Competition, Cooperation, and Corporate Culture

Published in RAND Journal of Economics (2011), 42, (1), 23-43.

Competition, Cooperation, and Corporate Culture∗ Michael Kosfeld

Ferdinand von Siemens

Goethe-University Frankfurt & IZA

University of Amsterdam & CESifo

Abstract Cooperation between workers can be of substantial value to a firm, yet its level often varies substantially between firms. We show that these differences can unfold in a competitive labor market if workers have heterogeneous social preferences and preferences are private information. In our model workers differ in their willingness to cooperate voluntarily. We show that there always exists a separating equilibrium in which workers self-select into firms that differ in their monetary incentives as well as their level of worker cooperation. Our model highlights the role of sorting and worker heterogeneity in the emergence of heterogeneous corporate cultures. It also provides a new explanation for the co-existence of non-profit and for-profit firms. JEL classification: C72, C90, D82, M50 Keywords: conditional cooperation, self-selection, corporate culture, non-profit firm, competition, asymmetric information

∗

This article is part of the University Research Priority Program on “Foundations of Human Social Be-

havior: Altruism and Egoism” at the University of Zurich and the EU-TMR Research Network ENABLE (MRTN CT-2003-505223). Financial support by the University of Zurich, the Swiss State Secretariat for Education and Research, the German Science Foundation (DFG) through the SFB/TR 15, and the PostdocProgramme of the German Academic Exchange Service (DAAD) is gratefully acknowledged. Contact details: [email protected] and [email protected]

“Here you don’t communicate. And sometimes you end up not knowing things. ... Everyone says we need effective communication. But it’s a low priority in action. ... The hardest thing at the gates when flights are delayed is to get information.” Customer service supervisor, American Airlines “There is constant communication between service and the ramp. When planes have to be switched and bags must be moved, customer service will advise the ramp directly or through operations.” If there’s an aircraft swap “operations keeps everyone informed. ... It happens smoothly.” Customer service supervisor, Southwest Airlines1

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Introduction

Teamwork, cooperation, and helpfulness between workers can be of substantial value to a firm. There are many examples — workers with complementary skills can increase output and productivity by helping each other on individual tasks. Similarly, communication and the sharing of relevant information between different workers or work groups often greatly enhance the efficiency of production. While cooperation between workers is beneficial to the firm, the exertion of cooperative effort is usually costly to a worker. Moreover, it is typically hard to identify, let alone to verify, whether or not a worker helped a co-worker or shared information. Hence, monetary incentives for cooperation are difficult to provide. Unless workers have non-material motives to cooperate, firms should therefore face low levels of cooperation among workers.

However, both empirical evidence and carefully designed experiments suggest that workers sometimes do cooperate even in the absence of monetary incentives for cooperation. A large part of this cooperation is driven by so-called “conditional cooperation”, that is, a preference 1

Quotes taken from Gittell (2003).

1

to cooperate conditional on the cooperation of others.2 Could it be that the existence of conditional cooperators mitigates (or even solves) the above described cooperation problem within firms? The answer is not immediately clear since the same empirical evidence suggests that workers’ preferences are heterogeneous, where a substantial fraction of workers reveal purely selfish behavior. In case cooperation is efficient and labor markets are competitive, firms that employ cooperative workers should pay high wages due to high output. This attracts selfish types. Consequently, it has been conjectured in the literature (Lazear 1989, Kandel and Lazear 1992) that no separating equilibrium in which workers self-select into different firms can exist if preferences are private information. Since conditionally cooperative workers no longer voluntarily exert cooperative effort in the presence of sufficiently many selfish types, labor market competition should eventually erode all cooperation.

While the above arguments are intuitive, we show in this paper that the conjecture is wrong. We prove that workers in a competitive labor market with hidden information separate in equilibrium, thereby leading to the emergence of heterogeneous “corporate cultures” with regard to employment contracts and worker cooperation. The key element of our model is that screening and self-selection shape equilibrium contracts. In this regard, monetary incentives play an important dual role: They directly affect workers’ behavior in firms, but they also determine the motivational characteristics of the pool of workers that are attracted.

In a nutshell, our model runs as follows. Consider a competitive labor market where firms can employ teams of two workers. Each worker chooses individual and cooperative effort. Firms can provide monetary incentives for individual effort, but cooperative effort is noncontractible. There exist two types of workers: selfish workers and conditionally cooperative workers. A selfish worker only responds to monetary incentives and hence exerts individual effort only if monetary incentives are high. Since cooperation is non-contractible, a selfish worker never exerts cooperative effort. A conditionally cooperative worker also responds to monetary incentives with respect to individual effort. However, he might also exert cooperative effort in case his co-worker cooperates as well because mutual cooperation yields him positive non-monetary utility. Types are private information. We model competition between firms using Rothschild and Stiglitz’s (1976) notion of a competitive equilibrium under adverse selection. We extend their equilibrium concept to account for workers’ contract and 2

Laboratory experiments include Fischbacher, G¨ achter, and Fehr (2001), G¨ achter and Th¨ oni (2005) and

Fischbacher and G¨ achter (2010). See G¨ achter (2007) for an overview. Field evidence is provided by Frey and Meier (2004), Heldt (2005), Ichino and Maggi (2000), and Bandiera, Barankay, and Rasul (2005).

2

effort choices by requiring workers to play a perfect Bayesian equilibrium given any set of offered contracts.

We show that there always exist a separating equilibrium in which workers self-select into different firms. In this equilibrium firms offer two different types of contracts: selfish contracts and particular cooperative contracts. Selfish contracts provide strong monetary incentives for individual effort and are accepted by selfish workers. No cooperation is observed in firms offering these contracts, and firms make zero profit in equilibrium. Cooperative contracts are exclusively accepted by conditional cooperators, who cooperate in equilibrium. As long as output from cooperative effort is not too small, cooperative contracts entail muted incentives where workers are paid only a fixed wage. Conditional cooperators then exert cooperative effort but no individual effort. Since firms offering cooperative contracts in a separating equilibrium have to ensure that selfish workers do not accept their contract, they may not be able to pay out all surplus to their workers. It is thus possible that cooperative contracts yield positive profit in equilibrium. We further show that separation is the unique equilibrium outcome if workers have no direct benefit from co-workers’ cooperation. This also holds if workers’ types are observable. In the latter case, firms can condition contracts on a worker’s type and are not constrained by separation incentives. In consequence, all firms make zero profit.

Our results can account for a number of empirical findings. First, firms, even within the same industry, often develop remarkably different corporate cultures with respect to the level of cooperation and team work. In the airline industry, for example, Southwest Airlines has become the prototype of a strong cooperative corporate culture, where employees are known to help each other and team spirit and good relationships between different work units play a key role (Gittell 2000, Gittell 2003). In contrast to Southwest, most other U.S. airlines have been unable to achieve similar levels of worker cooperation and have often been characterized by a culture of conflict rather than cooperation (Gittell, von Nordenflycht, and Kochan 2004). The findings are not unique to the airline industry. Ichniowski, Shaw, and Prennushi (1997), e.g., report similar differences for the steel industry, documenting a strong heterogeneity in human resource management practices (including team work, training, hiring, supervision, etc.) in a sample of 36 U.S. production lines all of which operate in the same steel finishing business. Second, firms that enjoy high levels of cooperation tend to be more productive than firms without or with only low levels of cooperation. Southwest Airlines, for example, announced its 31st consecutive year of profitability in 2004 and Fortune magazine has called

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it “the most successful airline in history” (Brooker 2001). Ichniowski et al., e.g., find that production lines using innovative work practices, which include high levels of team work, are significantly more productive than lines with the traditional approach where team work does not play an important role. The positive impact of team work on productivity is also confirmed by Hamilton, Nickerson, and Owan (2003) who analyze the effects of a “cultural change” from individual to team production in a U.S. garment plant. Finally, high levels of cooperation seem to be associated with weak individual incentives. Encinosa, Gaynor, and Rebitzer (2007), for example, show in a U.S. sample of medical group practices that incentive pay reduces help activities among physicians such as mutual consultations about cases. Consistent with this finding, Burks, Carpenter, and G¨otte (2009) show in a sample of Swiss and U.S. bicycle messenger companies that firms that pay for performance employ significantly less cooperative workers than those that pay hourly wages or are organized as cooperatives. Unlike the studies above which rely on firm data, Burks et al. elicit workers’ preferences for cooperation in a controlled field experiment. In the light of our model, the empirical findings can be seen as a consequence of labor market competition with private information about workers’ preferences for cooperation. By choosing different contracts workers in equilibrium self-select into different firms, thus leading to heterogeneous corporate cultures of team work with corresponding differences in incentives and firm productivity.

Our results also provide a new explanation for the existence of non-profit and governmentowned firms (Rose-Ackerman 1996). Compared to for-profit firms, non-profit and governmentowned firms are typically more constrained in their possibility to redistribute surplus. In our setting, this can turn into a competitive advantage vis-`a-vis for-profit firms. The reason is that constrained firms have a credible commitment not to pay out residual surplus. In consequence, they can better attract conditional cooperators because they are able to commit not to pay wages that lure selfish types. Our results suggest that the competitive advantage of non-profit firms will be particularly relevant when cooperation is important for production, some workers care for cooperation, and a worker’s preferences are private information. The explanation is different from previous explanations that focus on the non-contractibility of output or effort (Hart, Shleifer, and Vishny 1997, Glaeser and Shleifer 2001, Francois 2000, Francois 2001) or assume that workers are mission motivated and hence have a preference to work for a particular employer (Besley and Ghatak 2005, Auriol and Brilon 2010).

The article complements a strand of research emphasizing the role of separation and selfselection in labor markets. Other articles analyze the possible sorting of workers differing by

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skill (Kremer and Maskin 1996, Saint-Paul 2001, Grossmann 2007), liability (Dam and PerezCastrillo 2006), vision (Van Den Steen 2005), or by mission (Besley and Ghatak 2005, Auriol and Brilon 2010). To the best of our knowledge, our model is the first to demonstrate the separation of workers who differ in their preference for conditional cooperation.3 Further, firms in our model compete under incomplete information about the type of worker accepting a particular contract. In this respect, our model is related to Prendergast (2007), Delfgaauw and Dur (2007, 2008), and Francois (2007) who investigate incentives and selection of workers who differ in their intrinsic motivation. Contrary to our model, these articles assume that workers’ intrinsic motivation is independent of other workers’ effort choices. None of the above mentioned articles studies cooperation within teams.

Finally, our article adds to the, still rather small, economic literature on corporate culture. Previous articles have argued that different corporate cultures are the result of coordination (Kreps 1990), shared knowledge (Cr´emer 1993), asymmetric equilibria in the product market (Hermalin 1994), differences in firms’ initial social capital (Rob and Zemsky 2002), or shared beliefs (Van Den Steen forthcoming). We show that labor-market competition can itself sustain — via sorting — firm heterogeneity with regard to work organization, incentives, and cultures of cooperation. In so far as cooperative firms are likely to be non-profit or government-owned firms our results are also related to the literature on non-profit vs. for-profit firm differences, in particular with regard to wages and employee motivation. See, e.g., Leete (2001) for a recent discussion and empirical evidence. Finally, firms hiring conditionally cooperative workers in our model sometimes provide weaker incentives than those that employ selfish workers. Our results thus offer a rationale for the potential optimality of muted incentives. Unlike Holmstrom and Milgrom (1991) or Baker, Gibbons, and Murphy (2002) muted incentives in our setting are not directly driven by multi-tasking or repeated interaction, but result from competition under adverse selection.

2

Model

Our model of a competitive labor market under adverse selection is based on Rothschild and Stiglitz’s (1976) analysis of the insurance market. In addition, we account for workers’ effort choices by modeling the interaction between workers within firms as a Bayesian game. 3

Cabrales and Calv´ o-Armengol (2008) and von Siemens (forthcoming) look at worker segregation by ability

if workers are also inequity averse. They assume that effort is contractible and do not analyze cooperation within teams. Recent experimental articles that analyze the interaction between incentives, sorting, and social preferences are Eriksson and Villeval (2008) and Dohmen and Falk (forthcoming).

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Suppose a given number of firms operate in the market offering contracts to a countably infinite number of workers. Concerning worker behavior there is the following sequence of actions. First, workers simultaneously choose among the set of offered contracts. Conditional on the contract chosen, workers are assigned a corresponding firm and randomly matched into teams of two. In case a worker decides not to accept any offered contract or remains unmatched, he earns an outside-option utility normalized to zero. Second, workers in each team produce output by simultaneously exerting effort. Finally, wages are paid and payoffs are realized. We next fill in all the necessary details.

Workers and Firms Let us start with team production. When matched into a team of two, worker i can produce output by exerting a two-dimensional effort ei = (ei1 , ei2 ). Effort is costly and binary in both dimensions, ei1 ∈ {0, 1} and ei2 ∈ {0, 1}. The effort components ei1 and ei2 differ in the way they contribute to team output, and thus differ in the extent to which their output is individually attributable. Effort ei1 encompasses all activities that generate individually attributable team output. Effort ei2 smoothes the overall team production process through cooperation. It thereby increases joint production of the team, but this increase is difficult to attribute. We call ei1 a worker’s individual effort and ei2 his cooperative effort. A key assumption in our model is that it is easier for a firm to provide monetary incentives for individual effort as compared to cooperative effort. By the very nature of team production it is difficult to identify who is responsible for what share of the team output that is generated by cooperative effort. The resulting free-rider problem complicates the provision of incentives. Cooperative effort may also increase team output via informal communication, mutual help, or quality improvements. In that case cooperative effort might possibly be observable but hard to quantify or verify. Since the focus in this article is on the adverse selection rather than the moral hazard problem, we abstract from the underlying details regarding the incentive provision problem. Instead, we simply assume that cooperative effort is non-contractible, either because it is impossible to provide monetary incentives or because the involved agency costs are too high. Individual effort, however, is observable, verifiable, and hence contractible. A contract w then consists of two elements, a fixed wage f and a bonus b each worker gets if and only if he exerts individual effort. Fixed wage and bonus are assumed to be weakly positive. Let W denote the set of all possible contracts.4 4

Contracts also contain firm identity numbers so that workers’ acceptance choices are always well defined.

In most of our analysis we suppress them for notational simplicity.

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Firms sell output at a price normalized to one. Given contract w = (b, f ) and workers’ effort choices (ei , ej ) let
π (w, ei , ej ) = x(ei1 ) + x(ej1 ) + 2 y(ei2 , ej2 ) − 2 f + ei1 b + ej1 b

(1)

describe the firm’s profit generated by the considered team. The term in brackets captures the firm’s wage payments. In addition to his fixed wage, a worker receives the bonus if and only if he exerts individual effort. The remaining three terms determine total team output. The latter consists of the individual outputs x and the joint team production y per worker. We set x(1) equal to some strictly positive constant X > 0 and normalize x(0) to zero. To model effort complementarity in team production we set y equal to some strictly positive constant Y > 0 if and only if both workers provide cooperative effort. For all other effort choices we normalize y to zero. Firms can hire any number of teams. To capture the idea of a common corporate culture we assume that firms offer a single contract that applies to all its teams. Firms maximize expected profit per team.

Our second key assumption is that workers differ in their willingness to contribute cooperative effort. There are two types: each worker is either selfish or conditionally cooperative. Selfish workers never exert any cooperative effort because effort is costly and entails no private benefits. Conditionally cooperative workers, however, might contribute cooperative effort if they believe their team colleague to do the same. Let θ ∈ {s, c} denote a worker’s type. The utility of worker i who is of type θ, chooses effort vector ei = (ei1 , ei2 ), and is matched in a team with worker j who exerts effort ej = (ej1 , ej2 ) is defined as   b e + f − ψ(e , e ) + ξ(e ) i1 i1 i2 j2 uiθ (w, ei , ej ) =  b ei1 + f − ψ(ei1 , ei2 ) + ξ(ej2 ) + γ(ei2 , ej2 )

if

θ = s,

if

θ = c.

(2)

Worker i’s utility consists of up to three components. First, he enjoys utility b ei1 + f from his wage. Second, exerting effort causes him effort costs ψ(ei1 , ei2 ). The cost function ψ is weakly positive and strictly increasing in both effort dimensions. We also assume that ψ is supermodular so that ψ(1, 1) + ψ(0, 0) > ψ(0, 1) + ψ(1, 0).

(3)

Individual and cooperative effort are thus substitutes in a worker’s cost function, i.e., marginal costs of effort in one dimension are an increasing function of effort in the other dimension. We further assume that ψ(0, 0) ≥ 0. In addition, worker i receives utility ξ(ej2 ) depending on whether the other worker in his team exerts cooperative effort or not. We set ξ(1) equal to some positive constant Ξ ≥ 0 and normalize ξ(0) to zero. Ξ can be interpreted as a worker’s 7

reduction in effort costs if he is helped by his co-worker. We assume that Ξ ≤ ψ(0, 0)

(4)

ensuring that workers do not have a positive willingness to pay for a job only because they receive help from their co-worker. With Ξ = 0 cooperative effort thus only affects team output, whereas with Ξ > 0 cooperation also generates additional utility for workers.5 Finally, conditionally cooperative workers enjoy intrinsic satisfaction γ(ei2 , ej2 ) from cooperation. On the one hand, they are willing to reciprocate help provided by the other worker. On the other hand, they might value highly to participate in a team production process that runs smoothly due to the cooperation of all team members. To model conditional cooperation, we set γ equal to some strictly positive constant Γ > 0 if and only if both workers contribute cooperative effort. For all other effort choices γ is normalized to zero. The assumption that conditionally cooperative workers enjoy some intrinsic satisfaction from joint cooperation is consistent with Rabin’s (1993) model of fairness. Hamilton, Nickerson, and Owan (2003) provide empirical evidence for the hypothesis that some workers receive intrinsic satisfaction from team work. Analyzing team participation in a garment plant, they find that “some workers joined teams despite an absolute decrease in pay, suggesting that teams offer nonpecuniary benefits to workers.”6 We assume that workers’ types are private information, yet it is common knowledge that they are independently distributed with each worker being conditionally cooperative with some prior probability λ ∈]0, 1[. All workers maximize expected utility.

We make the following assumptions with regard to the efficiency of individual and cooperative effort. First, we assume that X > ψ(1, 1) − ψ(0, 1).

(5)

This ensures that marginal output always exceeds marginal costs of individual effort. Further, we assume that X − ψ(1, 0) > 0 so that accepting employment and exerting individual effort is efficient. Second, we assume that Γ > ψ(1, 1) − ψ(1, 0). 5

(6)

For example, suppose workers assemble products during a given working day. With Ξ = 0 workers can

assemble more products when they cooperate, but cooperation does not lead to lower effort costs. With Ξ > 0 cooperation has additional cost-reducing effects for workers, for example via skill improvements. 6

The assumption is also supported by numerous studies from organizational psychology. See Tyler and

Blader (2000) and Organ, Podsakoff, and MacKenzie (2006) for an overview.

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Workers’ benefit from exerting cooperative effort exceeds the respective marginal effort costs even if individual effort is chosen. Because of the intrinsic satisfaction Γ from cooperation, cooperation is thus always efficient in a team of conditionally cooperative workers, independent of team output Y and the benefits from helping Ξ.

Equilibrium Behavior The objective of our article is to investigate whether labor market competition can lead to a separation of workers according to their type, whether conditionally cooperative workers cooperate in equilibrium, and whether, thus, heterogeneous corporate cultures can emerge in an ex-ante symmetric environment. To answer these questions we follow the tradition in the literature of competition under adverse selection and analyze the equilibrium set of offered contracts, abstracting from the identity and the explicit decision making of individual firms. At the same time, we model workers’ interactive behavior within firms as a Bayesian game.

As in other models of adverse selection, workers’ contract choices might reveal information on their preferences. We assume beliefs not to differ across individuals. Let µ(θ|w, W ) ∈ [0, 1] denote the probability with which firms and all other workers believe a worker to be of type θ ∈ {s, c} if the latter accepts contract w out of a set of offered contracts W. We require that individuals form such beliefs for all contracts w ∈ W and all possible sets of offered contracts W ⊆ W.

To describe workers’ behavior let aiθ (w, W ) ∈ [0, 1] denote the type-dependent probability with which worker i accepts contract w out of a set of offered contracts W. Equally, let eiθ (w, W ) ∈ {0, 1} × {0, 1} denote worker i’s type-dependent effort choice given that he is assigned a team within a firm that employs contract w from a set of offered contracts W. We require workers to have fully specified strategies that determine type-dependent behavior for all contracts w ∈ W and all possible sets W ⊆ W of offered contracts. We focus on symmetric equilibria throughout the article where all workers share the same type-dependent equilibrium strategies.7 Thus, we skip subindices i, j in the following.

In a competitive equilibrium, we require workers’ strategies and beliefs to form a perfect Bayesian equilibrium given all possible finite sets of offered contracts W ⊆ W. Details are 7

In addition, we impose the following cutoff rule: whenever a worker is indifferent between exerting effort

and not exerting effort in some dimension, he exerts effort in that dimension while keeping his effort decision in the other dimension fixed.

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given in the appendix (Definition 1). Intuitively, workers behave optimally within a firm if they maximize their expected utility given the firm’s contract, the equilibrium strategies of the other workers, and the common beliefs. In a perfect Bayesian equilibrium, workers’ effort choices must form a Bayesian equilibrium in every subgame and thus for all possible contracts w. Based on these equilibrium outcomes each worker computes his expected utility upon accepting a particular contract and chooses the contract that maximizes his utility over the set of offered contracts. Although preferences are private information, contract choices might serve as a signal. Equilibrium beliefs are required to be consistent with worker’s acceptance decisions and Bayes’ rule whenever this is possible.

Given that workers behave optimally, a competitive equilibrium is a finite set of offered contracts W ∗ satisfying the following requirements (see Definition 2 in the appendix). First, the equilibrium set of offered contracts W ∗ contains no contracts that are not accepted in equilibrium. Second, no firm offers a contract that makes expected losses in equilibrium. Otherwise, the firm could increase its expected profit by withdrawing the offered contract. Third, no firm can enter the market by offering a new contract w ˜ 6∈ W ∗ that attracts workers and yields positive expected profit. Competition is thus modeled by free entry.8

A competitive equilibrium has the following properties. Since firms can employ any number of teams, there is no rationing. Because there is an infinite number of workers, workers in a symmetric equilibrium can be sure to find a co-worker when accepting a contract that is accepted with strictly positive probability. Consequently, all workers of the same type receive the same equilibrium utility. Let u∗θ denote the equilibrium utility of a type-θ worker. We call a competitive equilibrium a separating equilibrium if there exists no offered contract that is accepted with strictly positive probability by both types of workers. A worker’s type can then perfectly be inferred from his contract choice. In all other equilibria there exists at least one contract that is accepted by both types of workers with strictly positive probability. In this case we say that there is pooling in equilibrium. Whether a set W ∗ of contracts can be supported as a competitive equilibrium depends on workers’ reaction towards a newly offered contract w ˜ 6∈ W ∗ . This reaction in turn depends on workers’ beliefs upon accepting the new contract, on the Bayesian equilibria they expect to be played within firms, and on whether they expect to find a colleague. As common in 8

By definition the equilibrium set of offered contracts is finite. Further, we only consider profitable market

entry by a single firm. In all relevant cases workers thus choose among finite sets of offered contracts, so that optimal acceptance and effort decisions are well defined.

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screening and signalling models, there exist multiple equilibria that are based on particular out-of-equilibrium beliefs. Further, there can arise coordination problems concerning the Bayesian equilibria within firms and workers’ acceptance decisions. To rule out implausible competitive equilibria we employ the following equilibrium refinements.9 First, we apply Cho and Kreps’s (1987) intuitive criterion requiring that workers do not believe a worker accepting a new contract to be of a particular type if this type will always get strictly less than his equilibrium utility whereas the other type might get weakly more (Refinement 1). Second, suppose that workers believe that a new contract might only be accepted by a certain type and that acceptance is only optimal for this type if a particular Bayesian equilibrium is played within the firm. Then we assume that workers accepting the new contract successfully coordinate on that equilibrium (Refinement 2).10 Finally, we impose the following technical refinement: workers do not reject a new contract yielding a higher expected utility only because they expect no other worker to accept this contract; further, workers stick to their old contract and effort choices if a new contract is offered but not accepted by any workers (Refinement 3).

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Results

In this section, we proceed as follows. We first analyze the set of Bayesian equilibria within firms, that is, the set of Bayesian equilibria in all possible subgames which are defined by a particular contract and a particular belief about the co-worker’s type. We then explore the competitive equilibria in our model. We start by analyzing the complete-information situation as a benchmark case. We then show under the assumption of private information, that there always exists a separating equilibrium, that in all separating equilibria conditionally cooperative workers cooperate, and that all separating equilibria are Pareto-efficient. Finally, we show that if the benefit from being helped is zero, no pooling can arise in equilibrium.

Bayesian Equilibria within Firms Consider a Bayesian game within a firm that is characterized by a particular contract and corresponding beliefs about the co-worker’s type. There are two types of equilibria: selfish equilibria, in which no type of worker cooperates, and cooperative equilibria, in which the conditionally cooperative workers contribute cooperative effort. Concerning selfish equilibria we get the following result. 9 10

Again, details are provided in the appendix. Refinement 2 is related to the forward induction argument of van Damme (1989).

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Lemma 1 (Selfish Equilibrium) Consider the Bayesian game given by contract w ∈ W and beliefs µ(θ|w, W ) for θ ∈ {s, c}. There always exists a selfish equilibrium in which there is no cooperation and   (0, 0) eSE (b) =  (1, 0)

if

b < ψ(1, 0) − ψ(0, 0)

if

b ≥ ψ(1, 0) − ψ(0, 0)

(7)

describes the equilibrium effort choices of both types of workers θ ∈ {s, c}. The simple proof is left to the reader. Since a conditionally cooperative worker in a selfish equilibrium correctly anticipates that his co-worker never contributes cooperative effort, the same is optimal for him. Given that there is no cooperation, selfish and conditionally cooperative workers have the same utility function. Consequently, both types of workers exert individual effort if and only if the bonus exceeds the associated increase in their effort costs. The following lemma shows that there also exists a cooperative equilibrium depending on workers’ beliefs. Lemma 2 (Cooperative Equilibrium) Consider the Bayesian game given by contract w ∈ W and beliefs µ(θ|w, W ) for θ ∈ {s, c}. There exists a cooperative equilibrium in which conditionally cooperative workers exert cooperative effort if and only if   if b < ψ(1, 0) − ψ(0, 0)   ψ(0, 1) − ψ(0, 0)   µ(c|w, W ) Γ ≥ b + ψ(0, 1) − ψ(1, 0) if b ∈ ψ(1, 0) − ψ(0, 0), ψ(1, 1) − ψ(0, 1)    ψ(1, 1) − ψ(1, 0) if b ≥ ψ(1, 1) − ψ(0, 1).

(8)

In a cooperative equilibrium, selfish workers choose the same efforts eSE (b) as in the selfish equilibrium, whereas   (0, 1) eCE (b) =  (1, 1)

if

b < ψ(1, 1) − ψ(0, 1)

if

b ≥ ψ(1, 1) − ψ(0, 1)

(9)

describes the equilibrium effort choices of the conditionally cooperative workers. All proofs can be found in the appendix. Whether a cooperative equilibrium exists depends on workers’ belief. If a conditional cooperator believes that a conditionally cooperative colleague exerts cooperative effort, exerting cooperative effort increases his expected utility by µ(c|w, W ) Γ. If this expected benefit of cooperation exceeds the associated effort costs, there exists a cooperative equilibrium. The costs of contributing cooperative effort, however, depend on the individual effort choice, which in turn depends on the contract. Given a high bonus, conditionally cooperative workers exert individual effort. Since individual effort and 12

cooperative effort are substitutes in the worker’s cost function, this makes the contribution of cooperative effort more costly. Individual incentives thus render the conditions on the belief for the existence of a cooperative equilibrium more restrictive. In this sense, high incentives can crowd-out cooperation as in Holmstrom and Milgrom (1991).

However, cooperation is efficient. Assumption (6) ensures that for every contract there always exists a cooperative equilibrium if workers believe their colleagues to be conditionally cooperative and to cooperate with probability one. Further, workers’ effort costs imply that the minimum bonus needed to implement individual effort by conditional cooperators in a cooperative equilibrium is strictly higher than the corresponding minimum bonus in a selfish equilibrium. Since conditional cooperators cooperate in equilibrium, they need stronger incentives for individual effort than selfish workers who do not cooperate and thus save on effort costs. In contrast to Holmstrom and Milgrom (1991), high incentives thus do not crowd out cooperation in teams that primarily consist of conditionally cooperative workers.

Complete Information Benchmark Having characterized the Bayesian equilibria within firms, we next introduce competition between firms. We start with complete information as a benchmark case. In general, conditionally cooperative workers want to be separated from selfish workers, as only then they can reap the benefits from mutual cooperation. If preferences are observable, firms can target particular types of workers and thus ensure separation. Consider conditionally cooperative workers in this situation. By assumption exerting both individual and cooperative effort is efficient. Therefore, the best that can happen to a conditionally cooperative worker is to join a firm that implements individual effort, pays out all profit to its workers, and attracts only conditionally cooperative workers who coordinate on the cooperative equilibrium. By Lemma 2 we define the set of optimal cooperative contracts as  W c = w ∈ W : b ∈ [ψ(1, 1) − ψ(0, 1), X + Y ] and f = X + Y − b .

(10)

Consider next selfish workers. Given that they are separated from conditionally cooperative workers, selfish workers cannot free-ride on the cooperation of conditional cooperators. The best that can then happen to a selfish worker is to join a firm that implements individual effort and pays out the entire profit X to each worker. Hence, we define the set of optimal selfish contracts under separation as  W s = w ∈ W : b ∈ [ψ(1, 0) − ψ(0, 0), X] and f = X − b . We get the following result. 13

(11)

Lemma 3 (Complete Information Equilibrium) Suppose workers’ preferences are observable. Then there always exists a competitive equilibrium. Further, any competitive equilibrium can be characterized as follows: (i) Selfish workers accept a contract from W s . (ii) Conditionally cooperative workers accept a contract from W c and exert cooperative effort. (iii) All firms make zero profit. The intuition for this result is straightforward (the simple proof is again left to the reader). Conditionally cooperative workers do not want to work in firms attracting selfish workers. If workers’ types are observable, the labor market thus splits into two, one market for each type. In combination with our refinements, competition ensures that selfish workers get a contract from set W s whereas conditionally cooperative workers get a contract from set W c . Equally, cooperative workers cooperate in equilibrium. All firms make zero profit.

Incomplete Information: Separating Equilibrium If information about workers’ preferences is private, the situation in Lemma 3 can no longer form an equilibrium. The reason is that selfish workers have an incentive to infiltrate cooperative firms in order to get a higher wage and enjoy the benefits from being helped. Yet, we now show that also with private information a separating equilibrium exists. Further, cooperative workers cooperate in any separating equilibrium, and any separating equilibrium is Pareto-efficient.

Before deriving our main result, it is helpful to first describe the Pareto-efficient separating contracts. Selfish workers receive contracts from W s that maximize their utility. Conditionally cooperative workers receive what we call constrained optimal cooperative contracts that maximize their utility when cooperating without causing the firm losses or attracting selfish types. Denote W cc the set of these contracts. It contains all contracts (b, f ) that maximize CE b eCE 1 (b) + f + Ξ + Γ − ψ(e1 (b), 1)

14

(12)

satisfying the constraints eCE 1 (b) (X − b) + Y − f

≥ 0

SE X − ψ(1, 0) ≥ b eSE 1 (b) + f + Ξ − ψ(e1 (b), 0) CE b eCE 1 (b) + f + Ξ + Γ − ψ(e1 (b)) ≥ X − ψ(1, 0)

f, f + b ≥ 0.

(13) (14) (15) (16)

CE Recall that eSE 1 (b) is the individual effort in a selfish equilibrium and e1 (b) is the individual

effort of a conditionally cooperative worker in a cooperative equilibrium. The zero profit constraint (13) ensures that firms offering a contract w ∈ W cc make no losses if they attract only cooperative workers who coordinate on the cooperative equilibrium. By the screening constraints constraints (14) and (15) selfish and conditionally cooperative workers prefer their respective contracts to the contracts for the other type. Constraint (16) is the limited liability constraint. Define Y1 = X − ψ(1, 1) + ψ(0, 1) − Ξ and Y2 = X − ψ(1, 0) + ψ(0, 0) − Ξ. We get the following result. Lemma 4 (Constrained Optimal Cooperative Contracts) The set of constrained optimal cooperative contracts W cc contains the following contracts: (i) Suppose Ξ ≤ X − ψ(1, 1) + ψ(0, 1). (a) If Y ≤ Y1 , then b ∈ [ψ(1, 1) − ψ(0, 1), X − Ξ] and f = X − Ξ − b. (b) If Y ∈ [Y1 , Y2 ], then b ∈ [0, X − Y − Ξ] and f = Y . (c) If Y ≥ Y2 , then b ∈ [0, ψ(1, 0) − ψ(0, 0)] and f = X − ψ(1, 0) − Ξ + ψ(0, 0). (ii) Suppose Ξ > X − ψ(1, 1) + ψ(0, 1). (a) If Y ≤ Y2 , then b ∈ [0, X − Y − Ξ] and f = Y . (b) If Y ≥ Y2 , then b ∈ [0, ψ(1, 0) − ψ(0, 0)] and f = X − ψ(1, 0) − Ξ + ψ(0, 0). Lemma 4 is illustrated in Figure 1 by mapping a conditionally cooperative worker’s utility uc under a constrained optimal cooperative contract as a function of per-worker cooperative output Y . Although contracts cannot condition on Y , the latter is important as it determines whether or not a constrained optimal cooperative contract implements individual effort from conditionally cooperative workers. Figure 1 considers case (i) in Lemma 4 where a worker’s direct benefit from cooperative effort Ξ is small. Case (ii) looks similar, except that there is only one cut-off point Y2 because it is not feasible for firms to implement individual effort from conditional cooperators and to ensure separation. 15

XXX Insert Figure 1 about here. XXX In general, conditionally cooperative workers want to earn high wages. In case per-worker team output Y is smaller than Y1 = X − ψ(1, 1, ) + ψ(0, 1) − Ξ, a constrained optimal cooperative contract implements individual effort. Otherwise, output would be too small to pay out high wages. However, conditionally cooperative workers also contribute cooperative effort to get the intrinsic benefit from cooperation. They thereby produce additional output per worker. Paying out this additional output is not incentive compatible as otherwise selfish workers are attracted by the high wages. Further, selfish workers are attracted by the utility from being helped Ξ. To ensure separation wages for cooperative workers must not exceed X − Ξ. This implies that if conditionally cooperative workers are to exert individual effort, they cannot benefit from their cooperative effort other than by their intrinsic satisfaction from cooperation. In consequence, they receive utility X + Γ − ψ(1, 1) including their utility from being helped Ξ.

Now suppose that team output Y is intermediate and thus sufficiently high to generate high wages, but that Y is still smaller than Y2 = X −ψ(1, 0)+ψ(0, 0)−Ξ. It is then no longer optimal to implement individual effort. To see this note that if conditional cooperators exert only cooperative effort, the firm can pay out all output Y as wages without violating selfish workers’ screening constraint since Y is still relatively small. Not implementing individual effort then has the following consequences. First, cooperative workers receive lower wages as output Y is small. Second, conditionally cooperative workers incur no costs from individual effort. Third, conditionally cooperative workers save on their costs from cooperation because individual and cooperative effort are substitutes in their cost function. The reduction in wages and individual effort costs – the first and second effect – equally affect conditionally cooperative and selfish workers. But selfish workers who infiltrate cooperative firms do not enjoy the third effect – the additional cost reduction for cooperation – because they never exert cooperative effort. By reducing monetary incentives, firms can thus pass on this additional cost reduction to conditionally cooperative workers without violating selfish workers’ incentive constraint. If this additional cost reduction exceeds the loss in wages, ψ(1, 1) − ψ(0, 1) > X − Ξ − Y , it is optimal not to implement individual effort. For intermediate team output Y conditionally cooperative workers thus exert only cooperative effort and earn utility Y + Ξ + Γ − ψ(0, 1). Finally, as per-worker team output Y exceeds the threshold Y2 = X −ψ(1, 0)+ψ(0, 0)−Ξ, the selfish workers’ incentive constraint becomes binding again. Therefore, firms cannot increase 16

wages of the conditional cooperative workers any further. Workers still exert cooperative effort but their utility is capped at X − ψ(1, 0) + ψ(0, 0) + Γ − ψ(0, 1).

Lemma 4 implies that firms offering contracts in W cc might make strictly positive expected profit. The reason is the screening constraint for selfish workers: whenever it binds the respective firms cannot pay out all profits without attracting selfish workers. As shown above, the incentive constraint is binding whenever contracts in W cc implement both cooperative effort and individual effort (Y < Y1 ), or when contracts in W cc implement only cooperative effort but Y is sufficiently large (Y > Y2 ). Building on our findings sofar, we can now present our main result. Proposition 1 (Separating Equilibrium) Suppose workers’ preferences are private information. Then there always exists a separating equilibrium for all λ ∈]0, 1[. Moreover, any separating equilibrium can be characterized as follows: (i) Selfish workers accept a contract from W s . (ii) Conditionally cooperative workers accept a contract from W cc and coordinate on the cooperative equilibrium. (iii) Firms employing selfish workers make zero profit. Firms employing conditionally cooperative workers make strictly positive profits unless max{Y1 , 0} ≤ Y ≤ Y2 . Given selfish workers get an optimal selfish contract, at least one contract from the set W cc of constrained optimal cooperative contracts must be offered in equilibrium. Otherwise, a firm could attract conditionally cooperative workers by offering a contract that is optimally accepted only by conditionally cooperative workers who then cooperate. Our refinements ensure that market entry with such a contract is indeed profitable. By essentially the same argument cooperative workers cooperate in any separating equilibrium.

At first sight it might seem puzzling that there can exist a competitive equilibrium in which some firms make strictly positive profit whereas other firms just break even. Why do those firms that make zero profit not imitate the more profitable firms? It might appear that our abstraction from firm identity and explicit profit maximization drives this result. This is not the case. With firm identity made explicit, workers must condition their acceptance decisions not only on the contract a firm offers, but also on the firm’s identity. If a new firm offers a contract that is already offered by another firm, workers cannot increase their utility by switching to the new firm. Consequently, optimal acceptance decisions can be chosen such 17

that the new firm cannot attract workers unless it promises them a strictly higher utility.11 In equilibrium, imitating the most successful firms is thus no profitable option. Yet due to incomplete information, paying a higher wage is not possible without attracting selfish workers and thus making losses. The same argument holds concerning firms that are already active on the market but want to change the contract they provide.

It might also seem puzzling that there always exists a separating equilibrium even if the fraction of conditionally cooperative workers becomes arbitrarily large. The existence result is based on the following intuition. In our model the utility a worker expects when accepting a particular contract depends on the expected effort choices of his colleague. These effort choices might depend on the colleague’s type. Contrary to standard screening models, workers’ acceptance and effort decisions thus depend on their beliefs and the Bayesian equilibria they expect to be played within firms. The following arguments show that this allows us to specify worker behavior that is consistent with out refinements but renders market entry with a pooling contract unattractive. Since by definition a pooling contract can attract both types of workers, the intuitive criterion (Refinement 1) cannot restrict beliefs. Depending on the newly offered pooling contract, there are two cases. First, suppose the pooling contract promises very high wages such that both types of workers are attracted, no matter what Bayesian equilibrium they expect to be played. In this case, coordination on the selfish equilibrium may not be ruled out by our refinements. Then the contract must make losses, as even in this case it promises workers a higher utility than optimal selfish contracts. Second, suppose the pooling contract might attract conditional cooperators only if the latter expect the cooperative equilibrium to be played. Then workers may still consistently believe that only selfish workers accept the new contract, in which case the selfish equilibrium is the only equilibrium. Because the new contract is no longer a pooling contract, it cannot upset the Pareto-efficient separating equilibrium by construction.12

Incomplete Information: Pooling Equilibrium So far we have focused on separating equilibria. In this section we show that if workers’ direct benefit from cooperative effort Ξ is zero, there exists no pooling equilibrium. 11

For example, suppose that firms are numbered consecutively. Then, workers behave optimally if they

always accept a contract that maximizes expected utility, and in case of indifference choose the firm with the lowest identity number. 12

von Siemens and Kosfeld (2009) provide a more general analysis of equilibrium existence in competitive

markets with adverse selection if there are externalities between workers.

18

Proposition 2 (No Pooling) Suppose workers’ preferences are private information. If Ξ = 0, then there is no pooling in any competitive equilibrium. The intuition for the result is as follows. In most cases, firms can skim off conditionally cooperative workers by offering a new contract. The reason is that pooling on some contract w is not very attractive for conditional cooperators: either the presence of selfish workers destroys cooperation, or conditionally cooperative workers do not benefit from their cooperative effort with the probability µ(s|w, W ) with which their respective colleague is selfish. A new firm can then enter the market and skim off the conditionally cooperative workers by offering a contract that promises a slightly lower monetary payoff. As they get less pay under the new contract, selfish workers stick to the old contract. By Refinement 1 conditionally cooperative workers can thus be sure to be among conditional cooperators when accepting the newly offered contract. Moreover, the new contract can be chosen such that by Refinement 2 conditionally cooperative workers coordinate on the cooperative equilibrium upon accepting the new contract. Since their cooperative effort is now never wasted on a selfish colleague, the associated increase in their expected benefit from cooperation overcompensates their small monetary loss. Consequently, firms can enter the market and attract conditionally cooperative workers. Since wages are low they thereby make positive profit.

While it is typically the conditionally cooperative workers that can be attracted by a newly offered contract, we show in the proof of Proposition 2 that, in fact, there also exists a case where firms can skim off selfish workers. This happens when the pooling contract specifies an intermediate bonus and workers coordinate on the cooperative equilibrium such that only selfish workers exert individual effort. If expected team production by conditionally cooperative workers does not cover the fixed wages, conditionally cooperative workers are subsidized by selfish workers. A new firm can then enter the market and skim off selfish workers by offering a contract which implements individual effort, pays out all profit, and thereby grants selfish workers the subsidy which previously went to conditionally cooperative workers.

Proposition 2 shows that pooling cannot arise in equilibrium if workers have no direct benefit from cooperative effort by their co-worker, Ξ = 0. This situation arises if any benefit from cooperation is fully absorbed by higher team output. Unfortunately, if workers benefit also directly, Ξ > 0, the situation is less clear. The reason is that selfish workers lose some direct benefit from cooperation if cooperative colleagues leave the firm. Selfish workers may then have an incentive to follow conditional cooperators. This makes skimming off conditional cooperators more difficult, as a new contract that keeps away selfish workers even if all 19

cooperative workers have left their old firms might no longer be attractive for conditional cooperators. In consequence, it is not clear whether pooling can be ruled out in equilibrium if Ξ > 0. However, Proposition 1 shows that for all values of Ξ there exists the Pareto-efficient separating equilibrium. This drives all empirical implications of our theory. We therefore leave a full exploration of possible pooling equilibria for future research.

4

Conclusion

We analyze a competitive labor market where workers differ in their intrinsic motivation for cooperation in teams. Even without monetary incentives for cooperation, some workers are willing to cooperate if their co-workers cooperate as well (conditional cooperators). Others only respond to monetary incentives (selfish workers). Workers’ types are private information and firms compete for workers by offering employment contracts that can provide monetary incentives for individual effort but not for cooperative effort. Selfish workers thus never cooperate, whereas conditionally cooperative workers might cooperate if matched into a team with another conditional cooperator. Our results show that there always exists a separating equilibrium in which workers sort into firms that differ from each other in several dimensions. Selfish workers are employed in firms that offer strong monetary incentives for individual effort but do not induce cooperation among workers. Conditionally cooperative workers are employed in firms where workers cooperate. In addition, if productivity effects of cooperation are high or workers have sufficiently large direct benefits from cooperation, monetary incentives are muted in firms employing conditional cooperators.

The assumption of conditionally cooperative preferences is based on solid empirical evidence showing that most cooperation observed in experiments and in the field is contingent on the cooperation of others (G¨ achter 2007). Interestingly, the sorting result in our model breaks down once we assume that cooperation is unconditional. Suppose cooperative workers get no extra non-pecuniary utility from cooperating with a cooperating worker and always exert cooperative effort independent of their colleague’s behavior. In a separating equilibrium all firms must pay out equal wages, since both cooperative and selfish workers now always prefer those firms that offer the highest wages. The firms attracting only cooperative workers then make strictly positive profits. But then a firm can enter the market with a pooling contract that offers slightly higher wages, because with such unconditional cooperation the new firm attracts all workers. Sorting under private information is therefore impossible.

20

In our model conditional cooperation is driven by workers’ intrinsic satisfaction from mutual cooperation. Alternatively, conditional cooperation could be modeled by equity concerns or feelings of guilt, where players suffer from failing to reciprocate cooperation (e.g. Fehr and Schmidt, 1999). In this case our sorting mechanism — conditional cooperators forgo higher wages to enjoy some warm glow from mutual cooperation — no longer works. However, the field evidence by Hamilton, Nickerson, and Owan (2003) suggests that mutual cooperation generates positive utility. Further, recent experiments shed doubt on inequity aversion driving conditional cooperation. Whereas inequity aversion predicts that conditional cooperators should trust less due to the additional disutility they experience in case trust is exploited, conditional cooperators actually trust more than selfish types, even when controlling for differences in beliefs (Altmann, Dohmen, and Wibral 2008, Blanco, Engelmann, Koch, and Normann 2009). While these findings are consistent with the intrinsic-satisfaction assumption, the existing data is not rich enough to fully disentangle the true motivation for conditional cooperation. This leaves the empirical investigation of conditional cooperators’ “real utility function” an interesting question for future research.

Our results show that firms employing conditional cooperators can make positive profits in equilibrium, whereas competition for selfish workers drives down profits of the corresponding firms to zero. This is because in a separating equilibrium employment contracts for conditional cooperators have to be sufficiently unattractive to selfish workers. Thus, firms may be constrained from paying out all surplus to the workers, because high wages lead selfish workers to invade the firm. In consequence, cooperation is destroyed and the firm incurs losses. This suggests that firms with a more cooperative corporate culture pay out relatively low wages. Following up on the quote at the beginning, this prediction is in line with Gittell, von Nordenflycht, and Kochan (2004), who document that wages at Southwest Airlines are about 12 percent below average and 33 percent below the best paying airline in a sample of ten mayor U.S. airlines.

As mentioned in the introduction, the positive-profit result has important implications for the existence of non-profit firms and the potential provision of goods and services by the government. Our model is agnostic about which firms will be “cooperative” and which will be “selfish” in equilibrium. Yet, because cooperative firms may have to withhold parts of their surplus in order not to attract selfish workers, firms that are better in committing not to redistribute profits will have a competitive advantage in attracting and retaining conditional cooperators. Non-profit firms, which are constrained in redistributing surplus due to their

21

commitment to devote parts of their resources to a non-profit goal, are one such example; government-owned firms that provide public service or goods out of the surplus created are another. Our model thus predicts that the existence of non-profit and government-owned firms is particularly likely whenever the separation constraint of selfish workers is binding, i.e., whenever cooperation plays a sufficiently important role in the production function and a worker’s preference is private information. Contrary to previous explanations focusing on non-contractibility or employee mission, in our model it is the ability to commit not to redistribute surplus which yields non-profit firms a competitive advantage.13 Ironically, our results show that non-profit firms might be more profitable than for-profit firms. However, if we add the assumption that conditional cooperators care also for the mission or the goal of a non-profit firm, competition eliminates any positive profit because it forces non-profit firms to spend all remaining surplus on the non-profit goal. In this case, all firms in our model make zero profit.14

Our model highlights the role of sorting and worker heterogeneity in the emergence and stability of different production processes, thereby providing an argument for the co-existence of non-profit and for-profit firms. Our model provides a similar argument for the co-existence of different corporate cultures. Contrary to research in management science stressing the importance of leadership (Schein 2004), firms in our model develop different cultures not because particular entrepreneurs create them, but because they are the outcome of competition for workers with heterogeneous preferences. We do not consider these two explanations to be contradictory but believe that they are, in fact, complements. In much the same way as non-profit firms will have a competitive advantage in hiring cooperative workers, it seems intuitive that entrepreneurs who trust in the motive of conditional cooperation will be better in implementing a culture of cooperation than entrepreneurs who believe that workers are selfish and do not provide effort without explicit incentives.15 Our results then show that labor market competition will not necessarily eliminate a potential ex-ante heterogeneity of firms due to entrepreneurs with different beliefs, goals or attitudes. Competition can indeed foster firm heterogeneity, thereby providing a basis for the survival of teamwork and cooperation within some firms. 13

In this sense our explanation connects to Francois (2000).

14

By a similar argument, all firms make zero profit if firms can offer benefits that are strictly preferred by

cooperative workers compared to selfish workers (for example gregarious events). 15

Falk and Kosfeld (2006) provide evidence for belief heterogeneity in a simple principal-agent experiment.

Their findings suggest that even in the absence of sorting principals’ beliefs can become self-fulfilling prophecies that reinforce the emergence of corporate cultures.

22

Appendix A: Definition of Equilibrium and Refinements Definition 1 (Workers’ Equilibrium Behavior) Workers behave optimally given a set of offered contracts W ⊆ W if and only if the corresponding beliefs µ and type-dependent  strategies e∗θ , a∗θ for θ ∈ {s, c} form a perfect Bayesian equilibrium, i.e., (i) the type-dependent effort choice e∗θ (w, W ) maximizes a type-θ worker’s expected utility
˜ W ), Eθ˜ uθ w, e∗θ (w, W ), e∗θ˜(w, W ) given beliefs µ(θ|w, (ii) the type-dependent acceptance decision a∗θ (w, W ) maximizes a type-θ worker’s expected utility over the set of offered contracts W given the outcomes as implied by the behavior characterized in (i) and the other workers’ acceptance decisions, and (iii) the beliefs µ(θ|w, W ) for θ ∈ {s, c} are consistent with workers’ acceptance decisions a∗θ (w, W ) and Bayes’ rule if the contract is accepted with strictly positive probability a∗θ (w) > 0 by at least one type θ ∈ {s, c}. Definition 2 (Competitive Equilibrium) Given that workers behave optimally for all possible finite sets of offered contracts, a competitive equilibrium is a finite set of offered contracts W ∗ , where (i) every contract w ∈ W ∗ is accepted with strictly positive probability by at least one type of worker, i.e., a∗θ (w, W ∗ ) > 0 for at least one type θ ∈ {s, c}, (ii) every contract w ∈ W ∗ generates non-negative expected profit given the equilibrium beliefs µ(θ|w, W ∗ ) and workers’ type-dependent equilibrium effort choices e∗θ (w, W ∗ ) for both types θ ∈ {s, c}, and (iii) no contract w ˜∈ / W ∗ can profitably attract workers, i.e., there exists no w ˜∈ / W ∗ such ˜ W ∗ ∪ w) ˜ > 0 for at least one type θ ∈ {s, c} and w ˜ yields strictly positive that a∗θ (w, expected profit given the beliefs µ(θ| w, ˜ W ∗ ∪ w) ˜ and workers’ type-dependent effort choices e∗θ (w, ˜ W ∗ ∪ w) ˜ for both types θ ∈ {s, c}. Refinement 1 (Beliefs) Consider a competitive equilibrium W ∗ and suppose an additional  ˜) for all (e, e ˜) contract w ˜ 6∈ W ∗ is offered. Suppose further that u∗θ > max(e,˜e) uθ (w, ˜ e, e ˜ Then µ(θ| w, ˜) ≥ u∗ for some (e, e ˜) with θ 6= θ. and u ˜(w, ˜ e, e ˜ W ∗ ∪ w) ˜ = 0. θ

θ˜

Refinement 2 (Coordination) Consider a competitive equilibrium W ∗ and suppose an additional contract w ˜ 6∈ W ∗ is offered. Suppose further that µ(θ|w, ˜ W ∗ ∪ w) ˜ = 1 and that there are unique equilibrium efforts e∗θ so that uθ (w, ˜ e∗θ , e∗θ ) ≥ u∗θ given belief µ(θ|w, ˜ W ∗ ∪ w) ˜ 23

and contract w. ˜ Then workers expect workers who accept contract w ˜ to choose equilibrium efforts e∗θ . Refinement 3 (Acceptance and Stability) Consider a competitive equilibrium W ∗ and suppose an additional contract w ˜ 6∈ W ∗ is offered. Given the corresponding beliefs and typedependent effort choices, let uθ (w) ˜ denote the expected utility of a type-θ worker who accepts w ˜ and is matched into a team. Then

a∗θ (w, ˜ W ∗ ∪ w) ˜ =

      

1

if uθ (w) ˜ > u∗θ ,

∈ [0, 1] if uθ (w) ˜ = u∗θ , and 0

if uθ (w) ˜ < u∗θ .

for both types θ ∈ {s, c}. Moreover, suppose that a∗θ (w, ˜ W ∗ ∪ w) ˜ = 0 for both types of workers θ ∈ {s, c}. Then workers do not change their acceptance and effort choices, that is, a∗θ (w, W ∗ ∪ w) ˜ = a∗θ (w, W ∗ ) and e∗θ (w, W ∗ ∪ w) ˜ = e∗θ (w, W ∗ ) for both types of workers θ ∈ {s, c} and all contracts w ∈ W ∗ .

Appendix B: Proofs Proof of Lemma 2 (Cooperative Equilibrium) Selfish workers never exert cooperative effort. A cooperative equilibrium thus exists if and only if it is optimal for a conditionally cooperative worker to exert cooperative effort given he expects his colleague to exert cooperative effort, as well. Given contract w ∈ W and belief µ(c|w, W ) there are two cases.

A) Suppose conditionally cooperative workers also exert individual effort in equilibrium. They then get expected utility b + f + µ(c|w, W ) (Γ + Ξ) − ψ(1, 1).

(17)

Exerting cooperative effort forms an equilibrium if there are no profitable deviations. It is optimal not to deviate and exert only cooperative effort if and only if (17) is weakly larger than f +µ(c|w, W ) (Γ+Ξ)−ψ(0, 1). This is equivalent to b ≥ ψ(1, 1)−ψ(0, 1). From the remaining two deviations, exerting only individual effort dominates exerting no effort at all by Assumption (3) and the above condition on the bonus b. Finally, it is optimal not to deviate and exert only individual effort if and only if (17) is weakly larger than b + f + µ(c|w, W ) Ξ − ψ(1, 0). This yields the condition on the belief µ(c|w, W ) for the case b ≥ ψ(1, 1) − ψ(0, 1).

24

B) Suppose conditionally cooperative workers only exert cooperative effort in equilibrium. They then get expected utility f + µ(c|w, W ) (Γ + Ξ) − ψ(0, 1).

(18)

By our cutoff rule conditionally cooperative workers do not deviate and exert both individual and cooperative effort if and only if (18) strictly exceeds b + f + µ(c|w, W ) (Γ + Ξ) − ψ(1, 1). This is equivalent to b < ψ(1, 1) − ψ(0, 1). From the remaining two deviations, exerting only individual effort dominates exerting no effort at all if and only if b+f +µ(c|w, W ) Ξ−ψ(1, 0) ≥   f + µ(c|w, W ) Ξ − ψ(0, 0). In case b ∈ ψ(1, 0) − ψ(0, 0), ψ(1, 1) − ψ(0, 1) , then the most attractive remaining deviation is to exert only individual effort. It is then optimal not to deviate if and only if (18) exceeds b + f + µ(c|w, W ) Ξ − ψ(1, 0). If b < ψ(1, 0) − ψ(0, 0), then the most attractive remaining deviation is to exert no effort at all. It is then optimal not to deviate if and only if (18) exceeds f + µ(c|w, W ) Ξ − ψ(0, 0). These inequalities yield the corresponding conditions on the belief µ(c|w, W ) given bonus b.

Q.E.D.

Proof of Lemma 4 (Constrained Optimal Cooperative Contracts) Whether or not conditionally cooperative workers are to exert individual effort in addition to cooperative effort determines the constrained optimal cooperative contracts. Consider first contracts that implement both individual and cooperative effort. Then Lemma 2 implies b ≥ ψ(1, 1) − ψ(0, 1). The minimum bonus does not violate the selfish workers’ screening constraint (14) if and only if Ξ ≤ X − ψ(1, 1) + ψ(0, 1).

(19)

For large Ξ this condition is violated. We are then in case (ii) and it is not possible to implement both individual and cooperative effort without attracting selfish workers. Optimal contracts then implement only cooperative effort. In the following suppose (19) is satisfied so that we are in case (i). Increasing bonus and fixed wage then increases the utility of conditionally cooperative workers, but tightens the zero profit constraint (13) and the selfish workers’ screening constraint (14). Only the latter is binding because paying out any cooperative output Y attracts selfish types. The binding (14) yields b + f = X − Ξ such that X + Γ − ψ(1, 1) is the equilibrium utility of a conditionally cooperative worker. This satisfies the screening constraint for cooperative workers (15).

Consider next contracts that implement only cooperative effort. Then the bonus has to satisfy b < ψ(1, 1) − ψ(1, 0) by Lemma 2. Since a conditionally cooperative worker exerts no 25

individual effort, he does not benefit from the bonus. An optimal contract thus specifies the highest possible fixed wage f . Doing so slackens the screening constraint for cooperative workers (15), but it tightens both the zero profit constraint (13) and the screening constraint for selfish workers (14). Which of these constraints binds determines optimal contracts; the interplay between these two constraints is also affected by the bonus b. Note that increasing b weakly tightens the screening constraint for selfish workers (14) but that the bonus has no direct impact on the objective function. Whether (13) or (14) is binding is ultimately determined by team output Y . There are two cases.

A) If Y ≤ Y2 = X − ψ(1, 0) + ψ(0, 0) − Ξ, then the zero profit constraint (13) is binding even if the bonus b is zero. An optimal contract specifies f = Y , where the bonus b can be positive and even exceed ψ(1, 0) − ψ(0, 0) as long as the screening constraint for selfish workers (14) remains non-binding. This is true as long as b ≤ X −Ξ−Y . An optimal contract yields conditionally cooperative workers utility Y + Ξ + Γ − ψ(0, 1). This utility satisfies the screening constraint for cooperative workers (15) if and only if Ξ ≥ X − ψ(1, 0) − Y − Γ + ψ(0, 1).

(20)

For very small Y this condition might not be satisfied, in which case there exists no contract that can separate conditionally cooperative and selfish workers if cooperative workers exert no individual effort. As we show below, it is optimal to implement both individual and cooperative effort if (20) is not satisfied.

B) If Y ≥ Y2 , then the screening constraint for selfish workers (14) is binding for all positive bonuses. Suppose b ≥ ψ(1, 0) − ψ(0, 0) so that the contract induces selfish workers who accept the contract to exert individual effort. Then the binding screening constraint yields as fixed wage f = X − b − Ξ. The fixed wage and thereby the objective function are decreasing in b as long as b induces selfish workers to exert individual effort if they accept the contract for conditionally cooperative workers. It is thus optimal to lower the bonus until b ≤ ψ(1, 0) − ψ(0, 0). An optimal contract specifies f = X − ψ(1, 0) + ψ(0, 0) − Ξ. This yields conditionally cooperative workers utility X − ψ(1, 0) + ψ(0, 0) + Γ − ψ(0, 1), which satisfies the screening constraint for cooperative workers (15).

Given the above cases, team output Y determines whether or not contracts in W cc optimally implement individual effort in addition to cooperative effort. Suppose Y ≥ Y2 . Conditional cooperators then get equilibrium utility X − ψ(1, 0) + ψ(0, 0) + Γ − ψ(0, 1) if they are to

26

exert no individual effort. By Assumption (3) this equilibrium utility exceeds the respective equilibrium utility X + Γ − ψ(1, 1) if they are to exert also individual effort. This holds even if (19) is satisfied so that it is actually feasible to implement individual effort without attracting selfish workers. It is thus always optimal not to implement individual effort if Y ≥ Y2 . Suppose next that Y < Y2 . If conditional cooperators are to exert no individual effort, they then get equilibrium utility Y + Ξ + Γ − ψ(0, 1). Optimal effort implementation depends on whether it is feasible or not to implement both individual and cooperative effort. If it is not feasible, we are in case (ii) and Ξ > X − ψ(1, 1) + ψ(0, 1). It is then optimal to implement only cooperative effort for all Y > 0. The equilibrium utility of conditionally cooperative workers satisfies their screening constraint (20) by Ξ > X − ψ(1, 1) + ψ(0, 1) and Assumption (6). If it is feasible to implement both individual and cooperative effort, we are in case (i) and Ξ ≤ X − ψ(1, 1) + ψ(0, 1). Then implementing both individual and cooperative effort is optimal if and only if Y ≤ Y1 . This holds even if (20) is satisfied so that it is feasible to only implement cooperative effort. For Y ≥ Y1 it is optimal to implement only cooperative effort. The equilibrium utility of conditionally cooperative workers then satisfies their screening constraint (20) by Y ≥ Y1 and Assumption (6).

Q.E.D.

Proof of Proposition 1 (Separating Equilibrium) We first show that in any separating equilibrium selfish workers receive a contract from W s and conditionally cooperative workers receive a contract from W cc . Suppose we are in any separating equilibrium. Consider first selfish workers. We show that these workers get equilibrium utility u∗s = X − ψ(1, 0). Since firms attracting selfish types make no losses in equilibrium, we have u∗s ≤ X − ψ(1, 0). Suppose u∗s < X − ψ(1, 0). A firm can then enter ˜ with fixed wage f˜ = 0 and bonus ˜b = X −  where the market by offering a new contract w  > 0 sufficiently small so that ˜b − ψ(1, 0) > u∗s . This contract implements high individual effort and makes strictly positive expected profits if accepted.

Our refinements guarantee that the contract must be accepted by some workers. To see this, suppose the contract was not accepted by any worker. Then by Refinement 3 all workers stick to their old contract and effort choices. Refinement 2 then implies that it is not consistent with Refinement 3 if selfish workers do not accept the new contract. It is possible to either construct an equilibrium in which selfish workers accept the new contract and conditional cooperators stick to their old contract and effort choices, or to construct an equilibrium in which all workers accept the new contract. The new contract generates positive profits whenever 27

accepted, thus the old situation cannot be a competitive equilibrium. Since selfish workers receive equilibrium utility u∗s = X − ψ(1, 0), they get a contract from set W s . Consider next conditional cooperators. We focus on the case in which contracts for cooperative workers implement no individual effort. The proof for the case in which they implement also individual effort is analogous. Suppose conditional cooperators do not receive any contract from W cc . They earn equilibrium utility u∗c . Selfish workers accept a contract ws ∈ W s . ˜ that specifies a positive fixed Suppose a firm now enters the market with a new contract w ˜ is equal to some contract from W cc except that the fixed wage f˜ wage f˜ and zero bonus. w ˜ is lower by  > 0. As all contracts in W cc satisfy the zero profit constraint (13) contract w generates strictly positive profits if it is accepted only by conditionally cooperative workers who then cooperate. For  sufficiently small, f˜ + Ξ + Γ − ψ(0, 1) > u∗ because contracts c

in W cc yield conditional cooperators their maximum utility in any separating equilibrium. Cooperative workers thus get more than their equilibrium utility if they accept the new contract and cooperate. Since  > 0 the relevant screening constraint for selfish workers (14) is slack. This has two implications. First, selfish workers strictly prefer their contract ws to ˜ even if they believe that the new contract is only accepted by cooperathe new contract w tive workers who cooperate. In any equilibrium at the acceptance stage selfish workers thus ˜ Second, conditionally cooperative workers strictly prefer never accept the new contract w. ˜ if they think that the new contracts is only the selfish contract ws to the new contract w accepted by workers who do not cooperate.

Again, our refinements guarantee that the contract is accepted by conditionally cooperators. To see this, suppose the contract was not accepted by any worker. Then by Refinement 3 all workers stick to their old contract and effort choices. However, by the above characteristics of the new contract, Refinement 1 and 2 imply that workers believe that any workers who accept the new contract are conditionally cooperative and cooperate. Then it is not consistent with Refinement 3 if conditionally cooperative workers do not accept the new contract. Equally, it cannot be an equilibrium that conditionally cooperative workers accept the new contract and do not cooperate. In this case both selfish and cooperative workers prefer the old contract of the selfish workers. Thus, in any equilibrium at the acceptance stage the new contract attracts conditionally workers who cooperate. As the contract makes strictly positive profit, there is market entry.

We next show that conditional cooperators cooperate in any separating equilibrium. Together

28

with the above, this implies that firms employing selfish workers make zero profit, that firms employing conditionally cooperative workers make profits as described in Lemma 4, and that the equilibrium is Pareto-efficient. Suppose this was not the case, i.e., there exists a separating equilibrium in which conditional cooperators do not cooperate. Because conditional cooperators behave like selfish workers, u∗c = u∗s = X − ψ(1, 0). A new firm can then enter the market with a contract w ˜ with bonus ˜b and fixed wage f˜ that implements the same individual effort as contracts from W cc but pays out slightly lower wages. By construction this contracts satisfies u∗s > us (w, ˜ eSE (˜b), e ˜)

(21)

for any effort e ˜ chosen by workers attracted by the new contract. However, uc (w, ˜ eCE (˜b), eCE (˜b)) > u∗c > uc (w, ˜ eSE (˜b), eSE (˜b))

(22)

for the conditionally cooperative workers. By essentially the same argument as above, our refinements then imply that there is profitable market entry.

Finally, we show that the separating equilibrium described in the Proposition exists. For this we need to show that there is no profitable market entry. By construction there cannot be profitable market entry by firms that attract only selfish types. Equally, there cannot be profitable market entry by firms that attract only conditional cooperators. Suppose a firm ˜ that specifies fixed wage f˜ and bonus ˜b. To enters the market with a pooling contract w complete the proof we only have to show that we can specify equilibrium worker behavior ˜ either attracts no workers or makes that does not violate our refinements whereas contract w losses. Before specifying workers’ equilibrium behavior note that u∗c ≥ u∗s because conditionally cooperative workers can always imitate the behavior of selfish workers and thereby get the same utility. Further, Refinement 1 does not restrict beliefs as by definition both types of workers ˜ We now specify that workers coordinate on the selfish equilibrium might be attracted by w. ˜ W ∗ ∪ w) ˜ = eSE (˜b) for θ ∈ {c, s}. In the when they accept the new contract, i.e., e∗ (w, θ

following we will show that such effort choices do not violate any of our refinements. As both types get the same utility in a selfish equilibrium, all workers get the same deviation utility ˜ u ˜ when accepting w. There are three cases. First, suppose u ˜ > u∗c ≥ u∗s . Our refinement regarding acceptance ˜ W ∗ ∪ w) ˜ = 1 for θ ∈ {c, s} so that µ∗ (c| w, ˜ W ∗ ∪ w) ˜ = λ. As workers then implies a∗θ (w, 29

are attracted even when expecting the selfish equilibrium to be played, coordinating on the ˜ must make losses because selfish equilibrium does not violate Refinement 2. But then w u ˜ > u∗s = X − ψ(1, 0). Second, suppose u∗c ≥ u∗s ≥ u ˜. Without violating Refinement 3 we can ˜ W ∗ ∪w) ˜ = 0 for θ ∈ {c, s}. Beliefs are not pinned down so that µ∗ (c| w, ˜ W ∗ ∪w) ˜ =0 set a∗θ (w, is possible. Given this belief the selfish equilibrium is the unique equilibrium. Acceptance choices are optimal and the firm cannot attract any workers. Third, suppose u∗c > u∗s and ˜ W ∗ ∪w) ˜ = 1 but setting a∗c (w, ˜ W ∗ ∪w) ˜ = u∗c ≥ u ˜ > u∗s . In this case Refinement 3 yields a∗s (w, ˜ W ∗ ∪ w) ˜ = 0. Given this belief the selfish 0 causes no violation. This implies µ∗ (c| w, equilibrium is the unique equilibrium. Refinement 2 thus has no bite and workers coordinate ˜ makes losses as u on the selfish equilibrium. In consequence, w ˜ > u∗s = X − ψ(1, 0). Q.E.D. Proof of Proposition 2 (No Pooling) Consider a competitive equilibrium W ∗. If there is pooling, there is a contract wp ∈ W ∗ that is accepted by both types of workers. Contrary to selfish workers, conditionally cooperative workers might contribute cooperative effort. As long as conditionally cooperative workers exert as much individual effort as selfish workers, they thus produce weakly more output. The latter is not the case if and only if bp ∈ [ψ(1, 0)−ψ(0, 0), ψ(1, 1)−ψ(0, 1)[ and conditionally cooperative workers coordinate on the cooperative equilibrium. In this situation µ(c|wp , W ∗ )( µ(c|wp , W ∗ ) Y − f ) + µ(s|wp , W ∗ )(X − b − f )

(23)

describes the firm’s expected profit per worker. Note that a conditionally cooperative worker produces team output only with the probability with which his colleague is also conditionally cooperative. (23) implies that selfish workers subsidize conditionally cooperative workers in a pooling equilibrium if and only if µ(c|wp , W ∗ ) Y < f such that a conditionally cooperative worker’s expected team output does not cover his fixed wage. The following proof is divided into two parts depending on whether or not selfish workers subsidize conditionally cooperative workers.

Suppose selfish workers subsidize conditionally cooperative workers. As argued above selfish workers then exert individual effort. They get both bonus and fixed wage and do not benefit from the cooperation of any cooperative colleagues because Ξ = 0. Selfish workers thus receive equilibrium utility u∗s = b + f − ψ(1, 0). Because conditionally cooperative workers are subsidized, wages are low so that b + f < X. This implies u∗s < X − ψ(1, 0) which is impossible in any competitive equilibrium by the arguments from the first part of the proof of Proposition 1. 30

Suppose now that selfish workers do not subsidize conditionally cooperative workers. We show that profitable market entry is possible. There are two cases.

A) Suppose workers coordinate on the selfish equilibrium after accepting pooling contract wp .
For any contract w ∈ W c we have uc w, eCE (bc ), eCE (bc ) > u∗c . If contract w attracts only conditionally cooperative workers who cooperate, it generates zero profit. We then consider market entry by a new firm that offers a contract w ˜ 6∈ W ∗ that resembles some contract in W c but specifies somewhat lower wages (see below). B) Suppose cooperative workers cooperate after accepting pooling contract wp with bonus
bp . Pooling implies µ(c|wp , W ∗) < 1 such that uc wp , eCE (bp ), eCE (bp ) > u∗c . Since selfish workers do not subsidize conditionally cooperative workers, wp generates at least zero profit if it is accepted only by conditionally cooperative workers who coordinate on the cooperative equilibrium. We then consider market entry by a new firm that offers a contract w ˜ 6∈ W ∗ that resembles the pooling contract wp but pays out somewhat lower wages (see below). We finally show that market entry with a contract w ˜ 6∈ W ∗ as specified above is profitable. The mentioned reduction in wages can be chosen such that w ˜ implements the same individual effort in the cooperative equilibrium as a contract from W c in case A) or the pooling contract wp in case B). By construction it generates strictly positive profit if it is accepted only by conditionally cooperative workers who then coordinate on the cooperative equilibrium. Further, the reduction in wages can be set such that

uc w, ˜ eCE (˜b), eCE (˜b) > u∗c > uc w, ˜ eSE (˜b), eSE (˜b) .

(24)

There are two cases. First, suppose u∗c = u∗s . By assumption conditionally cooperative workers who do not cooperate get the same utility as selfish workers. This and the above inequality then imply ˜ u∗s > us w, ˜ eSE (˜b), e




(25)

˜. Second, suppose u∗c > u∗s . Then the situation must be as in B) so that w for any e ˜ resembles

˜ > us w, ˜ for any e ˜ so that the pooling contract. This yields u∗ = us w, eSE (b), e ˜ eSE (˜b), e s

inequality (25) holds, as well. Our refinements then imply that market entry with contract w ˜ is profitable, where only because of Ξ = 0 we can be sure that it is an equilibrium that all cooperative workers accept the new contract whereas all selfish workers stick to their old contract and effort choices.

Q.E.D. 31

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uc

pppp ppppp

6

pp

Y + Ξ + Γ − ψ(0, 1)

pppp ppppp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp ppp p p p p p p p p p p p p p p p p p p p p p p p p X − ψ(1, 0) + ψ(0, 0) + Γ − ψ(0, 1) p p pppp p p p p  p p p pp p p p p p p p p p  p p p p p p p p p p p p p  p p p p  p ppp pp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p X + Γ − ψ(1, 1) p p p p p p p p p p p p p pp ppp p p p p p p p p ppppp ppppp -

Y1 = X − ψ(1, 1) + ψ(0, 1) − Ξ

Y

Y2 = X − ψ(1, 0) + ψ(0, 0) − Ξ

Figure 1: Equilibrium Utility of Conditionally Cooperative Workers

37