Team composition, worker effort and welfare

Team composition, worker effort and welfare∗ By Roland Bel†, Vladimir Smirnov and Andrew Wait‡ November 21, 2016

Abstract A successful organization needs the right team. We explore the optimal mix of familiar workers (who we call incumbents) and less familiar workers (newcomers) when production is group-based. If incumbents have a lower marginal return of effort, they will have less incentive to invest relative to newcomers. This is true, even when incumbents produce more for any given level of effort. This creates a tradeoff: less familiar agents will invest more whereas a more familiar team is inherently more productive. In our setup, the non-investing principal (weakly) prefers less familiar agents than the team that maximizes second-best surplus. On the other hand, (symmetric) agents prefer a more familiar teammate compared with the second-best option. These insights have implications for team composition, job rotation and worker tenure. Key words: experience, team composition, task allocation, job rotation, holdup. JEL classifications: D21, L23 ∗

We would like to thank Murali Agastya, Marco Alderighi, Nicolas de Roos, Robert Gibbons, Justin P. Johnson, Mark Melatos, Jonathan Newton, Hongyi Li, Simon Loertscher, Bonnie Nguyen, Suraj Prasad, Abhijit Sengupta, Kunal Sengupta, John Van Reenen, John Romalis, Don Wright, and participants at the ESAM Meetings 2011, the International Industrial Organization Conference in Boston 2011, the Australasian Theory Workshop 2013, the Organizational Economics Workshop 2013 and the 76th IAES Conference in Philadelphia. Roland Bel would like to thank University of Sydney for their support and hospitality. The authors are responsible for any errors. † KEDGE Business School, Marseille, France. [email protected]. ‡ School of Economics, University of Sydney, NSW 2006, Australia. [email protected]; [email protected].

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1

Introduction

What does it take for success on Broadway? A Broadway production needs: a composer; a lyricist; a librettist who writes the dialogue and plot; a choreographer; a director; and a producer. But who should fulfill these roles? Analyzing Broadway shows between 1877 and 1990, Uzzi & Spiro (2005) find that financial and critical success is increasing in the number of the team members who previously worked together (incumbents), but only up to a point; too many incumbents decrease the likelihood of a production’s success. Therefore, the most successful teams comprise of some incumbents, but also some newcomers. Familiar workers – who have worked together before – have some natural advantages. They understand how each other works; they know others’ strengths, weaknesses and communication idiosyncracies. Familiar team members often have a better sense of what the others will like and dislike, helping agents to avoid proposing and arguing for ideas that will never be part of the final product. Given the advantage of incumbents, why choose newcomers who have no experience of working with one another? Uzzi & Spiro (2005) emphasize that new team members bring fresh ideas to the collaborative effort, increasing overall quality. We focus on a similar rationale for using unfamiliar agents; previous experience of working together can decrease the incentive for agents to invest, ultimately reducing total output (or quality). In this framework we examine the preferred team composition (that is, the balance of familiar incumbents and unfamiliar newcomers) for the principal and the teammates themselves, and relate our findings to a variety of applications. To analyze the choice of team composition we make two key assumptions. First, contracts are incomplete, creating an underlying hold-up problem. Second, final output depends on both the team experience of the workers and the efforts they make. Specifically, we assume that both the total and marginal return depend on the workers’ familiarity. As a consequence, team familiarity will influence each worker’s equilibrium effort. In our model, two agents work in a team. The key choice when selecting a team is the level of familiarity between the workers; this choice is typically made by the principal. Worker familiarity helps determine effort and the output generated. For ease of exposition, we use the term incumbent for workers who are very familiar with each other. Alternatively, agents who have not had any previous work history are newcomers.1 1

Note, we are explicitly referring to experience or familiarity of agents working together; in this sense ‘experience’ means experience on a particular team, and not to their knowledge or skills in the profession more generally. Similarly, ‘newcomer’ refers to a new member of that team, not an inexperienced worker per se. We also allow familiarity to be a continuous

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Once the team composition is chosen, each agent makes a non-contractible effort. After these efforts are sunk, all parties negotiate and receive their share of ex post surplus. Teams made up of workers who are more familiar with each other, intrinsically produce greater output for any given level of effort. On the other hand, teams of unfamiliar agents, because they have never worked together, produce less surplus. We make the following important assumption: the additional surplus generated using incumbent agents relative to newcomers is decreasing in worker effort. In other words, unfamiliar workers can (partially) make up for the superior productivity of a familiar team by exerting effort. As a consequence, there is a tradeoff when choosing a team; while familiar workers produce more surplus for any given level of effort, unfamiliar workers put in more effort in equilibrium. This tradeoff gives rise to several results. First, if an enforceable contract can be written on effort, the principal will always opt for a team of incumbent agents (with the highest level of familiarity possible). This follows as surplus for any given effort level is always higher with more familiar agents. Second, if an enforceable contract cannot be written, the (second-best) welfare maximizing team may have workers with some level of unfamiliarity, just like many successful Broadway production teams. Third, we show that the different parties involved might have different preferences regarding team composition. If it is the principal who decides on the makeup of the team she will always choose a team that is more unfamiliar (or at least as unfamiliar) as would be chosen by an agent. Furthermore, a principal will choose a more unfamiliar team than required to maximize (second-best) surplus. On the other hand, if one of the agents decides on team composition, he could choose too much, too little or the second-best optimal level of familiarity with his teammate. If agents are symmetric, however, each worker always prefers too much familiarity amongst their co-workers relative to the second-best team structure. There are many applications of our model. The model is directly relevant to team composition and job rotation.2 Job rotation, by its very nature, breaks up old relationships and makes workers start afresh with at least some new members. This practice is used by many consulting firms; McKinsey & Company, for example, insists on rotating senior management roles. Our model suggests there are potential benefits from committing to a job-rotation policy, as the agents, left to their own devices tend to choose too many teamvariable, rather than a binary choice, reflecting the differing work experience histories agents can have with one another. 2 Other explanations for job rotation include: eliciting information from agents (Arya & Mittendorf 2004); avoiding worker boredom (Azizi et al. 2010); and limiting scope for corruption (Choi & Thum 2003).

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mates with whom they have previously worked, even if a more unfamiliar team would be more productive. This could also be suggestive as to why some firms use predetermined rotation rules so as to avoid influence costs (Milgrom & Roberts 1990). In a similar manner to the Broadway study mentioned above, Guimera et al. (2005) find that the inclusion of newcomers in research teams increases the probability of a successful scientific collaboration in social psychology, economics, ecology and astronomy. In another situation, some airlines integrate a permutation constraint in their cabin crew assignment algorithm that prevents familiar pilots from being assigned to the same flight.3 There is likely to be a potential advantage from using familiar workers – for instance, crew members who know each other well probably communicate more easily – but these airlines explicitly forgo this benefit. Familiar crew may be dissuaded from undertaking the same level of effort when teamed with each other (checking and cross checking and so on) than when teamed up with a stranger. With lower effort, the outcome (in terms of safety incidents) could be worse when familiar agents are paired together, despite their intrinsic advantage. Teams of doctors – attending physicians, residents, interns and so on – are also rotated, both within and across hospitals. The same tradeoffs between effective communication amongst familiar workers and providing additional incentives for effort (or, equivalently, to remain attentive) could arise in this case as in the cabin-crew example above. Likewise, manufacturing teams are often rotated, potentially foregoing the incumbents’ superior productive capability. In a similar way across a range of sports, even highly successful teams bring in new players (particularly in the off-season). Rotation of the team in our model does not rely on the need to replace deadwood or to achieve the right balance of skills; rather, our model suggests a tradeoff between experience and energy. The incumbent members know the team plays, structure, and so forth, so rotating the roster forgoes the complementarities that have developed over time. But new team members elicit greater effort from all players, old and new alike, and effort is crucial for sporting success. The foundation for our analysis is essentially a moral-hazard-in-teams model (see Alchian & Demsetz 1972, Holmstrom 1982 and Che & Yoo 2001). Given the externality between team members, there is always underinvestment in effort. In our paper, however, there is an additional consideration – the choice of team membership. This means that the principal (or the party deciding) also needs to take into account how the composition of the team affects agents’ incentives. Our analysis suggests that the additional output between familiar 3

Anonymous industry source.

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agents can foster (relative) indolence. If this is the case, choosing unfamiliar workers might be preferred as such an arrangement leads to greater levels of effort (and surplus).4 There is an existing literature that examines incentive-contract design when there are externalities between agents – see for example, Segal (1999, 2003), Bernstein & Winter (2012) and Winter (2004, 2006). These models typically investigate optimal contracting when the potential externalities are fixed (that is, they do not vary with effort). In contrast to most of these models, we focus on the case when the externality varies endogenously in equilibrium. The paper in this literature most similar to ours is Winter (2010), who examines how the structure of information inside a firm affects agents’ optimal incentive contracts. Specifically, he finds that creating an environment with greater information is beneficial when agents’ efforts are complementary. This is because the dissemination of information about agents’ effort (or lack thereof) can allow for effective punishment. The similarities in the two papers are that the work environment affects investment incentives. In Winter (2010) it is the flows of information; in our model it is the composition of the team. One key difference is that Winter explicitly considers individual incentive contracts based on output; in our model all parties receive a non-contractible share of group surplus ex post, as in Holmstrom (1982). In a different context, Franco et al. (2011) consider how worker types are matched, when this choice affects the optimal incentive contracts, which depend on type and output. They find that a principal might prefer to forgo technological complementarities (by not matching two low-cost workers together) if this allows for a better outcome in terms of effort and the cost of incentive compensation.5 Their result – that positive-associated matching need not hold once effort and incentive contracts are considered – parallels our prediction that new team members might be preferred. Note also that like in Franco et al. (2011) our focus here is on the incentives created when choosing different team structures; we do not focus on issues of matching that arise with unobservable worker type, such as in Jeon (1996), Newman (2007) and Thiele & Wambach (1999). 4

Team membership has also been explained by the technological complementarities between tasks (Brickley et al. 2008) or arising from a multi-tasking incentive problem (see Holmstrom & Milgrom 1991 and Corts 2006, for example). 5 Kaya & Vereshchagina (2014) make a similar point regarding moral hazard in partnerships.

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2

The Model

Consider a model with a principal P and two agents A1 and A2 .6 P owns an asset that is necessary for production and the agents use this asset to produce the final output or surplus. Each agent can expend some specific effort ei ∈ [0, ei )7 , for i = 1, 2, with a cost of Ci (ei ), as summarized in Assumption 1.8 Assumption 1. Ci (ei ) ≥ 0 and Ci (0) = 0. Ci (ei ) is twice differentiable. Ci0 (ei ) > 0 and Ci00 (ei ) ≥ 0 for ei ∈ [0, ei ). In addition, limei →ei Ci0 (ei ) = ∞ for i = 1, 2. The agents work together as a team to produce joint surplus v(e, s), where e = (e1 , e2 ). s is an index of familiarity between the two workers in the team; without loss of generality, s ∈ [0, 1]. The more experience agents have working with one another the higher the value of s.9 As the agents need to work as a team, their familiarity is one of the factors determining the team’s output (and the marginal productivity of effort). In the extreme, if s = 0 both agents are completely unfamiliar with one another; a team of two very familiar workers has an s = 1.10 Our next assumption summarizes the relationship between value, effort and the familiarity index s. Assumption 2. v(e, s) is twice differentiable with respect to all variables. 2 v(e,s) ∂ 2 v(e,s) 2 2 v(e,s) < 0, ∂ ∂e − ≥ 0, ∂v(e,s) > 0, ∂∂ev(e,s) ≥ 0, ∂ ∂e v(e, s) ≥ 0, ∂v(e,s) 2 2 ∂ei ∂s ∂e22 1 ∂e2 1 i  2 2 ∂ v(e,s) > 0 for i = 1, 2 and s ∈ [0, 1]. ∂e1 ∂e2 The intuition for the various parts of Assumption 2 is as follows. First, surplus is positive and non-decreasing in effort. This holds for all teams, whether they are comprised of two very familiar incumbents (s = 1), unfamiliar newcomers (s = 0) or agents with some intermediate level of familiarity with one another. Second, there is complementarity between the two efforts; an increase in one agent’s effort improves the marginal productivity of the other 6

Our insights can be generalized to a team of more than two players. Note that ei > 0 and it is possible that ei = ∞. 8 For ease of exposition the principal here does not incur any effort costs. Our key results hold provided the principal’s effort costs are relatively small. 9 We thank the editor for pointing out that s could be alternatively considered as a choice of technology or organizational structure that in turn determines the balance of familiar and unfamiliar workers. 10 As noted in the Introduction, the term ‘incumbents’ refers to agents who have an extensive work history together; on the other hand, ‘newcomers’ refers to agents with no experience working together in the same team. 7

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agent’s effort. Third, for a given level of effort incumbents always produce more output than a team of newcomers as familiarity helps workers mesh together more easily. Finally, for this team profit-maximization problem to be well defined, we assume that v(e, s) is concave in e for any s. The following assumption details how the relative advantage of incumbents is decreasing in effort. Assumption 3.

∂ 2 v(e,s) ∂ei ∂s

≤ 0 for i = 1, 2 and s ∈ [0, 1].

Even though the surplus is always non-decreasing in effort for any composition of the group (that is, for any s), the relative advantage of using more familiar workers is (weakly) decreasing in effort. This is because an agent has a higher marginal return of effort with lower level of s; in other words, a newcomer has a higher return on their effort than an agent working with more familiar teammates. This reflects that unfamiliar team members can partially overcome their relative disadvantage by learning the specific attributes of their fellow team members. That is, through their efforts newcomers can learn about the other agent’s strengths, weaknesses, how they communicate, and so on. As newcomers put in more effort, the relative advantage of incumbents already familiar with one another becomes smaller.

2.1

Timing and investment solution

The timing of the model is as follows. At date 0, the principal chooses the team composition in terms of the level of familiarity between teammates; specifically, the principal chooses s.11 At date 1, the selected agents choose their level of relationship-specific non-contractible effort. Finally, at date 2, the agents and principal bargain over their share of surplus. Figure 1 summarizes the timing. We assume that the principal and two agents are indispensable and receive shares of ex post surplus given by α0 and αi for i = 1, 2. We make the following standard assumption about these shares. P Assumption 4. αi > 0 for i = 0, 1, 2 and 20 αi = 1. Most property-rights models make a similar assumption; given the underlying hold-up problem the transacting parties share all of the ex post surplus according to some bargaining rule, be it the Shapley value (as in Hart & Moore, 1990), the Nash Bargaining solution (Grossman & Hart, 1986 and Hart, 1995) or a non-cooperative extensive-form bargaining game (Chiu 1998). 11

In Sections 3.2.2 and 3.2.3 we consider the agents’ and a social planner’s choice of team composition, respectively.

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t=0

t=1

6

• s chosen

t=2

6

6

• ei chosen

• bargaining over surplus • production

Figure 1: Timing of the game Our surplus-sharing rule is also consistent with the distribution of surplus in the moral-hazard-in-teams model with a balanced budget (Holmstrom 1982).

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Investment incentives

An agent’s incentive to invest depends on what team they find themselves in. In this section we investigate the implications of different levels of familiarity on investment incentives and welfare. To provide a benchmark, we first analyze the first-best team structure.

3.1

First-best incentives

Let us consider the first-best investment incentives and the choice of s. The total net surplus is 2 X W (e, s) = v(e, s) − Ci (ei ), (1) i=1

and the first-best investments e∗ solve: ∂v(e, s) ∂Ci (ei ) = , ∀ i = 1, 2 ∀ s ∈ [0, 1]. (2) ∂ei ∂ei Due to Assumptions 1 and 2, the solution for each agent’s investment choice exists and is unique. Given ∂v(e,s) > 0 ∀ e, both total and net surplus ∂s are always increasing in the degree of familiarity within the team. This means

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that the first-best team consists only of familiar workers (incumbents), or that s∗ = 1. Proposition 1. The surplus-maximizing team contains only incumbents; that is s∗ = 1. Total surplus is always higher when there is an increase in team familiarity for any level of effort. As a consequence, an increase in familiarity will increase gross surplus without altering costs. If an enforceable contract can be written on effort, the optimal team will consist of incumbents.

3.2

Second-best incentives

As a complete contract cannot always be written, consider the investment decision of the two workers taking into account ex post bargaining.12 The ex ante payoff for each agent is: Ai (e, s) = αi v(e, s) − Ci (ei ), ∀ i = 1, 2.

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Anticipating renegotiation, agents will set their effort to maximize their ex ante payoff. The equilibrium levels of effort e(s) solve the following first-order conditions: ∂Ci (ei ) ∂v(e, s) = , ∀ i = 1, 2. (4) ∂ei ∂ei Similar to the first-best case, the solution for each agent’s investment choice exists and is unique due to Assumptions 1 and 2. Moreover, given Assumption 3 the following proposition arises. αi

Proposition 2. An agent exerts a lower level of effort in a more familiar team; that is, ∂ei ≤ 0, ∀ i = 1, 2. (5) ∂s Proof: See Appendix A. Equilibrium effort – and cost – is always greater for an agent in a relatively unfamiliar team (as opposed, for example, in a team of incumbents). This follows as the marginal return from effort is lower with more familiar team members. 12

Note that given our assumption that a surplus-sharing contract or a contract on effort cannot be written at date 0, the principal in our model cannot act as a ‘budget breaker’ to ensure the first-best outcome, nor can she enforce a ‘target’ contract.

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3.2.1

The principal’s choice of team composition

Consider the principal’s choice of team composition taking into account the investment decision of the workers. The ex ante payoff for the principal is: P (e, s) = α0 v(e, s).

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Provided the second-order condition is satisfied for an interior solution, the choice of team composition s solves the following first-order condition: ∂v(e, s) ∂v(e, s) ∂e1 ∂v(e, s) ∂e2 + + = 0. (7) ∂s ∂e1 ∂s ∂e2 ∂s Note that we allow for corner solutions when the second-order condition does not hold. We label sp as the principal’s most profitable team structure (be it an internal or corner solution), anticipating each agent sets their effort so as to maximize their own ex ante payoff. 3.2.2

The agents’ choice of team structure

We now turn to the preferences of the agents as to the makeup of the team. The payoff for the agent i is given by equation 3. Taking into account the equilibrium levels of effort e(s) that satisfy condition 4, assuming the secondorder condition for an interior solution is satisfied, agent i’s choice of team structure is determined by the following first-order condition: ∂v(e, s) ∂v(e, s) ∂ej + =0 ∂s ∂ej ∂s

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for j 6= i. Anticipating that both agents in the team will set effort so as to maximize their own ex ante payoff, each individual agent – if they were able to – would choose the level of familiarity that maximizes his net return. Here we denote the agent’s choice of team structure as si .13 3.2.3

Second-best optimal choice of team structure

We now consider a social planner’s choice of team, where the goal of the planner is to maximize welfare subject to the equilibrium efforts of the agents. Assuming the second-order condition for an interior solution holds and given 13

In a similar manner to the principal’s preferred team structure, we allow this solution to be either an interior or corner solution, as we do for the second-best optimal choice of team structure, analyzed below.

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each agent’s equilibrium investment satisfies condition 4, the first-order condition for the second-best optimal choice of team structure is: ∂v(e, s) + ∂s



∂v(e, s) ∂C(e1 ) − ∂e1 ∂e1



∂e1 + ∂s



∂v(e, s) ∂C(e2 ) − ∂e2 ∂e2



∂e2 = 0. (9) ∂s

The intuition for this equation is as follows. Anticipating that agents set effort to maximize their ex ante payoff, the social planner will choose the team structure that provides the highest (second-best) surplus; we denote this team structure as ssb .

3.3

Comparison of team structure choices

Let us now compare the principal’s preferred team structure and what the agents themselves would prefer relative to both first- and second-best team compositions. We do this in the following proposition. Proposition 3. A comparison of team compositions yields that: (i) 0 ≤ sp ≤ si ≤ s∗ = 1 ∀ i = 1, 2; and (ii) 0 ≤ sp ≤ ssb ≤ s∗ = 1. Proof: See Appendix A. From the first part of the Proposition, it is evident that the principal (weakly) prefers less familiarity in the team than required for first best (sp ≤ s∗ ). As she does not make any non-contractible investment, the principal ignores the additional cost of effort borne by less familiar agents, while still sharing in any increase in surplus generated. Hence, a principal who captures a fraction of the benefits generated but who bears no effort costs will tend to prefer too little familiarity. Agents, for their part, take into account their own effort costs when considering their preferred team composition, but they do not factor in the effort costs of others. While each agent bears a higher individual effort cost in a team with a lower s, they gain from higher levels of effort from the other team member. Consequently, agents also weakly prefer less familiarity than the first-best level. An agent, however, always (weakly) prefers more familiarity than the non-investing principal who has no effort costs at all. This gives us the preferred ranking of familiarity of 0 ≤ sp ≤ si ≤ s∗ = 1. Now consider a social planner’s second-best choice of team composition. When deciding on her preferred team structure, a social planner considers the investment incentives and all of the agents’ effort costs. The social planner faces a tradeoff. A lower s is a credible way of inducing higher effort (as contracts are incomplete). As a result, the second-best choice of team structure 11

has a level of familiarity that is less than the first-best. On the other hand, less familiarity increases the effort costs endured by agents; consequently, compared with a non-investing principal, a social planer would choose a higher level of s. This yields the ranking in the second part of the Proposition: 0 ≤ sp ≤ ssb ≤ s∗ = 1. Proposition 3 compares the preferred team structures of the principal, agents and a social planner. With heterogenous team members there is, however, an ambiguous relationship between the preferred structure of any given agent (si ) and the second-best composition (ssb ). It is feasible, for example, that some agents might prefer more or less familiarity, given their differing marginal productivities and effort costs. To make a direct comparison we introduce the following definition. Definition 1. Agents are symmetric if both costs and value functions are symmetric with respect to efforts; that is, Ci (ei ) = C(ei ) ∀ i and v(e1 , e2 , s) = v(e2 , e1 , s). In addition, symmetric agents receive the same share of ex post surplus; α1 = α2 . This definition allows us to derive the following proposition. Proposition 4. If agents are symmetric, 0 ≤ sp ≤ ssb ≤ si ≤ s∗ = 1. Proof: See Appendix A. Proposition 4 states that if the agents are symmetric (in terms of their costs, productivity and claim on ex post surplus) they prefer too much familiarity compared with the second-best. Symmetry precludes the somewhat special case of an agent preferring less familiarity than the second-best choice when, for instance, an agent has the combination of: relatively low marginal effort costs; and a substantial claim on ex post surplus. To aid with the intuition for this Proposition, we temporarily relax Assumption 4 and consider the case when the principal has no claim on surplus (or is not present). Corollary 1. If agents are symmetric and α0 = 0, si = ssb . The intuition for this Corollary is as follows. As the principal has no claim on surplus and the agents are symmetric, each agent claims one-half of the surplus. Specifically, agent 1 considers their net surplus of 21 v − c(e1 ), while agent 2’s net surplus is 12 v − c(e2 ). In a symmetric equilibrium e1 (s) = e2 (s). This gives each agent exactly the same incentive regarding their choice of s as the social planner, who wishes to maximize v − c(e1 ) − c(e2 ). In a more general case when the principal has a positive claim on surplus (α0 > 0), each symmetric agent gets less than half of the surplus generated. In 12

contrast, each agent bears half of the team’s cost. As a result, compared with the social planner agents are concerned more about costs and place relatively less weight on the generation of extra surplus. This ensures symmetric agents will opt for a (weakly) more familiar team than would be chosen by the social planner (ssb ≤ si ). The discussion and the propositions outlined in this section are highlighted in the following example. Example 1. Assume that the joint surplus is v(e1 , e2 , s) = 3s(1 − e1 − e2 ) + 21(e1 + e2 ) − 80(e31 + e32 ). Each agent’s effort cost is Ci (ei ) = 6ei for i = 1, 2. In addition, we assume that s ∈ [0, 1] and the shares for the three parties are α0 = α1 = α2 = 1/3. In the first-best case, the optimal efforts are r 5−s ∗ ∗ . (10) e1 = e2 = 80 In this example, Assumptions 1-3 are satisfied for optimal choices for all of the team structures we consider. Specifically, the crucial assumption that ∂v(e,s) = 21 − 3s − 240(ei )2 ≥ 0 holds for ei = e∗i . In the second-best, effort ∂ei levels ei for i = 1, 2 will be smaller than in the first-best case, hence they will also satisfy ∂v(e,s) ≥ 0. For values ei > e∗i that are not optimal choices for any ∂ei team structure, we assume that alternative specific functional forms apply to ensure that these assumptions are satisfied. Net surplus in the first-best case is W (e∗1 , e∗2 , s) = 3s(1 − e∗1 − e∗2 ) + 15(e∗1 + e∗2 ) − 80((e∗1 )3 + (e∗2 )3 ).

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∂W (e∗ (s),e∗ (s),s)

1 2 One can show that > 0, confirming Proposition 1; that is, s∗ = 1. ∂s In the second-best case, the optimal efforts are r 1−s sb sb . (12) e1 = e2 = 80

Net surplus in the second-best case is sb 3 sb 3 sb sb sb sb sb W (esb 1 , e2 , s) = 3s(1 − e1 − e2 ) + 15(e1 + e2 ) − 80((e1 ) + (e2 ) ).

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The share that the principal gets is 1 sb sb sb sb sb P (esb 1 , e2 , s) = W (e1 , e2 , s) + 2(e1 + e2 ); 3 13

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4 3P

W

3A 3

2.6 0

s

sp

ssb

si

1

Figure 2: Welfare functions W , 3P and 3A while each symmetric agent receives 1 sb sb sb sb Ai (esb 1 , e2 , s) = W (e1 , e2 , s) − 2ei . 3

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Figure 2 illustrates the welfare functions. The respective maxima are achieved when sp ≈ 0.57, ssb ≈ 0.75 and si ≈ 0.81. This result is consistent with Proposition 4. 

3.4

Application: binary choice between incumbents and newcomers

Thus far we have considered a model when the principal could make fine adjustments to the level of familiarity s within a pairing; such a possibility might arise if the principal can draw on a large pool of workers with varying levels of familiarity with each other. Now consider the binary case when the choice of team composition is between: newcomers who have not worked together before (s = 0); and incumbents who have a work history (s = 1).14 This setup allows us to derive several interesting results. First, consider the case when the principal opts to use incumbent workers. 14

Note that provided the relevant composite functions given in equations 1, 3 and 6 are quasi-convex in s, the binary scenario we examine here is without loss of generality; see Example 2.

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Corollary 2. If a principal chooses incumbents, this choice is (second-best) welfare maximizing. Proof: See Appendix A. The intuition for this is as follows. The principal is concerned about their share of gross surplus from the two feasible alternative team structures. If the principal anticipates a higher payoff with incumbents, it must be the case that incumbents also maximize net surplus because, not only is incumbent gross surplus higher than newcomer gross surplus, but effort costs are lower with incumbents (Proposition 2). However, it is possible that the principal will choose newcomers when incumbents are second-best. This arises because the non-investing principal considers her share of ex post (gross) surplus but does not consider the increase in the agents’ ex ante costs, as these costs are borne by the agents themselves. Second, consider the team choice of the agents. Corollary 3. If symmetric agents choose newcomers, this choice is (secondbest) welfare maximizing. Proof: See Appendix A. A symmetric agent, bearing all of his own costs while only receiving a fraction of the surplus he generates, has an incentive to shy away from a team structure that raises equilibrium effort costs. If the agents prefer to be paired with a newcomer in spite of this disadvantage, we know that a social planner (who has in mind all costs and benefits) will also opt for a newcomer team. On the other hand, symmetric agents could prefer incumbents even if two newcomers produce greater net surplus. An agent receives a fraction of the ex post surplus but bears all of their effort costs. This asymmetry skews their incentives to choose a team structure with lower costs in equilibrium (an incumbent team), relative to a social planner, who has an eye on all costs and benefits. The discussion and the corollaries outlined in this section are highlighted in the following example. Example 2. Assume that the joint surplus is v(e1 , e2 , s) = 60sγ(1 − e1 − e2 ) + 96(e1 + e2 ) − 80(e31 + e32 ). The costs for both agents are Ci (ei ) = 6ei for i = 1, 2. Also assume s ∈ [0, 1] and that α0 = α1 = α2 = 1/3. In addition we assume that γ = 1 – this will be relaxed later. In the first-best case, the optimal efforts are r 1.5 − s ∗ ∗ . (16) e1 = e2 = 4 15

In this example, Assumptions 1-3 are satisfied for optimal choices for all of the team structures we consider. Specifically, the crucial assumption that ∂v(e,s) = 96 − 60s − 240(ei )2 ≥ 0 holds for ei = e∗i . In the second-best, effort ∂ei levels ei for i = 1, 2 will be smaller than in the first-best case, hence they will ≥ 0. For values ei > e∗i that are not optimal choices for any also satisfy ∂v(e,s) ∂ei team structure, we assume that alternative specific functional forms apply to ensure that these assumptions are satisfied. Net surplus in the first-best case is W (e∗1 , e∗2 , s) = 60sγ(1 − e∗1 − e∗2 ) + 90(e∗1 + e∗2 ) − 80((e∗1 )3 + (e∗2 )3 ).

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∂W (e∗ (s),e∗ (s),s)

1 2 One can show that > 0, confirming Proposition 1; that is, s∗ = 1. ∂s In the second-best case, the optimal efforts are r 1.3 − s sb . (18) esb 1 = e2 = 4

Net surplus in the second-best case is sb sb sb sb sb sb 3 sb 3 W (esb 1 , e2 , s) = 60sγ(1 − e1 − e2 ) + 90(e1 + e2 ) − 80((e1 ) + (e2 ) ). (19)

From this, the share that the principal gets is 1 sb sb sb sb sb P (esb 1 , e2 , s) = W (e1 , e2 , s) + 2(e1 + e2 ); 3

(20)

while each agent receives 1 sb sb sb sb Ai (esb 1 , e2 , s) = W (e1 , e2 , s) − 2ei . 3

(21)

Figure 3 illustrates the welfare functions. The respective maxima are achieved when sp = ssb = si = 1. Now we allow γ to vary in [0, 1] (effectively truncating Figure 3). Then we have the binary scenario described in this section. We can find the following critical values γ a = 0.44, γ sb = 0.59 and γ p = 0.88. When γ < γ a , the agent will always choose si = 0, the lowest level of familiarity possible.15 On the other hand, when γ > γ a , the agent opts for si = 1, the maximum possible level of familiarity. The second-best choice is ssb = 0 when γ < γ sb and ssb = 1 otherwise. As in Corollary 3, if γ > γ sb symmetric agents choose the secondbest efficient team composition; on the other hand, if γ < γ sb symmetric agents 15

Here, for simplicity we do not consider the equality cases, for example when γ = γ a .

16

72

3P

W 60

3A

0

s

γsb

γa

γp

1

Figure 3: Welfare functions W , 3P and 3A may choose a team of incumbents, even though total welfare would be higher with newcomers. Now consider the principal’s choice. The principal will choose sp = 0 when γ < γ p and sp = 1 otherwise. This is consistent with Corollary 2; if the principal opts for a pair of incumbents this is also the second-best welfare maximizing choice (as γ p > γ sb ). Conversely, the principal’s choice may or may not be second-best optimal if she chooses a pair of newcomers. Note that in this example, the value functions given by equations 19, 20 and 21 are quasi-convex in s. This ensures that the choice of team familiarity reduces to a binary choice between either incumbents or newcomers. See Footnote 14 for a discussion of this issue.  As noted in Corollaries 2 and 3, if they can, symmetric agents tend to choose to work in a team made up of incumbents, whereas a principal can have too much incentive to choose newcomers. If it is possible to make ex ante lumpsum transfers and trade the rights to decide the team composition, the parties should be able to achieve the highest feasible level of surplus; this could involve either the principal or the agents choosing the team, whichever produces more surplus. That is, depending on the situation, either centralization (with a principal making the decision) or decentralization (with symmetric agents choosing their team) can be the way of ensuring the second-best outcome. However, there are times when it is not possible to exchange decision-making rights and compensate those involved. If decision authority cannot be trans17

ferred, a firm might institute rules to ensure the selection of the right team. For example, partnerships might adopt a rotation policy where newcomers are routinely included in teams. Conversely, there could be times when the underlying issue is that the principal has too much incentive to choose newcomers. In this case principal-formed teams might also need guarantees about agent autonomy or worker tenure so as to prevent too much principal-driven churn.

4

Discussion and concluding comments

Getting the right team matters a great deal; simply choosing teammates with the most potential is not always the best option. Rather, teams made up of intrinsically less productive agents sometimes perform better. This could be why work groups often include new members. It is said that a champion team will beat a team of champions. Our analysis suggests that even if a team of champions performs better for a given level of effort, a team of journeymen could outperform them precisely because the less lauded unit has a greater incentive to put in effort (both in training and in the game). There are several key elements driving our result. First, there is contractual incompleteness in the model in that parties cannot write contingent contracts on either surplus or effort ex ante. If parties have the ability to contract on effort, the first-best outcome can be achieved and incumbent workers will always be used together. However it is often difficult to contract on effort, be it in producing a musical, playing sport or in business. Second, we assume that the incumbents’ productivity advantage is decreasing in the effort or investment made; this means that the marginal return from effort – while always positive – is lower for incumbents than it is for newcomer workers. The combination of these effects can together mean that (second-best) surplus is higher with newcomers than with incumbents. What the model suggests is that a team’s true productivity needs to be considered in equilibrium – agents that end up producing more output might be the team members who do not necessary intrinsically work best together.16 While we explicitly only focus on team production, this idea has much in common with what social psychologists refer to as ‘social loafing’ or the ‘Ringelmann effect’ (see West 2004), a phenomenon observed in experiments by Ringelmann in the late 19th Century. Ringelmann found that people put in 75 per cent of the effort pulling a rope when they thought they were a member 16

To make things clear, this model examines this tradeoff in a relatively simple setting. In particular, we consider a static model and abstract from the choice of team composition over time. We leave this issue for further research.

18

of a team of seven (they were blindfolded) compared with their effort as an individual undertaking the same activity (see Kravitz and Martin 1986).17 This literature also makes a second observation: the types of individuals in the group make a difference to observed effort. For instance, Stroebe et al. (1996) examine teams working on complex problems. While team performance exceeded that of individuals, performance was further improved when highability team members thought they were matched with a low-ability partner.18 While in a different context, there are similarities to the predictions of our model, namely that: (i) agents underinvest when they work as part of a team; and (ii) an agent’s effort depends on the makeup of the team. It is worth noting that our focus has been on the choice between incumbents and newcomers, both of whom enjoy a positive return on effort. However, the same point can be made for other types of agents; it is the relative difference in the marginal product of effort that helps determine team structure. For example, a principal may have to choose a team with complementary agents (producing more together, for the same effort level, than working separately) or agents with no team complementarity at all (independent workers). Similarly, the choice could be between independent workers and those who are substitutes at the margin (see Bel 2013). For both of these two new alternatives, provided that the marginal return of effort is lower for complementary agents than for independent agents (in the first example), or for the independents rather than substitutes workers (in the second case), the relative differences between the two options are qualitatively the same as the incumbent/newcomer tradeoff examined in our model. For instance, it could be that a firm or principal chooses to hire substitute workers even though they intrinsically produce less output for a given level of effort, provided this choice induces greater effort.19 While not claiming our key insights apply universally, there are many other applications of the model. As noted, if incumbency reduces the incentive to invest, it could be that a policy of job rotation increases total surplus (in con17

Other experiments have found similar effects: people shouted in a team with only 74 per cent of the effort as they did individually (Ingham et al. 1974); when solving mathematical problems individuals took on average five minutes, groups of 2 individuals took an average of 3-and-a-half minutes per-person and groups of 4 averaged 12 minutes per-person (Shaw 1932). 18 Social psychologists have invoked various explanations for this phenomenon, including: individuals feeling that they would be embarrassed if it were revealed they put in more effort than others; coordination failures; loud talkers drowning out others; and individuals reducing their effort if they feel it will not be adequately recognized (West 2004). 19 One example we have in mind is a firm hiring a difficult or obnoxious worker – their presence might naturally hinder output relative to a more congenial colleague, but this could be made up for if their presence on the team creates an additional incentive to work harder.

19

sulting projects, musical productions or in research projects). When deciding the composition of a team, an organization might wish to choose workers that do not otherwise have the highest possible potential. Note that our result differs from the usual moral-hazard-in-teams underinvestment outcome with a balanced budget. In Holmstrom (1982) for example, agents in a team underinvest because they do not receive the full marginal return from their effort. In our model, we also have underinvestment, however, we have the additional consideration that the level of underinvestment depends on team composition. This framework also has implications for the allocation of decision-making rights. If the principal’s effort costs are relatively small, and newcomers invest more than incumbents in equilibrium: a principal can have a tendency to choose newcomers too often. On the other hand, symmetric agents prefer familiar teammates, even if a team with more newcomers produces greater net surplus. This suggests that when inducing greater effort is more important, the choice of team should be made centrally by the principal. However, when taking advantage of intrinsic productive capability is more important, it might be advantageous for the principal to commit not to get involved, having the decision decentralized to the agents themselves. Our model could also apply to the joint use or co-location of assets. Using some specific assets together rather than separately might generate additional surplus, but this could also change parties’ incentives to engage in ex ante investment. The standard predictions in the property-rights models (Hart and Moore 1990 and Hart 1995 for example) suggest that complementary assets should be owned together, so as to provide the best possible incentives for ex ante investment. Our model generates results in the same vein as Bel (2013), who finds that assets with a positive synergy need not be owned together when they are substitutes at the margin. Unlike Bel (2013), however, we do not assume that the investments by incumbents are substitutes at the margin. In our model all agents – familiar or not – enjoy a positive return that is non-decreasing with effort. Rather, it is the relative difference in the marginal returns that drives our result.

20

Appendix A Proof of Proposition 2 Applying the Implicit Function Theorem to the system of equations in 4 yields:  2  2 ∂ 2 v(e,s) ∂ 2 v(e,s) ∂ 2 v(e,s) 1 ∂ Cj (ej ) − + ∂∂ev(e,s) 2 2 ∂ei ∂s αj ∂ej ∂ej ∂ei j ∂s ∂e1 ∂e2 = (22) 2    2 ∂s ∂ 2 v(e,s) ∂ 2 v(e,s) ∂ v(e,s) 1 ∂ 2 C1 (e1 ) 1 ∂ 2 C2 (e2 ) − ∂e2 − ∂e2 − ∂e1 ∂e2 α1 α2 ∂e2 ∂e2 1

1

2

2

for j 6= i. Assumptions 1-3 together guarantee that

∂ei ∂s

≤ 0. 

Proof of Proposition 3 Applying a similar concept to a revealed-preference argument, let us first prove by contradiction that sp ≤ si . Assume to the contrary that sp > si . Use equations 3 and 6 to derive αi Ai (esb (s), s) = P (esb (s), s) − Ci (esb (23) i (s)), α0 where esb (s) is implicitly defined by equation 4. Given that sp is the principal’s choice, it follows that P (esb (sp ), sp ) ≥ P (esb (si ), si ). On the other hand, from Proposition 2 and the assumption made earlier that sp > si , it follows that sb p p sb i i p sb i Ci (esb i (s )) < Ci (ei (s )). Consequently, Ai (e (s ), s ) > Ai (e (s ), s ), which p means the agent’s choice is s . This is a contradiction. To prove that sp ≤ ssb we use similar logic as in the argument above. Notice that 1 sb (24) W (esb (s), s) = P (esb (s), s) − C1 (esb 1 (s)) − C2 (e2 (s)). α0 To the contrary, assume that sp > ssb . Given sp is the principal’s choice, it follows that P (esb (sp ), sp ) ≥ P (esb (ssb ), ssb ). On the other hand, from Proposition p sb sb 2 and the assumption that sp > ssb , it follows Ci (esb i (s )) < Ci (ei (s )). Consequently, W (esb (sp ), sp ) > W (esb (ssb ), ssb ), which contradicts the supposed optimality of ssb .  Proof of Proposition 4 To prove that si ≥ ssb we use a similar argument to the one used in the proof of Proposition 3. Notice in the symmetric case that W (esb (s), s) =

1 1 − 2α1 A1 (esb (s), s) + C1 (esb 1 (s)). α1 α1 21

(25)

Assume to the contrary that si < ssb . Given si is the agent’s choice, it follows that A1 (esb (si ), si ) ≥ A1 (esb (ssb ), ssb ). On the other hand, from Proposition 2 and the assumption made earlier that si < ssb , it must be the i sb sb case that Ci (esb i (s )) > Ci (ei (s )). In addition, from Assumption 4 α1 < 1/2. Consequently, W (esb (si ), si ) > W (esb (ssb ), ssb ), contradicting the supposed optimality of ssb . This finding concludes the proof.  Proof of Corollary 2 To prove that if sp = 1 it must also be the case that ssb = 1, we use a similar argument to that used in the proof of Proposition 3. Assume to the contrary that sp = 1 and ssb = 0. This means that P (esb (1), 1) ≥ P (esb (0), 0) and W (esb (1), 1) ≤ W (esb (0), 0). On the other hand, from Proposition 2 it must sb be the case that Ci (esb i (1)) < Ci (ei (0)). Consequently, given relationship 24 we have a contradiction.  Proof of Corollary 3 To prove that if si = 0 it is necessarily the case that ssb = 0, we again use a proof by contradiction. Assume that si = 0 and ssb = 1. If this holds it follows that A1 (esb (1), 1) ≤ A1 (esb (0), 0) and W (esb (1), 1) ≥ W (esb (0), 0). sb On the other hand, from Proposition 2 we know that Ci (esb i (1)) < Ci (ei (0)). Consequently, given relationship 25 we have a contradiction. 

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