

Table 5 .

 5 Effort: Upwards and Downwards Comparisons

	Note: Robust standard errors in parentheses. *** significant at the .01 level; ** at the .05 level; * at the .1
	level.

Table 6 .

 6 Effort and Past Income in the Experimental Information Treatment

	Variable	(1)	(2)	(3)
	Income	0.116***	0.108***	0.105***
		(0.013)	(0.014)	(0.014)
	Normalized Income Rank	1.865**	2.232**	2.585***
		(0.862)	(0.904)	(0.968)
	Running Minimum Income	-0.002		0.004
		(0.013)		(0.017)
	Running Maximum	-0.045***		-0.013
	Income	(0.012)		(0.017)
	Running Minimum Rank		1.217	-0.148
			(0.755)	(1.166)
	Running Maximum Rank		-2.39**	-2.920**
			(1.25)	(1.504)
	Male	1.182***	-0.717*	0.839**
		(0.452)	(0.371)	(0.407)
	Period 10	-0.249	-0.384	-0.316
		(0.468)	(0.456)	(0.471)
	Experience	-0.564	-1.433***	-0.541
		(0.519)	(0.498)	(0.683)
	Constant	-3.138***	-3.652***	-2.777*
		(1.077)	(1.288)	(1.693)
	Observations	338	338	338
	Log-Likelihood	-364.14	-360.44	-363.37

Note: Standard errors in parentheses; *** statistically significant at the .01 level; ** at the .05 level: * at the .10 level

  Transportation cost = 20 points Relationship between the quantity of work and the associated cost

	Quantity of	.1	.2	.3	.4	.5	.6	.7	.8	.9	1
	work
	Associated	0	1	2	4	6	8	10	12	15	18
	cost

In the rest of the paper, when we evoke income comparisons in the experimental data, we define income as the wage offered by the employer to the employee, i.e. we do not take into account the cost of effort that depends on the level of effort chosen by the employee.

Formally, d 2 U i /da i da j = 0.

Most of the experimental work on the impact of others' income tests for inequality aversion and focus on distribution decisions through choices over tax rates, transfers or the distribution of income (see Cowell

See for example[START_REF] Van De Stadt | The Relativity of Utility: Evidence from Panel Data[END_REF]. A summary of the whole Leyden research programme in poverty and well-being is provided by[START_REF] Van Praag | The Measurement of Welfare and Well-Being: The Leyden Approach[END_REF].

This cell-average approach does not suffer from the identification problems which occur when y* is predicted in a regression framework, as the cell-average income is not a linear function of the X variables (the variables which define the cells -here country, age, gender and education).

If we consider individuals instead of decisions, we observe that only a minority of subjects behave selfishly. Defining as selfish individuals those subjects who choose the minimum effort in at least 8 periods out of 10, we have 35% of selfish people in the Benchmark and 27.5% in the Information Treatment. We cannot however determine whether this difference is inherent to the very nature of the subjects involved in

The ISSP results are largely unchanged when we drop the 20% of observations which are found in reference groups with 30 observations or less, or if we use a less aggregated reference group by dropping education, or by age.

This result concurs with that in[START_REF] Brown | Does Wage Rank Affect Employees' Wellbeing?[END_REF], where income rank is shown to outperform average reference group income in three satisfaction equations (influence over the job, achievement, and supervisor's respect). For the fourth dependent variable, satisfaction with pay, both rank and reference group income attract significant coefficients.

[START_REF] Kagel | Handbook of Experimental Economics[END_REF] report many examples in which experience (of the game itself or of the same class of games) leads the subjects to play strategies closer to the equilibrium, although not systematically. This is particularly the case in auctions or public goods games. The impact of experience, considered here as learning how to play a particular class of game, has been particularly discussed in the case of voluntary contribution games. In our experiment, nobody had participated to a gift exchange game in the past but may have participated in a public goods game.

These are only calculated for countries where income is measured continuously, rather than in bands (Hungary, Italy, Sweden, Poland and Switzerland).

Data from period 1 are dropped as income (income rank) and running maximum/minimum income (income rank) necessarily coincide in this period.

grateful to R. Zeiliger for programming the experiment presented in this paper, to K. Ben Abbes and M. Bouamoud for assistance in the lab, and to the MiRe -DREES (French Ministry of Social Affairs) for a grant to support this research.

The ISSP data were collected by independent institutions in each country, and documented and made available by the ZENTRALARCHIV FUER EMPIRISCHE SOZIALFORSCHUNG, Köln. Neither the original collector of the dataset nor the Archive bear any responsibility for the analyses or interpretations presented here. We are also

Appendix: Experimental Instructions in the Information treatment

General information

You are going to participate in an experiment on the labor market for the MiRE-Ministry of Social Affairs. If you read these instructions carefully, you can earn a considerable amount of money. The amount of your earnings depends not only on your decisions, but also on the decisions of the other participants you will interact with. During this session, your earnings will be calculated in points, with 100 points = 4 Euros At the end of the session, all the profits you have made in each period will be added up and converted into Euros. In addition, you will receive a show-up fee of 4 Euros. Your earnings will be paid to you in cash in a separate room in order to preserve confidentiality.

At the beginning of the session, each of the 20 participants will be assigned one of two roles: 10 participants will be "employees" and 10 participants will be "firms". Your computer screen will inform you about your role. You will keep the same role throughout the session. You will never be informed of the identity of the participants you will interact with.

The labor market consists of 10 periods.

Decision-making in each period

Each period consists of two stages.

-In the first stage, each firm is paired randomly and anonymously with an employee. Each firm makes a income offer to his employee. The employee is informed of the income offer made by his firm and he is also informed of the income offers made by 4 other firms randomly chosen in the room.

The employee can accept to work for the income offered by his firm or not to accept his firm's offer. If the employee accepts the offer, he proceeds to the second stage.

-In the second stage, the employees who have accepted an offer must decide on their quantity of work.

The details of the procedure are explained below.

Please note that in each new period, the firm-employee pairs are reshuffled. You are sure not to interact more than once with the same firm or with the same employee if you are a firm.

Information about the labor market in each period

1. At the beginning of the period, the firm makes an offer to the employee. This income is between 20 and 120 points. Information about this income offer will be communicated to 4 other employees.

2. The employee is informed about both the income offer made by his firm and the income offers made by 4 other firms to their employees. These firms are chosen randomly.

3. The employee can accept the offer from his firm and work. He can reject the offer and, in this case, he does not work: both he and the firm earn nothing for the current period. Only the firm is informed about the acceptance or the rejection of his offer by his employee.

4. If the employee has accepted the income offer, he receives his income and must decide on his quantity of work. The firm is informed about this quantity of work but neither other firms nor other employees are informed about it. The employee must bear a transportation cost of 20 points.


David Spector
email: spector@pse.ens.fr

Bundling, tying and collusion

Keywords: bundling, tying, collusion bundling, tying, collusion. JEL classiÞcation: L13

Tying a good produced monopolistically with a complementary good produced in an oligopolistic market in which there is room for collusion can be proÞtable if some buyers of the oligopoly good have no demand for the monopoly good. The reason is that a tie makes part of the demand in the oligopolistic market out of the reach of the tying Þrm's rivals, which decreases the proÞtability of deviating from a collusive agreement. Tying may thus facilitate collusion. It may also allow the tying Þrm to alter market share allocation in the collusive oligopolistic market.

Introduction

This paper shows that a Þrm enjoying a monopoly in market A while also being active in an oligopolistic market B may Þnd it proÞtable to tie products in both markets in order to facilitate collusion in market B. 1 We consider a model in which a fraction of buyers wants goods A and B in Þxed proportions while other buyers want only product B and do not care about product A. In such a context, the "Chicago critique" (or "single monopoly critique") of the leveraging theory, which stresses that raising the price of product A is a sufficient instrument for the product A monopolist to extract consumers' surplus, breaks down, because raising the price of product A does not allow the monopolist to extract surplus from the consumers who only want product B.

One way to extract surplus from these consumers is to collude in market B, and this paper shows that tying may help the product A monopolist to extract surplus by colluding in market B. We prove two results. First, for some parameter values, tying can make collusion in market B possible, which beneÞts both the tying Þrm and its competitors in market B, and harms consumers. Second, for parameter values such that collusion in market B is feasible irrespective of whether products A and B are tied, tying may allow the market A monopolist to increase its market share in market B, and thus its proÞt.

The mechanism behind this result is very simple. Collusion is feasible to the extent that each Þrm is deterred from deviating from the collusive agreement by slightly undercutting competitors. This is where tying helps: if the product A monopolist makes it mandatory for purchasers of product A to also buy product B from it, it insulates a fraction of the demand (that emanating from the fraction of consumers who want both products) from competition. As a consequence, tying makes it less attractive for market B Þrms (other than the product A monopolist) to deviate from any given collusive agreement, because the consumers who want both products are out of their reach anyway. This has two consequences. First, since the incentive compatibility constraint facing the Þrms other than the product A monopolist is relaxed, the set of parameter values inducing the existence of a collusive equilibrium expands: collusion becomes more likely. Second, for parameter values inducing collusion both with and without tying, tying expands the set of market share allocations which can be sustained in a collusive agreement. More precisely, since all Þrms' (except the market A monopolist's) incentive compatibility conditions are relaxed, collusion becomes feasible for market share divisions which are more favorable to the market A monopolist. Thus, tying a good in a monopolistic market with a complementary good in a collusive market may allow the tying Þrm to increase its market share in the collusive market. These results are proved Þrst in the case where the tying good monopolist has the option to irreversibly tie its products before the repeated oligopolistic interaction takes place (sections 2 and 3). We then show that they carry over to the case where the tying good monopolist cannot make any once-and-for-all commitment to bundle, but instead can decide in every period whether or not to offer a bundle (section 4).

Relation to the literature

Two strands of the vast literature on bundling2 are related to the present paper. One of them, initiated by Whinston (1990) has shown that in the presence of economies of scale, a tying-market monopolist can use tying in order to deter entry in the tied market by preventing potential entrants from reaching the minimum viable scale. This strategy can be proÞtable if some tied-market customers have no demand for the tying product: in this case, the Chicago critique breaks down, because the extension of monopoly to the tied market increases the tying good monopolist's proÞt by allowing it to exploit these consumers. 3 The present paper is based on a similar demand structure: some consumers in the tied market also want the tying product, some do not, and tying allows the tying market monopolist to increase the proÞt it extracts from the latter group. However, in the present paper the proÞtability of bundling does not result from its entry-deterring or exit-inducing virtues, but rather from its impact on the feasibility of collusion. Accordingly, the results of this paper do not rely on any assumption about economies of scale in the tied market.

A second strand of literature has shown that bundling may allow a Þrm to mitigate competition by increasing differentiation. 4 While the idea in the present paper is also that bundling may mitigate competition, the mechanism is different. The existing literature compares non-cooperative equilibria with and without bundling, while the present paper shows that bundling may allow Þrms to shift from non-cooperation to cooperation, or may change the division of market shares when the tied market is collusive anyway. Two other differences distinguish the present paper from that literature. First, the present results rely on the assumption that the two goods are complements. Second, in the present paper, the tying Þrm sells both a bundle and the tied product alone, and this point is crucial, because it is through the sale of the tied product alone that the tying Þrm can reap the beneÞts of collusion. On the contrary, the results showing that bundling can mitigate competition by increasing differentiation usually consider the situation in which the Þrm offering a bundle does not also offer the tied product alone. This is because the differentiation between different Þrms' offerings drives the results: some of them offer the bundle alone, while others offer products on a standalone basis.

The model 2.1 Agents, preferences and costs

There are two homogeneous goods, labeled A and B. Firm M can produce both goods at zero cost, while Firm E can produce only product B, at zero cost as well. Product B is not differentiated: the varieties produced by Firms M and E are perfect substitutes for consumers.

A continuum of consumers of total mass one exists and is subdivided into two groups. A mass b of consumers want to consume goods A and B in Þxed amounts (one unit of each). All these consumers have a valuation V 1 for the pair consisting of one unit of each good. A mass (1b) of consumers has no utility for product A. These consumers have a unit demand for product B and a common valuation V 2 < V 1 .We assume that price discrimination is not possible since the good can be resold.

The rate of time preference of both Þrms is denoted δ.5

The game

We consider the following timing.

• In period 0, Firm M decides whether or not it will tie products A and B. This decision is irreversible and cannot be made at any later stage.

If it decides to tie, Firm M will be able to offer two products at the later stages of the game: (i) a pair containing one unit of product A and one unit of product B, and (ii) product B alone. If it decides not to tie, it will be able to offer two products: (i) product A alone, and (ii) product B alone.

• In periods 1,2,3,... until inÞnity, both Þrms simultaneously set prices for the products they sell (i.e. Firm M sets the prices of its two products, which are either products A and B, or an A -B pair and product B, and Firm E sets the price of the product B it sells). In addition, we assume that each Þrm announces, in each period, the maximum amount of product B it is willing to sell. Then, if both Þrms set the same price for product B, consumers who want to purchase product B are allocated to both Þrms pro rata the announced amounts.

The last assumption (i.e. the assumption that each Þrm sets the maximum amount of product B it is willing to sell) is there to take into account the possibility of different divisions of market B.6 3 Tying and collusion

Conditions for the existence of a collusive equilibrium

In order to solve the above game by backward induction, we investigate the feasibility of collusion depending on whether Firm E chose to tie products A and B in period zero. The following lemma, proved in the appendix, characterizes optimal equilibria.

Lemma Assume that the repeated game has collusive equilibria (i.e. equilibria with payoffs different from those of the one-shot game), and consider a Pareto-optimal subgame perfect equilibrium EQ (i.e. one which is optimal from the point of view of the two Þrms, within the set of all subgame perfect equilibria). Then (i) this equilibrium is such that the price of product B is equal to V 2 in every period; and there exists a subgame perfect equilibrium yielding the same payoffs as EQ and such that (ii) each player's equilibrium payoff is identical across periods, and (iii) after any deviation off the equilibrium path, each Þrm's action in the continuation game is the one prescribed by the Nash equilibrium of the one-shot game.

This lemma means that the analysis can be carried out by limiting our attention to simple equilibria, namely those in which the equilibrium path involves both players quoting the monopoly price V 2 and a constant market share allocation, while any deviation from these strategies leads each player to earn the proÞt it would earn in the equilibrium of the one-shot game forever after the period during which the deviation took place. This simpliÞcation allows us to analyze the possibility of collusion as follows, depending on whether Firm M decided to tie both products or not in period 0.

Case 1: Firm M chose not to tie products A and B in period 0.

The above lemma means that a hypothetical collusive equilibrium is equivalent to one characterized by the following "tit-for-tat" strategies for some α between zero and one (denoting the share of the total market for product B attributed to Firm M):

• If no deviation occurred in any past period (i.e. if the actions described hereafter have been played by both Þrms in all previous periods) then both Þrms play as follows:

-Firm M sets the price of product A at V 1 -V 2 and offers product B at price V 2 . It announces that it wants to sell at most α of product B.

-Firm E offers product B at price V 2 . It announces that it wants to sell at most 1α of product B.

• If either Þrm deviated from the above-mentioned actions in any previous period, then -Firm M sets the price of product A at V 1 and offers product B at price zero.

-Firm E offers product B at price zero.

These strategies are equilibrium strategies only if, for each Þrm, the shortterm gain from deviating is smaller than the long-term loss from Bertrand competition in market B. For Firm M, the long-term beneÞt from collusion comes from the fact that without collusion, Firm M would only extract surplus from the group of consumers (of total mass b) who want both products, earning per-period proÞts equal to bV 1 , while collusion allows it to extract the entire surplus of a population of mass α, earning

b(V 1 -V 2 ) + αV 2 = bV 1 + (α -b)V 2 per period. If it deviates, Firm M earns bV 1 + (1 -b)V 2
during one period and bV 1 in all subsequent periods. With a rate of time preference δ, the condition for Firm M not to deviate is thus

α -b ≥ (1 -b)(1 -δ).
( 1 )

Similarly, Firm E earns a per-period proÞt of (1-α)V 2 if no Þrm ever deviates, while if it deviates it earns V 2 for one period and zero ever after. The condition for Firm E not to deviate is thus

1 -α ≥ 1 -δ. ( 2 )
Collusion with a division of market B according to the proportions (α, 1α) is feasible only if conditions (1) and (2) hold, i.e.

1 -(1 -b)δ ≤ α ≤ δ.
(3)

Case 2: Firm M chose to tie products A and B in period 0.

Just as in Case 1, the lemma means that a hypothetical collusive equilibrium is equivalent to one characterized by the following "tit-for-tat" strategies for some α between zero and one (denoting the share of the total market for product B attributed to Firm M, including the part that is tied to market A):

• If no deviation occurred in any past period (i.e. if the actions described hereafter have been played by both Þrms in all previous periods) then both Þrms play as follows:

-Firm M sets the price of the A -B pair at V 1 and offers product B at price V 2 . It announces that it wants to sell at most α-b of product B on a standalone basis (amounting to total sales of product B equal to α once the sales of the A -B pair are accounted for).

-Firm E offers product B at price V 2 . It announces that it wants to sell at most 1α of product B.

• If either Þrm deviated from the above-mentioned actions in any previous period, then -Firm M sets the price of the A -B pair at V 1 and offers product B at price zero.

-Firm E offers product B at price zero.

The incentives facing Firm M are the same as in the previous case: not deviating is optimal if condition (1) holds. But the incentives facing Firm E are different from those arising in the previous case: if it deviates by slightly lowering its price, Firm E captures the entire set of consumers who want only product B (their total mass is 1b), but no more. As a consequence, the one-period proÞt from deviating is not V 2 anymore, but (1b)V 2 instead. The condition for Firm E not to deviate when Firm M tied both products is thus

1 -α ≥ (1 -b)(1 -δ). ( 4 )
Collusion with a division of market B according to the proportions (α, 1α) is feasible only if conditions (1) and ( 4) both hold, i.e.

1 -(1 -b)δ ≤ α ≤ δ + b(1 -δ).
(5)

Main results

Tying in order to facilitate collusion

Conditions (3) and ( 5) yield the following result.

Proposition 1 If the rate of time preference δ is greater than 1 2 but smaller than 1 2-b , then collusion is possible only if Firm M decides to tie products A and B before the repeated pricing game took place.

This proposition is in accordance with the intuition explained in the introduction: tying decreases the proÞtability of deviation for Firm E because it insulates the tied part of market B from competition. This in turn makes it easier to deter it from deviating from collusive equilibria.

Tying in order to increase market share in a collusive market

Even in situations in which collusion would be feasible both with and without tying (i.e. when both conditions (3) and ( 5) hold for some values of α), tying may be proÞtable because because it increases the maximal market share which can be allocated to Firm M in a collusive equilibrium. In order to see this, we simply assume that after the decision whether to tie is made, Þrms bargain over the choice of a collusive equilibrium (i.e., they bargain over α), without making any side payments -this last assumption is very common in the analysis of collusion, because side payments are often more difficult to conceal than price-Þxing or market share allocation.

We assume that Þrms coordinate on a collusive outcome using Nash bargaining, i.e. they choose an equilibrium maximizing θΠ M + (1θ)Π E , where Π i denotes the expected discounted sum of Firm i's proÞts, and the parameter θ denotes Firm M's bargaining power relative to Firm E. The above characterization of feasible market share allocations in collusive equilibria (conditions (3) and ( 5) above, corresponding respectively to the case without and with tying) yields the following result.

Proposition 2 If 1 2-b < δ < θ, then collusion is feasible both without and with tying, but tying causes the tying Þrm's market share to rise from δ to Min(θ, δ + b(1δ)), and its proÞt to increase by

Min(θ -δ, b(1 -δ))V 2 .
This result is related to the familiar idea that tying may help a Þrm to proÞtably increase its market share in the market in which it does not hold a monopoly. If, absent tying, Firm M's market share in the tied market is limited not by its bargaining power, but by the need to leave Firm E a large enough market share (so as to deter it from deviating), then tying allows Firm M to increase its market share, and thus its proÞt, because it relaxes Firm E's incentive compatibility constraint.

Bundling and collusion

We assumed so far that the choice whether to tie products A and B was made once and for all before competitive interaction takes place. This restriction means that the above results, stated as such, can only apply to situations involving an irreversible (or at least costly enough) technical tie, or when a credible commitment to bundle (economically equivalent to a technical tie) is possible. However, many real-world situations involve mere commercial bundling, which often is easily reversible. We show in this section that the above results extend to commercial bundling without commitment.

We modify the game in order to model a situation in which the choice whether to bundle products A and B is made not once and for all before competitive interaction takes place, but rather in every period, simultaneously as decisions over prices. More precisely, we assume that in every period, both Þrms simultaneously make and announce the following choices:

• Firm M announces which products it wants to sell (i.e. A and B, or the A -B pair and B) and at what price, as well as the maximum amount of product B it is willing to sell should both Þrms offer it at the same price.

• Firm E sets the price at which is sells product B as well as the maximum amount of product B it is willing to sell should both Þrms offer it at the same price.

One can easily check that, if parameters induce the existence of a collusive equilibrium with tying under the assumptions of the previous section, then an equilibrium with bundling in every period and exactly the same prices and market shares exists under the assumptions of the previous section. Consider the following strategies:

• If no deviation occurred in any past period (i.e. if the actions described hereafter have been played by both Þrms in all previous periods) then both Þrms play as follows:

-Firm M decides to offer only an A -B bundle priced V 1 as well as product B priced V 2 , and it announces that it wants to sell at most α-b of product B when sold alone (amounting to total sales of product B equal to α once the sales of the A -B pair are accounted for).

-Firm E offers product B at price V 2 and announces that it wants to sell at most 1α of product B.

• If either Þrm deviated from the above-mentioned actions in any previous period, then -Firm M offers product A at price V 1 , and product B at price zero.

-Firm E offers product B at price zero.

The conditions for deviation to be proÞtable are exactly the same as in the game analyzed in the previous section, assuming that in period 0 Firm M had decided to tie both products. Indeed, if Firm E deviates, its proÞt during the deviation period is (1b) V 2 rather than V 2 , because bundling by Firm M prevents Firm E from selling anything to a mass b of consumers.

The fact that bundling may arise in equilibrium is not in itself very interesting, because equilibrium multiplicity in repeated games is a general phenomenon. The interesting result is that the possibility of bundling (i) increases the set of parameters for which collusion is feasible, and (ii) increases the set of market share divisions which are compatible with a collusive equilibrium. Indeed, the above reasoning implies that the results of the previous section carry over to the present one: the word "tying" can be replaced by "bundling" in Propositions 1 and 2.

Conclusion

This paper shows that bundling or tying may be a proÞtable strategy because it may facilitate collusion in the tied market or it may allow the tying Þrm to alter market share allocation in the collusive tied market. Contrary to much of the recent literature, the anticompetitive use of bundling or tying is possible even in the absence of economies of scale or scope in the tied market. The only conditions for the above model to be relevant is that collusion be feasible in the tied market, i.e. in particular that this market be transparent enough, that quick retaliation be feasible, and that entry be difficult enough to make the collective exercise of market power possible. How relevant this simple model is to real markets is an issue left for future empirical research.

Abreu, D., 1988, "On the Theory of InÞnitely Discounted Games with Discounting", Econometrica, 56, 383-396. Athey, S., and K. Bagwell, 2001, "Optimal collusion with private information", Rand Journal of Economics 32, 428-465.

Proof of the lemma.

In order to simplify notations, we consider the case where b = 0, which avoids the need to consider tying; the proof easily carries over to the case b > 0 (the important assumption is that each Þrm can guarantee itself at least its one-shot Bertrand proÞt whatever its rivals' strategies are). Consider a hypothetical collusive equilibrium (i.e. an equilibrium such that aggregate expected proÞts are strictly positive) which is Paretooptimal (from the Þrms' viewpoint) within the set of subgame perfect equilibria. We introduce the following notations:

• U it denotes the proÞt earned in equilibrium by agent i (i.e. Firm M or E) in period t.

•

V it = (1 -δ) X t 0 ≥t+1
δ t 0 -t-1 U it 0 denotes agent i's average equilibrium per-period proÞt from period t + 1 on.

• p t denotes the equilibrium price in the period t.

• P t denotes the average equilibrium price in periods t + 1 and after, deÞned as

P t = (1 -δ) X t 0 ≥t+1 δ t 0 -t-1 p t 0
• α it denotes Firm i's equilibrium market share in period t.

These notations imply the identities α Mt +α Et = 1 and U it = α it p t . We can apply the results about optimal penal codes derived in Abreu ( 1988): all equilibria are equivalent (in terms of payoffs) to equilibria such that out-of-equilibrium strategies involve optimal punishments. In the simple case of price competition with identical marginal cost, the optimal punishment is simply to play the strategies corresponding to the equilibrium of the one-shot game, because each player can guarantee itself the corresponding payoffs (i.e., zero) whatever the other player's strategy. We thus limit our attention to equilibria such that after any deviation, both players forever play the strategies corresponding to the one-shot game (i.e., they set a zero price).

In equilibrium, no Þrm has an incentive to deviate from its equilibrium strategy by slightly undercutting its rival. Therefore, for i = E and M:

U jt = (1 -α it )p t ≤ δ 1 -δ V it . (IC it )
We prove Þrst that for every t, p t is equal to the monopoly price V 2 . Assume that this is not the case, and that for some t 0 , p t 0 < V 2 .

Two cases must be distinguished, according to whether p t 0 is smaller or greater than P t 0 .

First case: p t 0 ≥ P t 0 . We deÞne p = (1δ)p t 0 + δP t 0 and α i by the equality

α i p = (1 -δ) X t≥t 0 δ t-t 0 α it p t .
These deÞnitions imply in particular that α M + α E = 1.We consider now the following change to the original equilibrium strategies: in period t and in all subsequent periods, Firm i announces a price p and a market share α i ; out-of-equilibrium strategies are the same as in the original equilibrium. Clearly, these new strategies do not affect conditions (IC it ) for t < t 0 since they leave each Þrm's expected discounted payoff between t 0 and inÞnity unchanged. In order to check whether these strategies can be equilibrium strategies, it is necessary to check whether the condition (IC it ) still holds for t ≥ t 0 . This condition is equivalent to

α i ≥ 1 -δ. (6)
But (IC it 0 ) implies that α i p ≥ (1δ)p t 0 , implying that (6) holds, because the inequality p t 0 ≥ P t 0 implies that p ≤ p t 0 . Also, p > 0 : if the equilibrium under consideration yields strictly positive aggregate proÞts, then it is not possible that all prices be equal to zero after a certain period (otherwise by backward induction prices could be shown to be zero in all periods), so that P t 0 > 0, and thus p > 0. Consider now the following modiÞed strategies: in every period t, Firm i announces a price V 2 and a market share α i . Clearly, these strategies are compatible with an equilibrium: if t ≥ t 0 , (IC it ) still holds since the left-hand and right-hand term are both multiplied by V 2 /p; if t < t 0 , (IC it ) still holds since the left-hand side is unchanged while the right-hand side increases. But the corresponding equilibrium yields each Þrm a greater proÞt than the original one, since the average price after period t 0 is multiplied by V 2 /p, which is greater than one (since p t 0 < V 2 ). Therefore the initial equilibrium could not be optimal from the point of view of Þrms.

Second case: p t 0 < P t 0 . Notice Þrst that (IC Et 0 ) and (IC Mt 0 ) are necessarily both binding: if (IC it 0 ) were not binding, a small price increase in period t 0 , together with a market share reallocation leaving Firm i's proÞt unchanged (and thus slightly increasing Firm j's) would be a Pareto improvement and would leave all the (IC it ) and (IC jt ) constraints satisÞed. Therefore,

(1 -α Et 0 )p t 0 = δ 1 -δ V Et 0 and (1 -α Mt 0 )p t 0 = δ 1 -δ V Mt 0 , implying that p t 0 = δ 1 -δ (V Et 0 + V Mt 0 ) = δ 1 -δ P t 0
and thus that δ 1δ = p t 0 P t 0 < 1.

In this introduction, we use both the words "tying" and "bundling", because, as is shown below, the results hold both in the case of an irreversible technical tie and in the case of a reversible (e.g., contractual) bundling strategy.

The literature on bundling is surveyed inNalebuff (2002), which in particular summarises the main critiques of the single monopoly critique.

3 Carlton and[START_REF] Waldman | The Strategic Use of bundling to Preserve and Create Market Power in Evolving Industries[END_REF] present a modiÞed version of this argument in the case where there are intertemporal economies of scale or network

effects.4 Carbajo et al. (1990[START_REF] Seidmann | Bundling as a Facilitating Device: A Reinterpretation of Leverage Theory[END_REF],[START_REF] Chen | Equilibrium Product Bundling[END_REF].

The results of this paper would carry over to the case of heterogeneous rates of time preference. The assumption of a uniform rate is made for the sake of simplicity (with heterogeneous rates, the optimal collusive equilibria may not be stationary). For a general analysis of collusion between Þrms with heterogeneous discount rates, see[START_REF] Harrington | Collusion Among Asymmetric Firms: The Case of Different Discount Factors[END_REF].

This is similar to the modeling of market share proposals inAthey and Bagwell (2001).

But conditions (IC Et ) and (IC Mt ) imply that for every t,

) is a weighted average of all values of (U Et 0 + U Mt 0 ) in periods t 0 ≥ t + 1, this implies that for any t, there exists t 0 > t such that U Et 0 +U Mt 0 U Et +U Mt ≥ 1-δ δ > 1. This implies that there exists a sequence t n such that

¢ n , which tends toward inÞnity as n does. This is clearly a contradiction, since U Et + U Mt is bounded above by the one-period monopoly proÞt V 2 . This proves that the price in any optimal collusive equilibrium is equal to V 2 in every period. Then, let us deÞne α i by the equality

Condition (IC i0 ) is equivalent to α i ≥ 1δ, which implies that there exists an equilibrium such that in every period the price is equal to V 2 and market shares are α M , α E . This equilibrium yields the same payoffs as the original one. This completes the proof: an optimal collusive equilibrium is such that the price is equal to the shortrun monopoly price in every period; there necessarily exists an equilibrium yielding the same payoffs and characterized by constant market shares, and such that any deviation is followed by the repetition of the short-run the Bertrand equilibrium.