OSIPP Discussion Paper: DP-2005-E-002

Incentives in Academics: Collaboration under Weak
Complementarities

March 31, 2005
Junichiro Ishida
Associate Professor, Osaka School of International Public Policy (OSIPP)【キーワード】Collaboration; Complementarities; Incentives; Contracts; Peer sanction.
【要約】This paper considers a contracting relationship with multiple agents in a
repeated setting under voluntary team formation. In each period, an opportunity tocollaborate arrives stochastically but whether this opportunity has arrived is the agents'
private information. The principal thus cannot simply tell them when to collaborate;
she must instead guide them through incentives. In this situation, there may arise an
incentive for the principal to encourage collaboration in the repeated setting as it allows
her to exploit the agents' ability to monitor each other. We then argue that low-powered
incentives offered for non-collaborative works, as often observed in our profession, canbe seen as an important part of optimal incentive schemes in an attempt toendogenously raise the cost of shirking.

We also show that this type of compensation

scheme is socially inefficient and detrimental to the agents when the discount factor isnear unity.

Financial support from the Japanese Ministry of Education, Culture, Sports, Science andTechnology (Grants-in-Aid for Young Scientists (B), 14730013, 2002) is gratefully
acknowledged.

Correspondence to: Junichiro Ishida, OSIPP, Osaka University, 1-31

Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: jishida@osipp.osaka-u.ac.jp

1

Introduction

Collaboration is a crucial aspect of academic research in many fields. This tendency is certainly moreprevalent in natural sciences, where a bulk of research projects are conducted by multiple researches.
The situation is, however, not far different in social sciences and especially in economics. A recent
tendency is that the number of collaborative works in economics increases rapidly and now occupies
a substantial portion of all published papers.1
There are presumably many reasons why researchers collaborate. From their viewpoint, though,it ultimately comes down to how the amount of output is related to pecuniary (and sometimes nonpecuniary) compensations such as job offers, tenures, grants, promotions and raises in base salary. To
illustrate this point, consider the world of no complementarity where writing one single-authored paper
requires exactly the same amount of energy as writing two coauthored papers (when collaborated by
two researchers). In this world, a ‘fair’ way to compensate is to count one single-authored paper
as equivalent to two coauthored papers. If two coauthored papers do count more heavily for somereasons, self-interested researchers actually have an incentive to collaborate, just for their own sake.
Although it is difficult to verify this claim empirically, we take it rather uncontroversial that there
is a tendency to reward lengthy curriculum vitae in academics (or at least in economics), implyingthat two coauthored papers indeed do count more heavily than one single-authored paper of identicalquality.2 Alternatively, this implies that researchers are given relatively low-powered incentives for
non-collaborative works conducted individually.
Given this, the question we need to ask at this juncture is why the compensation scheme in our
profession is designed in such a way that seems to encourage collaboration. An obvious reason forthis is perhaps the presence of complementarities: the contract designer (the principal) encourages
collaboration to reap the benefit of complementarities. Researchers can most likely benefit from
1 See, for instance, Heck and Zaleski (1991), Hudson (1996) and Engers (1999) et al. Also, see Laband and Tollison(2000) for the difference in the trend of collaboration between economics and biology.
2 Due to the nature of the issue, empirical evidence on this is relatively sparse and hardly conclusive: for instance,
Sauer (1988) fails to reject that n coauthored papers (with n authors) counts more heavily than a single-authored paper
in determining salaries of academic ecnomists, using cross-section data. It is documented in survey data, however, that
universities do not typically fully discount collaborative work by the number of authors (McDowell and Smith, 1992).

1

complementarities when a task requires various specialized skills. In this case, collaboration is valuable
as it realizes the efficient division of labor by allowing each participant to focus on narrowly definedtasks. A benefit of collaboration may also arise even between researchers with a similar set of skills
because collaboration may lead to an idea none could have come up with individually.3 This effectis more important when two or more specialists in the same fields work jointly, as we often observein economics. When the effect of complementarities is present, there arise potential benefits for thecontract designer to encourage collaboration since it leads to more output, given the same amount ofeffort.
While this reasoning sounds fairly convincing, collaboration also poses a cumbersome problem
when each researcher’s effort is not directly observable, as is normally the case. Under collaboration,observable signals of effort are more likely to become less sensitive to each researcher’s effort now
that they are also dependent on the collaborator’s effort as well. The cost of inducing effort fromeach researcher naturally increases since each can free-ride on the collaborator’s effort. To justify thecompensation scheme which seems to encourage collaboration, the degree of complementarities in the
production process needs to be strong enough to compensate for the loss arising from the free-rider
problem: in fact, under plausible circumstances, the degree of complementarities needed to outweighthis free-rider problem is fairly substantial. We argue that this is doubtful, especially in economics
where skill complementarities do not appear that critical.4 Indeed, empirical evidence on the effect ofcollaboration on academic output in economics is mixed at best.5
Provided that the presence of complementarities alone does not necessarily justify the current
compensation scheme, we focus on another potential source of benefit, that is, each researcher’s
desire to maintain reputation in the academic community. The critical aspect of the model is thatcollaboration allows researchers to work closely with each other and thus to observe what the contract
3 This effect is often referred to as synergies to distinguish them from skill complementarities. In the paper, however,
we do not make any particular distinction between them.
4 This may be in sharp contrast to natural sciences where the division of labor is clearer, compared to social sciences.
5 McDowell and Smith (1992) finds no significant evidence that collaboration enhances productivity while Durden
and Perri (1995) finds evidence that collaboration indeed has a positive effect. Recently, using panel data, Hollis (2001)shows that the effect of collaboration on output is even negative after discounting for the number of authors.

2

designer cannot observe, namely each researcher’s effort level. Since no one would like to collaborate
with lazy and dishonest colleagues, each researcher has an additional motivation, absent when working
individually, to present himself as hard-working and honest. This motivation becomes stronger whencollaboration is so much more attractive for researchers: in this case, it can simply be too costly toshirk and destroy a cooperative relationship with able colleagues. The contract designer can thenexploit this incentive to maintain reputation by offering low-powered incentives for non-collaborative
work in order to endogenously raise the cost of shirking.
To formalize this situation we consider a contracting situation in an infinitely repeated setting
under voluntary team formation. There is now a growing body of literature on so-called relational
contracts, which is an attempt to explicitly incorporate repeated interactions into the contract theory.6
The idea that the agents’ ability to monitor each other can be exploited by appropriately designingcompensation schemes is first pointed out and formulated by Che and Yoo (2001).7 This paper builds
on this framework and extends it to a situation under voluntary team formation. The situation wewill analyze in this paper is as follows. Consider a situation where a principal hires two agents andlets them perform a task either individually or jointly. In each period, an opportunity to collaborate
arrives stochastically, but whether this opportunity has arrived is the agents’ private information. Not
knowing when collaboration is feasible, the principal cannot simply tell them when to collaborate; shemust instead provide incentives to collaborate when she wants them to by appropriately designing thecompensation scheme.
In this framework there arises a link between incentives for collaborative works and those fornon-collaborative works. Consider a trigger strategy where an agent’s defection leads to perpetual
retaliation by the other agent. The cost of defection is certainly small if each agent can earn sufficiently
large rents from non-collaborative works because he can always work individually. The principal thenhas an incentive to offer low-powered incentives for non-collaborative works so that the agents cannot
earn sufficient rents from it. We identify conditions under which it is indeed optimal to offer this6 To name a few, Bull (1987), MacLeod and Malcomson (1989), Baker, Gibbons and Murphy (1994, 2002), Bernheim
and Whinston (1998), MacLeod (2003) and Levin (2003) explore the optimal form of contract in long-term relationships.
7 Also, see Rayo (2003) and Ishida (2004) on team incentives in long-term relationships.

3

type of contract under arbitrarily weak complementarities. We also show that this strategy is sociallyinefficient and especially detrimental to the agents when the discount factor is near unity. This implies
that our desire to maintain reputation in the academic community may backfire and actually hurt us
when the compensation scheme is designed to take advantage of it.
The rest of the paper is organized as follows. The next section outlines the basic environment.
Section 3 analyzes the static version of the model as a benchmark. Section 4 extends the analysis to
the repeated setting and show how the optimal compensation scheme from that in the static setting
and its effect on social welfare. Finally, section 5 offers some concluding remarks.

2

Model

2.1

Environment

Consider a model with a principal (female) and two agents (male, i = A, B) in an infinitely repeatedsetting. Time is discrete and denoted by t ∈ T ≡ {0, 1, 2, . . .}. All parties are assumed to be risk
neutral. The common discount factor is given by δ ∈ (0, 1).
In each period, the principle lets A and B engage in a task. Each task is characterized by the
number of agents to work on the task, which we refer to as the mode of production. The modeof production is denoted by superscript j ∈ {c, n} where j = c if the agents work jointly (c forcollaboration) and j = n if they work individually (n for non-collaboration).
An opportunity to collaborate arrives stochastically. With probability θ ∈ (0, 1), the opportunity
arrives and the agents must decide whether to take this opportunity and work jointly.8 The agentscan work jointly only if both of them agree to do so. The agents work individually if either (i) the
opportunity does not arrive or (ii) they fail to reach an agreement. Whether this opportunity has
arrived is the agents’ private information and cannot be observed by the principal.
8 There are many potential interpretations for the parameter θ. One is to regard this as the probability of finding asuitable collaborator. It is also possible to interpret this as the probability that the agents successfully come up with asuitable research idea on which both of them can agree to collaborate.

4

2.2

Production

In each period, each agent must exert costly effort to produce output. The effort choice is binary anddenoted by ei,t ∈ {0, 1}. The cost of effort is given by dei,t for non-collaborative work and γdei,t ,γ ∈ (0, 1), for collaborative work. The parameter γ measures the degree of complementarities in the
production process, and its effect disappears as γ → 1. The effort choice is not observable to theprincipal.
The task is either successful or not, and its outcome is denoted by yi,t ∈ {0, 1}. For practical
purposes, yi,t can be interpreted as the number of publications per author in a given period.9 Whenthe agents work individually, the probability of a success is given byprob(yi,t = 1 | ei,t ) = p1 ei,t + p0 (1 − ei,t ), p1 > p0 > 0.
The output of agent i thus depends solely on his own effort level. Define ∆p ≡ p1 − p0 .
When the agents work jointly, on the other hand, the probability of a success is given byprob(yi,t = y−i,t = 1 | ei,t , e−i,t ) ≡ f (ei,t , e−i,t ), i 6= −i.
For the analysis we consider a simple additive technology to emphasize that the only source of complementarities is the difference in the cost of effort: f (1, 1) = p1 , f (0, 0) = p0 and f (1, 0) = f (0, 1) =0.5(p0 + p1 ). Under this specification, an agent’s marginal value of effort is independent of the otheragent’s effort level: that is,f (1, 1) − f (0, 1) = f (1, 0) − f (0, 0) = 0.5∆p.

(1)

In each period, the principal earns a private benefit V (yA,t + yB,t ), depending on the outcome. Noticethat the principal simply values the number of publications per author so that there is no inherent
need to reward collaboration more heavily in this sense.
9 That is, yi,t = 1 means either one single-authored publication (if agent i works individually) or two coauthoredpublications (if he collaborates).

5

2.3

Contract

The principal makes a take-it-or-leave-it offer to the agents in order to maximize the expected profit.10
The reservation payoff for both parties is assumed to be zero. Throughout the analysis, we focus on acase where the contract offered initially applies for all subsequent periods. The principal can observe
the mode of production and the outcome of the task. There are thus four possible cases, and thecontract offered by the principal specifies the wage for each contingency. Let wj = (w0j , w1j ), j = c, n,denote a contract where wyj ∈ [0, ∞) is the wage when the mode of production is j ∈ {c, n} and theoutcome is y ∈ {0, 1}: for expositional purposes we sometimes refer to wn as the individual contract
and to wc as the team contract. Also, let w = (w0n , w1n , w0c , w1c ). Finally, there is a liquidity constraint
such that the principal must offer nonnegative wages.
Before we proceed further, we need to make an assumption concerning the possibility of ‘fake collaboration.’ Since the principal is unable to observe when collaboration is feasible, there is a possibility
that the agents claim to have engaged in collaboration even though they in fact worked individually.
This possibility may severely limit the effectiveness of a contract which rewards collaboration moreheavily. While it is conceptually possible, however, this behavior is a clear violation of a professional
code and we hence rule out this possibility at the outset. There are indeed several reasons why theagents do not falsely pretend to collaborate. First, if there is some (possibly very small) probability
that fake collaboration is detected by the principal, and the potential penalty for such a violation(in a broad sense) is very large, it is not beneficial to falsely pretend to collaborate. Second, there
may arise some psychological cost (or guilt) associated with being credited in a work that s/he has
nothing to do with. In any event, we do not believe that this (fake collaboration) is a serious problem
in the academic community, and we thus take the stance that there must be some mechanisms which
prevent researchers from falsely pretending to collaborate.
10 In other words, we assume that the principal possesses full bargaining power.

6

3

Static benchmark

We first consider a static version of the model as a benchmark. When the agents work individually,each agent exerts effort if
p1 w1n + (1 − p1 )w0n − d ≥ p0 w1n + (1 − p0 )w0n .

(2)

The incentive compatibility constraint for non-collaborative work is then given by∆p(w1n − w0n ) ≥ d.

(3)

For expositional purposes, we say that a contract is high-powered if it induces effort from each agent;on the other hand, we say that a contract is low-powered if it does not.
Now suppose that the agents choose to engage in collaboration. Taking the other agent’s effort
level e−i,t as given, each agent exerts effort if
f (1, e−i,t )w1c + (1 − f (1, e−i,t ))w0c − γd ≥ f (0, e−i,t )w1c + (1 − f (0, e−i,t ))w0c .

(4)

The incentive compatibility constraint for collaborative work is then given by(f (1, e−i,t ) − f (0, e−i,t ))(w1c − w0c ) = 0.5∆p(w1c − w1c ) ≥ γd.

(5)Given these, each agent must decide whether to collaborate with the other agent when the opportunity arrives. Let uj denote the expected stage-game payoff for each agent, conditional on the modeof production. Upon the arrival of the opportunity, the agents agree to collaborate if uc ≥ un .
It is exceedingly tedious to completely characterize the optimal form of contract for a full range ofparameters without yielding much insight. We thus confine our attention to a subset of parametersby the following assumptions.
Assumption 1 ∆pV ≥d.
∆pAssumption 2 γ > 0.5.

7Assumption 1 indicates that effort is sufficiently valuable for the principal (d is sufficiently small
relative to V ) so that the optimal contract is always high-powered in the static setting. The assumption
also implies that it is efficient to exert effort. Assumption 2 is what we refer to as the condition of
weak complementarities, which limits the lower bound for γ. Under those assumptions we can state
the following result (the role of the assumptions will be made clearer in the proof of proposition 1).
Proposition 1 Under Assumptions 1 and 2, the optimal static contract is given byw = (0,d, 0, 0).
∆p

Proof: See Appendix.

It can be seen from above that it is more costly for the principal to induce effort when theagents work jointly, because an agent can free-ride on the other agent’s effort. In other words, theprincipal must offer more to induce any given level of effort because each agent’s payoff is now lesssensitive to his behavior when the agents work jointly. The proposition indicates that the principal
never encourages collaboration in the static setting unless the production process exhibits sufficiently
strong complementarities. The magnitude of complementarities may need to be quite substantial tocompensate for the loss arising from the free-riding problem. In the present setup, collaboration ismore profitable only when γ < 0.5.11

4

Collaboration through repeated interactions

4.1

Analysis

We now extend the analysis to a repeated setting and examine how the principal should modify thecompensation scheme to exploit the possibility of mutual monitoring between the agents. In therepeated setting, the agents work closely with each other when they choose to work jointly. As a
11 In the present setup, γ = 0.5 implies that writing four coauthored papers is equivalent to writing one single-authoredpaper in terms of the amount of effort needed.

8consequence, there arises a possibility that the agents sustain mutual cooperation by adopting a typeof trigger strategy.
Suppose that collaboration allows each agent i to observe the other agent’s effort level e−i,t . Leta history in period t be a sequence of observable effort choices (i.e., the other agent’s effort level
when they collaborate) up to period t − 1. Then, a strategy profile is a sequence of functions which
map from any possible history in each period into a probability distribution over effort choices inthat period.12 Within this framework, the equilibrium concept we adopt is similar to Che and Yoo(2001): given some contract, the agents select an equilibrium which yields the highest total payoff
among all possible subgame perfect equilibria. Now define two states of the economy, S ∈ {C, N } andconsider the following trigger strategy. The economy is initially in state C until a defection occurs:in state C, the agents always choose to exert effort whenever they work jointly (if the team contract
is high-powered). Once a defection occurs, the economy moves into state N where the agents simply
play the stage-game Nash equilibrium ever after. Taking this into account, each agent decides whether
to work jointly when the opportunity arrives.
First, even in the repeated setting, the incentive compatibility constraint for non-collaborative
works is unchanged because his choice of effort has no future implication under the trigger strategy
described above. This means that an agent exerts effort if ∆p(w1n − w0n ) ≥ d.
The incentive compatibility constraint for collaborative works needs to be modified, on the other
hand, because the principal may exploit the agents’ ability of mutual monitoring sustained throughrepeated interactions. Given this set of strategies, the expected payoff for each agent depends also on
the state of the economy. Let ucS denote the expected payoff for each agent in state S when the agentschoose to work jointly. Further, let US , S = C, N , denote the expected discounted sum of payoffs instate S:US =1
(θ max{un , ucS } + (1 − θ)un ).
1−δ

12 We thus do not consider strategies which depend on the history of outcomes y

9

i,t .

(6)

The incentive compatibility constraint for collaborative works in the repeated setting is then given byf (1, e−i,t )w1c + (1 − f (1, e−i,t ))w0c − γd + δUC ≥ f (0, e−i,t )w1c + (1 − f (0, e−i,t ))w0c + δUN .

(7)

With some algebra, this can be simplified to
0.5∆p(w1c − w0c ) + δ(UC − UN ) ≥ γd,

(8)

where
UC − UN =θ
(max{un , ucC } − max{un , ucN }).
1−δ

(9)

The incentive compatibility constraint for collaborative works is now weaker than in the static setting
as long as UC > UN . Note that the incentive compatibility constraint for collaborative works now
includes not only wc but also wn .
Given this, the agents must decide whether to collaborate when the opportunity arrives. The
condition for this is virtually the same as in the static setting, except that it is now dependent on thestate of the economy: in state S, the agents choose to work jointly if
ucS − un ≥ 0.

(10)

Finally, when collaboration is preferred by the agents and the team contract is high-powered, thecontract must be collusion-proof to implement the desirable outcome as a team equilibrium:
p1 w1c + (1 − p1 )w0c − γd ≥ p0 w1c + (1 − p0 )w0c ,(11)

p1 w1c + (1 − p1 )w0c − γd ≥ 0.5(p0 + p1 )w1c + (1 − 0.5(p0 + p1 ))w0c − 0.5γd.

(12)These constraints assure that the agents as a team cannot be made better off by deviating from theequilibrium. As it turns out, these constraints are identical and can be simplified to
w1c − w0c ≥

γd
.
∆p

(13)

We now investigate whether there are circumstances under which it is optimal for the principal tooffer a contract which encourages collaboration. There are generically three distinct types of contract10

that the principal can offer in this situation: (i) only the individual contract is high-powered; (ii) bothof the contracts are high-powered; (iii) only the team contract is high-powered.13 In what follows we
explore each case in turn.
Type 1: Only the individual contract is high-powered
Consider a case where the principal offers a high-powered individual contract combined with alow-powered team contract. This is the case if∆p(w1n − w0n ) ≥ dand γd > 0.5∆p(w1c − w0c ) + δ(UC − UN ).

We denote the expected profit that the principal earns from each agent in this case by πhl . If un > ucC ,the expected profit per agent is given by
πhl = p1 (V − w1n ) − (1 − p1 )w0n .

(14)Note that if the team contract is low-powered, no peer sanction mechanism can function because
ucC = ucN = p0 w1c + (1 − p0 )w0c . This implies that UC = UN and the problem is simply reduced to its
static version. The optimal contract in this case is exactly as described in proposition 1. The next
result directly follows from proposition 1 (the proof abbreviated).
Lemma 1 Consider a class of contracts in which only the individual contract is high-powered. Under
Assumptions 1 and 2, the optimal contract within this class is given byw = (0,d, 0, 0).
∆p

Type 2: Both of the contracts are high-powered
We now consider a case where both of the contracts are high-powered, i.e.,∆p(w1n − w0n ) ≥ dand0.5∆p(w1c − w0c ) + δ(UC − UN ) ≥ γd.

We denote the expected profit in this case by πhh . If ucC ≥ un , the expected profit per agent is givenbyπhh= θ(p1 (V − w1c ) − (1 − p1 )w0c ) + (1 − θ)(p1 (V − w1n ) − (1 − p1 )w0n )13 Under the maintained assumptions we can rule out the possibility that both of the contracts are low-powered.

11

= p1 (V − θw1c − (1 − θ)w1n ) − (1 − p1 )(θw0c + (1 − θ)w0n ).

(15)

Lemma 2 Consider a class of contracts in which both of the contracts are high-powered. Under
Assumptions 1 and 2, (i) if 2δθ(1 − γ) > (1 − δ)(2γ − 1), the optimal contract within this class is givenbyw = (0,dd, 0, α(γ, δ, θ)
),
∆p
∆p

where
α(γ, δ, θ) ≡2γ(1 − δ + δθ)∆p + 2δθp0;(1 − δ)∆p + 2δθp1(ii) if not, the optimal contract is given byw = (0,dd, 0, β(γ, δ, θ)
),
∆p
∆p

where
β(γ, δ, θ) ≡2γ(1 − δ + δθ).
1 − δ + 2δθ

Proof: See Appendix.

If 2δθ(1 − γ) ≤ (1 − δ)(2γ − 1), β(γ, δ, θ) ≥ 1 so the type 2 contract is always dominated by thetype 1 contract. This indicates that 2δθ(1 − γ) > (1 − δ)(2γ − 1) is the necessary condition for thetype 2 contract to be optimal. Note that this necessary condition is satisfied when δ is sufficiently
close to unity.

Type 3: Only the team contract is high-powered
Finally, we consider a case where only the team contract is high-powered, i.e.,
0.5∆p(w1c − w0c ) + δ(UC − UN ) ≥ γd

and d > ∆p(w1n − w0n ).

As above, we denote the expected profit per agent in this case by πlh whereπlh = θ(p1 (V − w1c ) − (1 − p1 )w0c ) + (1 − θ)(p0 (V − w1n ) − (1 − p0 )w0n ).
12

(16)

Lemma 3 Consider a class of contracts in which only the team contract is high-powered. Under
Assumptions 1 and 2, the optimal contract within this class is given byw = (0, 0, 0, β(γ, δ, θ)

d
).
∆p

Proof: See Appendix.

4.2

The optimal form of contract in the repeated settingWith those results we are now ready to determine the optimal form of contract in the repeatedsetting. Under the maintained assumptions, in the static setting, the optimal individual contract ishigh-powered while the optimal team contract is low-powered. We now show that this conclusion does
not necessarily hold in the repeated setting. To illustrate this point, we confine our attention to alimiting case where δ → 1. In this case we can show the following.
Proposition 2 (Collaboration under Weak Complementarities) In the limiting case where δ → 1, theoptimal team contract is necessarily high-powered.
Proof: See Appendix.

Collaboration is more profitable for the principal if and only if w1c < w1n . As δ → 1, this condition
becomeslim α(γ, δ, θ) =δ→1

γp1 + (1 − γ)p0
< 1,
p1

lim β(γ, δ, θ) = γ < 1.

δ→1

(17)
(18)

This indicates that in the limiting case, the degree of complementarities needed to make collaboration
more profitable can be arbitrarily small: that is, collaboration can be more profitable under arbitrarily
weak complementarities. Conversely, though, it is also true that the presence of complementarities isnecessary to make collaboration profitable while its magnitude can be arbitrarily small.

13

Given this result, the principal is now left with two alternatives, either type 2 or type 3. Equivalently, the question comes down to whether the optimal individual contract should be high-powered.
While the optimal individual contract is high-powered in the static setting by design, we show that
this conclusion may be overturned in the repeated setting.
Proposition 3 In the limiting case, there exists some θ̄ such that the optimal individual contract is
low-powered for all θ ∈ [θ̄, 1)
Proof: See Appendix.

The proposition indicates that there exists a situation where the optimal individual contract is
low-powered as long as: (i) the agents are sufficiently patient so that they have vested interests inmaintaining their reputation; and (ii) the opportunity to collaborate arrives sufficiently frequently.
The result is reasonably intuitive. First, the peer sanction mechanism is more effective when the agentscare more about the future. An extreme case is the static setting (δ = 0) where there is no room forthe effect of the peer sanction mechanism to take place. Second, the frequency of the opportunity
to collaborate also plays an important role on two accounts. On one hand, the more frequently
the opportunity arrives, the more the agents lose by defecting and thus destroying a cooperative
relationship with the collaborator. The peer sanction mechanism is hence more effective when θ issufficiently large. On the other hand, the cost of offering low-powered incentives for non-collaborative
work monotonically decreases as the agents can collaborate more frequently. One can then certainly
argue that an expansion in the size of the profession facilitates collaboration (an increase in θ), which
in turn accelerates the tendency to reward lengthy curriculum vitae.14

4.3

WelfareWe now examine the effect of infinitely repeated interactions on social welfare. In general, one cannot
assert whether the optimal dynamic contract promotes efficiency compared to its static counterpart.
14 As for the increase in the number of collaborative works, Hudson (1996) raises three potential reasons for the recent
increase in the number of collaborative works. One of them is the recent growth of the size of the profession.

14

In either case, an inefficiency may possibly arise but for different reasons.
Under the maintained payoff structure, it is socially efficient to (i) induce effort in all periods, and(ii) engage in collaboration whenever possible (because of complementarities γ < 1). First, consider
the optimal static contract. Under the optimal static contract, the agents always exert effort, and it issocially efficient in this sense. Under the maintained assumptions, however, the optimal static contract
does not encourage collaboration, which entails efficiency losses. The optimal static contract fails toachieve the socially efficient outcome because of the free-rider problem. Since the observable signal is
now less sensitive to each agent’s behavior, an agent can free-ride on the other agent’s effort, and thecost of inducing effort necessarily increases for collaborative work. Under plausible circumstances, thisadditional cost of inducing effort outweighs the benefit from collaboration: in this case, the principal
chooses to let the agents work individually even when it is socially efficient to let them work jointly.
Now consider the optimal dynamic contract. In the repeated setting, the socially efficient outcome
is achieved when type (ii) is optimal: in this case, repeated interactions allow the principal to overcome
the free-rider problem. When both δ and θ are sufficiently close to unity, however, only the teamcontract is high-powered. Apparently, this type of contract fails to implement the first-best effort
when the opportunity to collaborate does not arrive. This inefficiency (the failure to induce effort for
non-collaborative works) stems from the principal’s incentive to make the peer sanction mechanism
more effective, contrary to the case of the optimal static contract where the inefficiency arises fromthe free-rider problem. This indicates that the current compensation scheme in our profession, which
seems to reward collaboration more heavily, may be socially inefficient because it fails to provide
proper incentives to induce the first-best effort for non-collaborative works. The following conclusion
is immediately obtained from proposition 3.
Proposition 4 In the limiting case, the optimal contract in the repeated setting is socially efficient
if and only if θ ∈ (0, θ̄).
Note also that the effect of infinitely repeated interactions on the agents’ welfare is ambiguous. To
see this, define Uhl , Uhh and Ulh as the agent’s expected payoff under the type 1, 2 and 3 contract15

respectively. First, when only the individual contract is high-powered, we obtain
Uhl = p1 w1n − d =

p0 d
.
∆p

(19)

This is the agent’s expected payoff when δ is sufficiently small, and hence the expected payoff in thestatic setting. As δ approaches unity, either the type 2 or 3 contract emerges as the optimal contract.
When both of the contracts are high-powered, the expected payoff isUhh= θ(p1 w1c − γd) + (1 − θ)(p1 w1n − d)
Ã
!
d
p0 d
= θ p1 α(γ, δ, θ)− γd + (1 − θ)
.
∆p
∆p

(20)

if 2δθ(1 − γ) ≤ (1 − δ)(2γ − 1). We can show that the agents are made better off under the type 2
contract than under the type 1 contract, i.e., Uhh ≥ Uhl . This condition can be written as
p1 α(γ, δ, θ)p0 dd− γd ≥
,
∆p
∆p

(21)

α(γ, δ, θ) ≥γp1 + (1 − γ)p0
.
p1

(22)

which is further simplified to

It is straightforward to verify that this condition holds when γ > 0.5 as assumed. The expected payoff
is monotonically decreasing in δ and converges to that in the static setting Uhl as δ → 1.
Finally, when only the team contract is high-powered, the expected payoff in the limiting case isUlh= θ(p1 w1c − γd)Ã!
d= θ p1 β(γ, δ, θ)− γd .
∆p(23)

In this case, the agents may or may not be made better off. When δ is sufficiently close to unity,
however, they are made strictly worse off since limδ→1 β(γ, δ, θ) = γ. In this sense, the agents’ desire
to maintain reputation may backfire and actually be detrimental to them. The agents suffer in therepeated setting for two reasons. First, they suffer because the principal needs to offer less to induce
effort for collaborative work because she can effectively exploit the agents’ ability to monitor eachother in the repeated setting. Second, they also suffer because they are given high-powered incentives
16

only for collaborative works, but there is a positive probability that the opportunity to collaborate
does not arrive. We summarize this result as follows.
Proposition 5 In the limiting case, if θ ∈ [θ̄, 1), the agents are made strictly worse off in the repeatedsetting than in the static setting.

5

Conclusion

The paper examines the optimal provision of incentives under voluntary team formation. When collaboration allows the agents to mutually monitor each other, the principal may benefit from encouraging
collaboration even in the absence of strong productive complementarities. Moreover, under certain
conditions, it is optimal to offer low-powered incentives for non-collaborative works as it makes thepeer sanction mechanism more effective. We argue that seemingly low-powered incentives offered for
non-collaborative works, as often observed in our profession, can be seen as a critical ingredient of
optimal incentive schemes. While our model simply aims at providing an alternative view and is by
no means to exclude other potential explanations, we believe that the desire to maintain reputationis critical for academic researchers and the current compensation scheme is a fairly effective way to
exploit this desire.
As a final note, there are several unresolved issues that need to be settled. One possibility is toconsider the case with multiple principals who independently design compensation schemes. Although
the extension does not completely alter the nature of the problem as long as agents are not completely
mobile, it certainly makes the problem more complicated since this may allow us to endogenize theprobability with which an opportunity to collaborate arrives. A natural way to model this is thatthe probability is an increasing function of the number of principals offering a high-powered teamcontract. It is expected that there arises a type of strategic complementarity among the principals
because it is certainly not a good idea to offer a low-powered individual contract when others offer ahigh-powered individual contract and hence not many agents are willing to collaborate. With multiple
principals, competition among them also becomes an issue.

17Another potential avenue to extend the current analysis is to allow for more than two agents.
For instance, it is more realistic to construct a model where there are many agents and each agent
searches for a potential collaborator in a search environment. This type of setup may require some
sort of community enforcement, in the sense of Kandori (1992), to make the peer sanction mechanism
effective. In this context it naturally becomes crucial how each agent can establish (or tarnish) his
reputation or alternatively how the information spreads out in the community when a defection occurs.
The peer sanction mechanism is expected to be weak if there are many agents (so that the likelihoodof meeting the same agent is negligible) and the information transmission is rather slow. On the other
hand, the peer sanction mechanism can be highly effective if each of us belongs to a small community(a guild) from which to choose potential collaborators and the information is immediately sharedwithin the community. In future it is of some interest to pursue this possibility.

Appendix
Proof of Proposition 1:

We first examine the optimal individual contract. It is immediate tosee w0n = 0. It also follows from (3) that the optimal individual contract takes one of the two forms:
w1n = d/∆p, or w1n = 0. If the principal offers the high-powered contract w1n = d/∆p, the expected
profit is p1 (V − d/∆p); if she offers the low-powered contract w1n = 0, it is p0 V . The principal thuschooses the high-powered contract if
∆pV ≥

d
.
∆p

(24)

Under Assumption 1, therefore, it is optimal for the principal to offer the high-powered individual
contract.
We now shift our attention to the optimal team contract and see whether it can improve upon
the optimal individual contract. To this end, we first show that the low-powered team contract
can never yield more profit than the high-powered individual contract under Assumption 1. Notefirst that w0c = 0. Given this, if w1c = 0, the expected profit is p0 V . It then follows directly from theargument above that the low-powered team contract cannot improve upon the high-powered individual

18

contract. Now consider the high-powered team contract w1c = γd/∆f (1). Under this contract, theexpected profit is p1 (V − γd/∆f (1)). The team contract improves upon the individual contract if
Ã
!
!
Ã
2γd
d
p1 V −≥ p1 V −
,
(25)
∆p
∆p
which never holds under Assumption 2. This indicates that the principal should never encourage
collaboration: one of the team contracts that can achieve this is wc = (0, 0) while there are certainly
many others.
Q.E.D.
Proof of Lemma 2: Before we proceed, we first conjecture that 0.5∆p(w1c − w0c ) < γd so that it isstage-game Nash equilibrium to exert no effort for collaborative works, i.e., ucN = p0 w1c + (1 − p0 )w0c .
For the high-powered team contract to be meaningful, it is also necessary that the agents choose tocollaborate in state C when they can, i.e., ucC ≥ un . We suppose for now that these conditions holdand later verify that this is indeed the case. The incentive compatibility constraint for collaborative
works is then given by
0.5∆p(w1c − w0c )+δθ(p1 w1c + (1− p1 )w0c − γd − max{p1 w1n + (1− p1 )w0n − d, p0 w1c + (1− p0 )w0c }) ≥ γd.
1−δ
(26)

Provided that the individual contract is high-powered, it is immediate to see w0n = 0 and w1n = d/∆p.
The constraint can be written as
0.5∆p(w1c − w0c ) +δθd(p1 w1c + (1 − p1 )w0c − γd − max{p1− d, p0 w1c + (1 − p0 )w0c }) ≥ γd. (27)1−δ∆pThere are two distinct situations we need to consider: (Case 1) p1 d/∆p − d > p0 w1c + (1 − p0 )w0c and(Case 2) p1 d/∆p − d ≤ p0 w1c + (1 − p0 )w0c . We now examine each case in turn.

(Case 1): Suppose that p1 d/∆p−d > p0 w1c +(1−p0 )w0c , which means that the agents never collaborate
in state N . Under this condition, the constraint now becomes
0.5∆p(w1c − w0c ) +δθd(p1 w1c + (1 − p1 )w0c − γd − p1+ d) ≥ γd.
1−δ
∆p
19

(28)

We first show w0c = 0. To show this, suppose that the optimal team contract is given by wc = (a, b)
where a > 0. Note, however, that another contract wc = (0, (p1 b + (1 − p1 )a)/p1 ) yields the sameprofit level and satisfies the constraint with strict inequality. This is a contradiction and we can thus
conclude w0c = 0. Given this result, the constraint can be simplified to
0.5∆pw1c +δθd(p1 w1c − γd − p1+ d) ≥ γd.
1−δ
∆p

(29)

Solving this constraint, we obtain a candidate team contract
w0c = 0, w1c =2γ(1 − δ + δθ)∆p + 2δθp0d.
(1 − δ)∆p2 + 2δθp1 ∆p(30)

To show that this contract implements the desired outcome, it is necessary that (i) γd > 0.5∆pw1c
and (ii) ucC ≥ un , i.e., p1 w1c − γd ≥ p1 w1n . It is straightforward to see that the first condition is always
satisfied when γ > 0.5. The second condition can be written as
p1

d
p0 d
2γ(1 − δ + δθ)∆p + 2δθp0
d − γd ≥ p1−d=,(1 − δ)∆p2 + 2δθp1 ∆p
∆p
∆p

(31)

2γ(1 − δ + δθ)∆p + 2δθp0≥ γp1 + (1 − γ)p0 .
(1 − δ)∆p + 2δθp1(32)

which is reduced to
p1

With some algebra, one can show that this condition is always satisfied when γ > 0.5.
The optimal team contract must also satisfy p1 d/∆p − d > p0 w1c , which can also be written asd/∆p > w1c . This holds if
2γ(1 − δ + δθ)∆p + 2δθp0,(1 − δ)∆p + 2δθp1(33)

2δθ(1 − γ) > (1 − δ)(2γ − 1).

(34)

1>
which is further simplified to

Finally, we need to show that the candidate contract is collusion-proof, i.e.,∆pw1c ≥ γd.

(35)

2γ(1 − δ + δθ)∆p + 2δθp0≥ γ.
(1 − δ)∆p + 2δθp1(36)This condition can be written as20

With some algebra, one can show that this condition always holds. This implies that if (34) is satisfied,
the optimal team contract within this class is given by (30).

(Case 2): Now suppose that p1 d/∆p − d ≤ p0 w1c + (1 − p0 )w0c , which means that the agents choose
to collaborate even in state N . Provided that 0.5∆p(w1c − w0c ) < γd, the incentive compatibility
constraint for collaborative works becomes
0.5∆p(w1c − w0c ) +δθ(p1 w1c + (1 − p1 )w0c − γd − p0 w1c − (1 − p0 )w0c ) ≥ γd.
1−δ

(37)

This condition can further be written as
0.5∆p(w1c − w0c ) +δθ(∆p(w1c − w0c ) − γd) ≥ γd.
1−δ

(38)

Solving this we obtainw1c − w0c ≥2γ(1 − δ + δθ)d,(1 − δ + 2δθ)∆p(39)

which implies w0c = 0.
To show that this contract implements the desired outcome, it is necessary that (i) γd > 0.5∆pw1c
and (ii) ucC ≥ un , i.e., p1 w1c − γd ≥ p1 w1n − d. It is immediate to see that the first condition is always
satisfied. Note also that2γ(1 − δ + δθ)≥ 1,
1 − δ + 2δθ(40)

when 2δθ(1 − γ) ≤ (1 − δ)(2γ − 1). This implies that the second condition is also satisfied.
The optimal team contract must also satisfy p1 d/∆p − d ≤ p0 w1c + (1 − p0 )w0c , which can be
simplified to
w1c ≥

d
.
∆p

(41)

Note that this condition always holds when 2δθ(1 − γ) ≤ (1 − δ)(2γ − 1). It follows from this that thecandidate contract is given by
w0c = 0, w1c =2γ(1 − δ + δθ)d,(1 − δ + 2δθ)∆p

21

(42)

if 2δθ(1 − γ) ≤ (1 − δ)(2γ − 1).
Finally, it directly follows from (41) that this candidate contract is also collusion-proof, i.e., ∆pw1c ≥γd. This implies that if (34) is not satisfied, the optimal team contract within this class is given by(42).
Q.E.D.
Proof of Lemma 3:

As above, we first conjecture (i) 0.5∆p(w1c − w0c ) < γd and (ii) ucC ≥ un .

Given these, the incentive compatibility constraint for collaborative works becomes
0.5∆p(w1c −w0c )+δθ(p1 w1c +(1−p1 )w0c −γd−max{p0 w1n +(1−p0 )w0n , p0 w1c +(1−p0 )w0c }) ≥ γd. (43)1−δIf the individual contract is low-powered, there is no reason to offer a positive wage for noncollaborative works, implying w0n = w1n = 0 and un = 0. The incentive compatibility constraint thenbecomes0.5∆p(w1c − w0c ) +δθ(∆p(w1c − w0c ) − γd) ≥ γd,
1−δ

(44)

which leads to
w1c − w0c ≥2(1 − δ + δθ)γd.
(1 − δ + 2δθ)∆p(45)

2(1 − δ + δθ)γd,(1 − δ + 2δθ)∆p(46)

The candidate contract is thus given by
w0c = 0, w1c =which indeed satisfies γd > 0.5∆pw1c for any (δ, θ) ∈ (0, 1)2 as conjectured. It is also immediate to seethat ucC ≥ un = 0.
Finally, we need to verify that the candidate contract is collusion-proof, i.e.,2(1 − δ + δθ)γd≥ γd,1 − δ + 2δθ(47)2(1 − δ + δθ) ≥ 1 − δ + 2δθ.

(48)

∆pw1c =
which can be simplified toSince this evidently holds, the optimal team contract within this class is given by (46).
22

Q.E.D.
Proof of Proposition 2: To prove the proposition it suffices to show πhh > πhl when δ is arbitrarily close to unity. It follows from lemma 2 that as δ → 1, the optimal contract is (0, d/∆p, 0, α(γ, δ, θ)d)when both of the contracts are high-powered. Note thatlim α(γ, δ, θ) =δ→1γ∆p + p0
.
p1

(49)

This implies that as δ → 1,
πhh = p1Ã!
γ∆p + p0dV −θd − (1 − θ)
.
p1 ∆p
∆p

Note also that the expected profit under the type 1 contract is independent of δ:!
Ãdπhl = p1 V −
.
∆p

(50)

(51)

Given p0 > 0, it is immediate to see πhh > πhl as δ → 1.
Q.E.D.
Proof of Proposition 3:

The optimal individual contract is low-powered if πlh ≥ πhh . Noticethat
lim β(γ, δ, θ) = γ.

δ→1

(52)

As δ → 1, πlh ≥ πhh if
θp1

Ã

γd
V −
∆p

!

≥ p1

Ã

!
γ∆p + p0dV −θd − (1 − θ)
,
p1 ∆p
∆p

(53)

which is reduced toθ(1 − γ)p0 d + (1 − θ)p1 d≥ (1 − θ)V.
p1 ∆p

(54)

It is easy to see that this condition holds as θ → 1.
Q.E.D.

23

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