Review at least 4 articles on TEAM DYNAMICS and write power point presentation 12 slides

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Review of Economic Studies (2015) 82, 187–218 doi:10.1093/restud/rdu031 © The Author 2014. Published by Oxford University Press on behalf of The Review of Economic Studies Limited. Advance access publication 18 September 2014

Projects and Team Dynamics GEORGE GEORGIADIS

Boston University and California Institute of Technology

First version received November 2011; final version accepted June 2014 (Eds.)

I study a dynamic problem in which a group of agents collaborate over time to complete a project. The project progresses at a rate that depends on the agents’ efforts, and it generates a pay-off upon completion. I show that agents work harder the closer the project is to completion, and members of a larger team work harder than members of a smaller team—both individually and on aggregate—if and only if the project is sufficiently far from completion. I apply these results to determine the optimal size of a self-organized partnership, and to study the manager’s problem who recruits agents to carry out a project, and must determine the team size and its members’ incentive contracts. The main results are: (i) that the optimal symmetric contract compensates the agents only upon completing the project; and (ii) the optimal team size decreases in the expected length of the project.

Key words: Projects, Moral hazard in teams, Team formation, Partnerships, Differential games

JEL Codes: D7, H4, L22, M5

1. INTRODUCTION

Teamwork and projects are central in the organization of firms and partnerships. Most large corporations engage a substantial proportion of their workforce in teamwork (Lawler et al., 2001), and organizing workers into teams has been shown to increase productivity in both manufacturing and service firms (Ichniowski and Shaw, 2003). Moreover, the use of teams is especially common in situations in which the task at hand will result in a defined deliverable, and it will not be ongoing, but will terminate (Harvard Business School Press, 2004). Motivated by these observations, I analyse a dynamic problem in which a group of agents collaborate over time to complete a project, and I address a number of questions that naturally arise in this environment. In particular, what is the effect of the group size to the agents’ incentives? How should a manager determine the team size and the agents’ incentive contracts? For example, should they be rewarded for reaching intermediate milestones, and should rewards be equal across the agents?

I propose a continuous-time model, in which at every moment, each of n agents exerts costly effort to bring the project closer to completion. The project progresses stochastically at a rate that is equal to the sum of the agents’effort levels (i.e. efforts are substitutes), and it is completed when its state hits a pre-specified threshold, at which point each agent receives a lump sum pay-off and the game ends.

This model can be applied both within firms, for instance, to research teams in new product development or consulting projects, and across firms, for instance, to R&D joint ventures. More broadly, the model is applicable to settings in which a group of agents collaborate to complete a project, which progresses gradually, its expected duration is sufficiently large such that the agents discounting time matters, and it generates a pay-off upon completion. A natural example is the Myerlin Repair Foundation (MRF): a collaborative effort among a group of leading scientists in quest of a treatment for multiple sclerosis (Lakhani and Carlile, 2012). This is a long-term venture,

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progress is gradual, each principal investigator incurs an opportunity cost by allocating resources to MRF activities (which gives rise to incentives to free-ride), and it will pay off predominantly when an acceptable treatment is discovered.

In Section 3, I characterize the Markov perfect equilibrium (MPE) of this game, wherein at every moment, each agent observes the state of the project (i.e. how close it is to completion), and chooses his effort level to maximize his expected discounted pay-off, while anticipating the strategies of the other agents. A key result is that each agent increases his effort as the project progresses. Intuitively, because he discounts time and is compensated upon completion, his incentives are stronger the closer the project is to completion. An implication of this result is that efforts are strategic complements across time, in that a higher effort level by one agent at time t brings the project (on expectation) closer to completion, which in turn incentivizes himself, as well as the other agents to raise their future efforts.

In Section 4, I examine the effect of the team size to the agents’ incentives. I show that members of a larger team work harder than members of a smaller team—both individually and on aggregate—if and only if the project is sufficiently far from completion.1 Intuitively, by increasing the size of the team, two forces influence the agents’ incentives. First, they obtain stronger incentives to free-ride. However, because the total progress that needs to be carried out is fixed, the agents benefit from the ability to complete the project quicker, which increases the present discounted value of their reward, and consequently strengthens their incentives. I refer to these forces as the free-riding and the encouragement effect, respectively. Because the marginal cost of effort is increasing and agents work harder the closer the project is to completion, the free-riding effect becomes stronger as the project progresses. On the other hand, the benefit of being able to complete the project faster in a bigger team is smaller the less progress remains, and hence the encouragement effect becomes weaker with progress. As a result, the encouragement effect dominates the free-riding effect, and consequently members of a larger team work harder than those of a smaller team if and only if the project is sufficiently far from completion.

I first apply this result to the problem faced by a group of agents organizing into a partnership. If the project is a public good so that each agent’s reward is independent of the team size, then each agent is better off expanding the partnership ad infinitum. However, if the project generates a fixed pay-off upon completion that is shared among the team members, then the optimal partnership size increases in the length of the project.2

Motivated by the fact that projects are often run by corporations (rather than self-organized partnerships), in Section 5, I introduce a manager who is the residual claimant of the project, and he/she recruits a group of agents to undertake it on his/her behalf. His/Her objective is to determine the size of the team and each agent’s incentive contract to maximize his/her expected discounted profit.

First, I show that the optimal symmetric contract compensates the agents only upon completion of the project. The intuition is that by backloading payments (compared to rewarding the agents for reaching intermediate milestones), the manager can provide the same incentives at the early stages of the project (via continuation utility), while providing stronger incentives when the project is close to completion. This result simplifies the manager’s problem to determining the team size and his/her budget for compensating the agents. Given a fixed team size, I show that the manager’s optimal budget increases in the length of the project. This is intuitive: to incentivize

1. This result holds both if the project is a public good so that each agent’s reward is independent of the team size, and if the project generates a fixed pay-off that is shared among the team members so that doubling the team size halves each agent’s reward.

2. The length of the project refers to the expected amount of progress necessary to complete it (given a fixed pay-off).

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 189

the agents, the manager should compensate them more, the longer the project. Moreover, the optimal team size increases in the length of the project. Recall that a larger team works harder than a smaller one if the project is sufficiently far from completion. Therefore, the benefit from a larger team working harder while the project is far from completion outweighs the loss from working less when it is close to completion only if the project is sufficiently long. Lastly, I show that the manager can benefit from dynamically decreasing the size of the team as the project nears completion. The intuition is that he/she prefers a larger team while the project is far from completion since it works harder than a smaller one, while a smaller team becomes preferable near completion.

The restriction to symmetric contracts in not without loss of generality. In particular, the scheme wherein the size of the team decreases dynamically as the project progresses can be implemented with an asymmetric contract that rewards the agents upon reaching different milestones. Finally, with two (identical) agents, I show that the manager is better off compensating them asymmetrically if the project is sufficiently short. Intuitively, the agent who receives the larger reward will carry out the larger share of the work in equilibrium, and hence she/he cannot free-ride on the other agent as much.

First and foremost, this article is related to the moral hazard in teams literature (Holmström, 1982; Ma et al., 1988; Bagnoli and Lipman, 1989; Legros and Matthews, 1993; Strausz, 1999, and others). These papers focus on the free-rider problem that arises when each agent must share the output of his/her effort with the other members of the team, and they explore ways to restore efficiency. My article ties in with this literature in that it analyzes a dynamic game of moral hazard in teams with stochastic output.

Closely related to this article is the literature on dynamic contribution games, and in particular, the papers that study threshold or discrete public good games. Formalizing the intuition of Schelling (1960),Admati and Perry (1991), and Marx and Matthews (2000) show that contributing little by little over multiple periods, each conditional on the previous contributions of the other agents, mitigates the free-rider problem. Lockwood and Thomas (2002) and Compte and Jehiel (2004) show how gradualism can arise in dynamic contribution games, while Battaglini, Nunnari and Palfrey (2013) compare the set of equilibrium outcomes when contributions are reversible to the case in which they are not. Whereas these papers focus on characterizing the equilibria of dynamic contribution games, my primary focus is on the organizational questions that arise in the context of such games.

Yildirim (2006) studies a game in which the project comprises of multiple discrete stages, and in every period, the current stage is completed if at least one agent exerts effort. Effort is binary, and each agent’s effort cost is private information, and re-drawn from a common distribution in each period. In contrast, in my model, following Kessing (2007), the project progresses at a rate that depends smoothly on the team’s aggregate effort. Yildirim (2006) and Kessing (2007) show that if the project generates a pay-off only upon completion, then contributions are strategic complements across time even if there are no complementarities in the agents’ production function. This is in contrast to models in which the agents receive flow pay-offs while the project is in progress (Fershtman and Nitzan, 1991), and models in which the project can be completed instantaneously (Bonatti and Hörner, 2011), where contributions are strategic substitutes. Yildirim also examines how the team size influences the agents’ incentives in a dynamic environment, and he shows that members of a larger team work harder than those of a smaller team at the early stages of the project, while the opposite is true at its later stages.3

This result is similar to Theorem 2(i) in this article. However, leveraging the tractability of my

3. It is worth pointing out, however, that in Yildirim’s model, this result hinges on the assumption that in every period, each agent’s effort cost is re-drawn from a non-degenerate distribution. In contrast, if effort costs are deterministic,

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model, I also characterize the relationship between aggregate effort and the team size, which is the crucial metric for determining the manager’s optimal team size.

In summary, my contributions to this literature are 2-fold. First, I propose a natural framework to analyse the dynamic problem faced by a group of agents who collaborate over time to complete a project. The model provides several testable implications, and the framework proposed in this article can be useful for studying other dynamic moral hazard problems with multiple agents; for example, the joint extraction of an exhaustible common resource, or a tug of war between two teams (in the spirit of Cao, 2014), or a game of oligopolistic competition with demand that is correlated across time (as in Section IV of Sannikov and Skrzypacz, 2007). Moreover, in an earlier version of this article, I also analyse the cases in which the agents are asymmetric and the project size is endogenous (Georgiadis, 2011). Secondly, I derive insights for the organization of partnerships, and for team design where a manager must determine the size of his/her team and the agents’ incentive contracts. To the best of my knowledge, this is one of the first papers to study this problem; one notable exception being Rahmani et al. (2013), who study the contractual relationship between the members of a two-person team.

This paper is also related to the literature on free-riding in groups. To explain why teamwork often leads to increased productivity in organizations in spite of the theoretical predictions that effort and group size should be inversely related (Olson, 1965; Andreoni, 1988), scholars have argued that teams benefit from mutual monitoring (Alchian and Demsetz, 1972), peer pressure to achieve a group norm (Kandel and Lazear, 1992), complementary skills (Lazear, 1998), warm- glow (Andreoni, 1990), and non-pecuniary benefits such as more engaging work and social interaction. While these forces are helpful for explaining the benefits of teamwork, this paper shows that they are actually not necessary in settings in which the team’s efforts are geared towards completing a project.

Lastly, the existence proofs of Theorems 1 and 3 are based on Hartman (1960), while the proof techniques for the comparative statics draw from Cao (2014), who studies a continuous-time version of the patent race of Harris and Vickers (1985).

The remainder of this paper is organized as follows. Section 2 introduces the model. Section 3 characterizes the MPE of the game, and establishes some basic results. Section 4 examines how the size of the team influences the agents’ incentives, and characterizes the optimal partnership size. Section 5 studies the manager’s problem, and Section 6 concludes. Appendix A contains a discussion of non-Markovian strategies and four extensions of the base model. The major proofs are provided in Appendix B, while the omitted proofs are available in the online Appendix.

2. THE MODEL

A team of n agents collaborate to complete a project. Time t ∈ [0,∞) is continuous. The project starts at some initial state q0<0, its state qt evolves according to a stochastic process, and it is completed at the first time τ such that qt hits the completion state which is normalized to 0. Agent i∈{1,...,n} is risk neutral, discounts time at rate r>0, and receives a pre-specified reward Vi>0 upon completing the project.4 An incomplete project has zero value. At every moment t, each

then this comparative static is reversed: the game becomes a dynamic version of the “reporting a crime” game (ch. 4.8 in Osborne, 2003), and one can show that in the unique symmetric, mixed-strategy MPE, both the probability that each agent exerts effort, and the probability that at least one agent exerts effort at any given stage of the project (which is the metric for individual and aggregate effort, respectively) decreases in the team size.

4. In the base model, the project generates a pay-off only upon completion. The case in which the project also generates a flow pay-off while it is in progress is examined in Appendix A.1, and it is shown that the main results continue to hold.

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agent observes the state of the project qt , and privately chooses his/her effort level to influence the drift of the stochastic process

dqt = (

n∑ i=1

ai,t

) dt+σdWt ,

where ai,t ≥0 denotes the effort level of agent i at time t, σ >0 captures the degree of uncertainty associated with the evolution of the project, and Wt is a standard Brownian motion.5,6 As such, |q0| can be interpreted as the expected length of the project.7 Finally, each agent is credit constrained, his effort choices are not observable to the other agents, and his flow cost of exerting effort a is

given by c(a)= ap+1p+1 , where p≥1.8 At every moment t, each agent i observes the state of the project qt , and chooses his/her effort

level ai,t to maximize his/her expected discounted pay-off while taking into account the effort choices a−i,s of the other team members. As such, for a given set of strategies, his/her expected discounted pay-off is given by

Ji (qt)=Eτ [

e−r(τ−t)Vi − ∫ τ

t e−r(s−t)c

( ai,s ) ds

] , (1)

where the expectation is taken with respect to τ : the random variable that denotes the completion time of the project.

Assuming that Ji (·) is twice differentiable for all i, and using standard arguments (Dixit, 1999), one can derive the Hamilton–Jacobi–Bellman (hereafter HJB) equation for the expected discounted pay-off function of agent i:

rJi (q)=−c ( ai,t )+ ⎛ ⎝ n∑

j=1 aj,t

⎞ ⎠J ′i (q)+ σ 22 J ′′i (q) (2)

defined on (−∞,0] subject to the boundary conditions

lim q→−∞Ji (q)=0 and Ji (0)=Vi . (3)

Equation (2) asserts that agent i’s flow pay-off is equal to his/her flow cost of effort, plus his marginal benefit from bringing the project closer to completion times the aggregate effort of the team, plus a term that captures the sensitivity of his/her pay-off to the volatility of the project.

5. For simplicity, I assume that the variance of the stochastic process (i.e. σ ) does not depend on the agents’ effort levels. While the case in which effort influences both the drift and the diffusion of the stochastic process is intractable,

numerical examples with dqt = (∑n

i=1 ai,t ) dt+σ (∑ni=1 ai,t)1/2 dWt suggest that the main results continue to hold. See

Appendix A.3 for details. 6. I assume that efforts are perfect substitutes. To capture the notion that when working in teams, agents may

be more (less) productive due to complementary skills (coordination costs), one can consider a super- (sub-) additive

production function such as dqt = (∑n

i=1 a 1/γ i,t

)γ dt+σdWt , where γ >1 (0<γ <1). The main results continue to hold.

7. Because the project progresses stochastically, the total amount of effort to complete it may be greater or smaller than |q0|.

8. The case in which c(·) is an arbitrary, strictly increasing, and convex function is discussed in Remark 1, while the case in which effort costs are linear is analysed in Appendix A.5 The restriction that p≥1 is necessary only for establishing that a MPE exists. If the conditions in Remark 1 are satisfied, then all results continue to hold for any p>0.

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To interpret equation (3), observe that as q→−∞, the expected time until the project is completed so that agent i collects his/her reward diverges to ∞, and because r>0, his/her expected discounted pay-off asymptotes to 0. However, because he/she receives his/her reward and exerts no further effort after the project is completed, Ji (0)=Vi.

3. MARKOV PERFECT EQUILIBRIUM

I assume that strategies are Markovian, so that at every moment, each agent chooses his/her effort level as a function of the current state of the project.9 Therefore, given q, agent i chooses his/her effort level ai (q) such that

ai (q)∈argmax ai≥0

{ aiJ

′ i (q)−c(ai)

} .

Each agent chooses his/her effort level by trading off marginal benefit of bringing the project closer to completion and the marginal cost of effort. The former comprises of the direct benefit associated with the project being completed sooner, and the indirect benefit associated with influencing the other agents’ future effort choices.10 By noting that c′(0)=0 and c(·) is strictly convex, it follows that for any given q, agent i’s optimal effort level ai (q)= f

( J ′i (q)

) , where

f (·)=c′−1(max{0, ·}). By substituting this into equation (2), the expected discounted pay-off for agent i satisfies

rJi (q)=−c ( f ( J ′i (q)

))+ ⎡ ⎣ n∑

j=1 f (

J ′j (q) )⎤⎦J ′i (q)+ σ 22 J ′′i (q) (4)

subject to the boundary conditions (3). An MPE is characterized by the system of ordinary differential equations (ODE) defined by

equation (4) subject to the boundary conditions (3) for all i∈{1,...,n}. To establish existence of a MPE, it suffices to show that a solution to this system exists. I then show that this system has a unique solution if the agents are symmetric (i.e., Vi =Vj for all i �= j). Together with the facts that every MPE must satisfy this system and the first-order condition is both necessary and sufficient, it follows that the MPE is unique in this case.

Theorem 1. An MPE for the game defined by equation (1) exists. For each agent i, the expected discounted pay-off function Ji (q) satisfies:

(i) 0<Ji (q)≤Vi for all q. (ii) J ′i (q)>0 for all q, and hence the equilibrium effort ai (q)>0 for all q.

(iii) J ′′i (q)>0 for all q, and hence a′i (q)>0 for all q. (iv) If agents are symmetric (i.e. Vi =Vj for all i �= j), then the MPE is symmetric and unique.11

9. The possibility that the agents play non-Markovian strategies is discussed in Remark 5, in Section 3.2. 10. Because each agent’s effort level is a function of q, his/her current effort level will impact his/her and the other

agents’ future effort levels. 11. To simplify notation, if the agents are symmetric, then the subscript i is interchanged with the subscript n to

denote the team size throughout the remainder of this article.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 193

J ′i (q)>0 implies that each agent is strictly better off, the closer the project is to completion. Because c′(0)=0 (i.e. the marginal cost of little effort is negligible), each agent exerts a strictly positive amount of effort at every state of the project: ai (q)>0 for all q.12

Because the agents incur the cost of effort at the time effort is exerted but are only compensated upon completing the project, their incentives are stronger, the closer the project is to completion: a′i (q)>0 for all q. An implication of this result is that efforts are strategic complements across time. That is because a higher effort by an agent at time t brings the project (on expectation) closer to completion, which in turn incentivizes himself/herself, as well as the other agents to raise their effort at times t′> t.

Note that Theorem 1 hinges on the assumption that r>0. If the agents are patient (i.e. r =0), then in equilibrium, each agent will always exert effort 0.13 Therefore, this model is applicable to projects whose expected duration is sufficiently large such that the agents discounting time matters.

Remark 1. For an MPE to exist, it suffices that c(·) is strictly increasing and convex with c(0)= 0, it satisfies the INADA condition lima→∞c′(a)=∞, and σ 24

∫∞ 0

sds r ∑n

i=1 Vi+nsf (s) > ∑n

i=1Vi. If

c(a)= ap+1p+1 and p≥1, then the LHS equals ∞, so that the inequality is always satisfied. However, if p∈(0,1), then the inequality is satisfied only if∑ni=1Vi, r and n are sufficiently small, or if σ is sufficiently large. More generally, this inequality is satisfied if c(·) is sufficiently convex.

The existence proof requires that Ji (·) and J ′i (·) are always bounded. It is easy to show that Ji (q)∈ [0,Vi] and J ′i (q)≥0 for all i and q. The inequality in Remark 1 ensures that the marginal cost of effort c′(a) is sufficiently large for large values of a that no agent ever has an incentive to exert an arbitrarily high effort, which by the first-order condition implies that J ′i (·) is bounded from above.

Remark 2. An important assumption of the model is that the agents are compensated only upon completion of the project. In Appendix A.1, I consider the case in which during any interval (t, t+dt) while the project is in progress, each agent receives a flow pay-off h(qt)dt, in addition to the lump sum reward V upon completion. Assuming that h(·) is increasing and satisfies certain regularity conditions, there exists a threshold ω (not necessarily interior) such that a′n(q)≥0 if and only if q≤ω; i.e. effort is hump-shaped in progress.

The intuition why effort can decrease in q follows by noting that as the project nears completion, each agent’s flow pay-off becomes larger, which in turn decreases his/her marginal benefit from bringing the project closer to completion. Numerical analysis indicates that this threshold is interior as long as the magnitude of the flow pay-offs is sufficiently large relative to V .

Remark 3. The model assumes that the project is never “cancelled”. If there is an exogenous cancellation state QC<q0<0 such that the project is cancelled (and the agents receive pay-off 0) at the first time that qt hits QC, then statements (i) and (ii) of Theorem 1 continue to hold,

12. If c′ (0)>0, then there exists a quitting threshold Qq, such that each agent exerts 0 effort on (−∞,Qq], while

he/she exerts strictly positive effort on ( Qq,0

] , and his/her effort increases in q.

13. If σ =0, because effort costs are convex and the agents do not discount time, in any equilibrium in which the project is completed, each agent finds it optimal to exert an arbitrarily small amount of effort over an arbitrarily large time horizon, and complete the project asymptotically. (A project-completing equilibrium exists only if c′ (0) is sufficiently close to 0.)

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but effort needs no longer be increasing in q. Instead, there exists a threshold ω (not necessarily interior) such that a′n(q)≤0 if and only if q≤ω; i.e. effort is U-shaped in progress. See Appendix A.2 for details.

Intuitively, the agents have incentives to exert effort (i) to complete the project, and (ii) to avoid hitting the cancellation state QC . Because the incentives due to the former (latter) are stronger the closer the project is to completion (to QC), depending on the choice of QC , the agent’s incentives may be stronger near QC and near the completion state relative to the midpoint. Numerical analysis indicates that ω=0 so that effort increases monotonically in q if QC is sufficiently small; it is interior if QC is in some intermediate range, and ω=−∞ so that effort always decreases in q if QC is sufficiently close to 0.

Remark 4. Agents have been assumed to have outside option 0. In a symmetric team, if each agent has a positive outside option u>0, then there exists an optimal abandonment state QA>−∞ satisfying the smooth-pasting condition ∂∂q Jn(q,QA)

∣∣∣ q=QA

=0 such that the agents find it optimal to abandon the project at the first moment q hits QA, where Jn(·,QA) satisfies equation (4) subject to Jn(QA,QA)=u and Ji (0,QA)=Vi. In this case, each agent’s effort increases monotonically with progress.

3.1. Comparative statics

This section establishes some comparative statics, which are helpful to understand how the agents’ incentives depend on the parameters of the problem. To examine the effect of each parameter to the agents’ incentives, I consider two symmetric teams that differ in exactly one attribute: their members’ rewards V , patience levels r, or the volatility of the project σ .14

Proposition 1. Consider two teams comprising symmetric agents.

(i) If V1<V2, then all other parameters held constant, a1(q)<a2(q) for all q. (ii) If r1>r2, then all other parameters held constant, there exists an interior threshold �r

such that a1(q)≤a2(q) if and only if q≤�r . (iii) If σ1>σ2, then all other parameters held constant, there exist interior thresholds�σ,1 ≤

�σ,2 such that a1(q)≥a2(q) if q≤�σ,1 and a1(q)≤a2(q) if q≥�σ,2.15

The intuition behind statement (i) is straightforward. If the agents receive a bigger reward, then they always work harder in equilibrium.

Statement (ii) asserts that less patient agents work harder than more patient agents if and only if the project is sufficiently close to completion. Intuitively, less patient agents have more to gain from an earlier completion (provided that the project is sufficiently close to completion). However, bringing the completion time forward requires that they exert more effort, the cost of which is incurred at the time that effort is exerted, whereas the reward is only collected upon completion of the project. Therefore, the benefit from bringing the completion time forward (by exerting more effort) outweighs its cost only when the project is sufficiently close to completion.

14. Since the teams are symmetric and differ in a single parameter (e.g. their reward Vi in statement (i)), abusing notation, I let ai (·) denote each agent’s effort strategy corresponding to the parameter with subscript i.

15. Unable to show that J ′′′i (q) is unimodal in q, this result does not guarantee that�σ,1 =�σ,2, which implies that it does not provide any prediction about how the agents’ effort depends on σ when q∈[�σ,1,�σ,2]. However, numerical analysis indicates that in fact �σ,1 =�σ,2.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 195

Finally, statement (iii) asserts that incentives become stronger in the volatility of the project σ when it is far from completion, while the opposite is true when it gets close to completion. As the volatility increases, it becomes more likely that the project will be completed either earlier than expected (upside), or later than expected (downside). If the project is sufficiently far from completion, then Ji (q) is close to 0 so that the downside is negligible, while J ′′i (q)>0 implies that the upside is not (negligible), and consequently a1(q)≥a2(q). However, because the completion time of the project is non-negative, the upside diminishes as it approaches completion, which implies that the downside is bigger than the upside, and consequently a1(q)≤a2(q).

3.2. Comparison with first-best outcome

To obtain a benchmark for the agents’ equilibrium effort levels, I compare them to the first-best outcome, where at every moment, each agent chooses his effort level to maximize the team’s, as opposed to his individual expected discounted pay-off. I focus on the symmetric case, and denote by Ĵn(q) and ân(q) the first-best expected discounted pay-off and effort level of each member of an

n-person team, respectively. The first-best effort level satisfies ân(q)∈argmaxa {

anĴ ′n(q)−c(a) }

,

and the first-order condition implies that ân(q)= f (

nĴ ′n(q) )

. Substituting this into equation (2)

yields

rĴn(q)=−c (

f (

nĴ ′n(q) ))

+nf (

nĴ ′n(q) )

Ĵ ′n(q)+ σ 2

2 Ĵ ′′n (q)

subject to the boundary conditions (3). It is straightforward to show that the properties established in Theorem 1 apply for Ĵn(q) and ân(q). In particular, the first-best ODE subject to equation (3) has a unique solution, and â′n(q)>0 for all q; i.e. similar to the MPE, the first-best effort level increases with progress.

Proposition 2 compares each agent’s effort and his/her expected discounted pay-off in the MPE to the first-best outcome.

Proposition 2. In a team of n≥2 agents, an(q)< ân(q) and Jn(q)< Ĵn(q) for all q.

This result is intuitive: because each agent’s reward is independent of his/her contribution to the project, he/she has incentives to free-ride. As a result, in equilibrium, each agent exerts strictly less effort and he/she is strictly worse off at every state of the project relative to the case in which agents behave collectively by choosing their effort level at every moment to maximize the team’s expected discounted pay-off.

Remark 5. A natural question is whether the agents can increase their expected discounted pay- off by adopting non-Markovian strategies, so that their effort at t depends on the entire evolution path of the project {qs}s≤t . While a formal analysis is beyond the scope of this article, the analysis of Sannikov and Skrzypacz (2007), who study a related model, suggests that no, there does not exist a symmetric public perfect equilibrium (PPE) in which agents can achieve a higher expected discounted pay-off than the MPE at any state of the project. See Appendix A.4 for details.

It is important to emphasize, however, that this conjecture hinges upon the assumption that the agents cannot observe each other’s effort choices. For example, if efforts are publicly observable, then in addition to the MPE characterized in Theorem 1, using a similar approach as in Georgiadis et al. (2014), who study a deterministic version of this model (i.e. with σ =0), one can show that there exists a PPE in which the agents exert the first-best effort level along the

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equilibrium path. Such equilibrium is supported by trigger strategies, wherein at every moment t, each agent exerts the first-best effort level if all agents have exerted the first-best effort level for all s< t, while he/she reverts to the MPE otherwise.16

4. THE EFFECT OF TEAM SIZE

When examining the relationship between the agents’ incentives and the size of the team, it is important to consider how each agent’s reward depends on the team size. I consider the following (natural) cases: the public good allocation scheme, wherein each agent receives a reward V upon completing the project irrespective of the team size, and the budget allocation scheme, wherein each agent receives a reward V/n upon completing the project.

With n symmetric agents, each agent’s expected discounted pay-off function satisfies

rJn(q) = −c ( f ( J ′n(q)

))+nf (J ′n(q))J ′n(q)+ σ 22 J ′′n (q) subject to limq→−∞Jn(q)=0 and Jn(0)=Vn, where Vn =V or Vn =V/n under the public good or the budget allocation scheme, respectively.

Theorem 2 below shows that under both allocation schemes, members of a larger team work harder than members of a smaller team—both individually and on aggregate—if and only if the project is sufficiently far from completion. Figure 1 illustrates an example.

Theorem 2. Consider two teams comprising n and m>n identical agents. Under both allocation schemes, all other parameters held constant, there exist thresholds �n,m and �n,m such that

(i) am (q)≥an(q) if and only if q≤�n,m ; and ii) mam (q)≥nan(q) if and only if q≤�n,m.

By increasing the size of the team, two opposing forces influence the agents’ incentives: First, agents obtain stronger incentives to free-ride. To see why, consider an agent’s dilemma at time t to (unilaterally) reduce his/her effort by a small amount ε for a short interval . By doing so, he/she saves approximately εc′(a(qt)) in effort costs, but at t+ , the project is ε farther from completion. In equilibrium, this agent will carry out only 1/n of that lost progress, which implies that the benefit from shirking increases in the team size. Secondly, recall that each agent’s incentives are proportional to the marginal benefit of bringing the completion time τ forward: −d/dτVnE

[ e−rτ

]=rVnE[e−rτ ], which implies that holding strategies fixed, an increase in the team size decreases the completion time of the project, and hence strengthens the agents’ incentives. Following the terminology of Bolton and Harris (1999), who study an experimentation in teams problem, I refer to these forces as the free-riding and the encouragement effect, respectively, and the intuition will follow from examining how the magnitude of these effects changes as the project progresses.

It is convenient to consider the deterministic case in which σ =0. Because c′(0)=0 and effort vanishes as q→−∞, and noting that each agent’s gain from free-riding is proportional to c′(a(q)), it follows that the free-riding effect is negligible when the project is sufficiently far from completion. As the project progresses, the agents raise their effort, and because effort costs are convex, the free-riding effect becomes stronger. The magnitude of the encouragement effect

16. There is a well-known difficulty associated with defining trigger strategies in continuous-time games, which Georgiadis et al. (2014) resolve using the concept of inertia strategies proposed by Bergin and MacLeod (1993).

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 197

Figure 1

Illustration of Theorem 2.

The upper panels illustrate each agent’s expected discounted pay-off under public good (left) and budget (right)

allocation for two different team sizes: n=3 and 5. The lower panels illustrate each agent’s equilibrium effort.

can be measured by the ratio of the marginal benefits of bringing the completion time forward: rV2ne

− rτ2 rVne−rτ =

V2n Vn

e rτ 2 . Observe that this ratio increases in τ , which implies that the encouragement

effect becomes weaker as the project progresses (i.e. as τ becomes smaller), and it diminishes under public good allocation (since V2nVn =1) while it becomes negative under budget allocation (since V2nVn <1).

In summary, under both allocation schemes, the encouragement effect dominates the free- riding effect if and only if the project is sufficiently far from completion. This implies that by increasing the team size, the agents obtain stronger incentives when the project is far from completion, while their incentives become weaker near completion.

Turning attention to the second statement, it follows from statement (i) that aggregate effort in the larger team exceeds that in the smaller team if the project is far from completion. Perhaps surprisingly, however, when the project is near completion, not only the individual effort, but also the aggregate effort in the larger team is less than that in the smaller team. The intuition follows by noting that when the project is very close to completion (e.g. qt =−�), this game resembles the (static) “reporting a crime” game (ch. 4.8 in Osborne, 2003), and it is well known that in the unique symmetric mixed-strategy Nash equilibrium of this game, the probability that at least one agent exerts effort (which is analogous to aggregate effort) decreases in the group size.

The same proof technique can be used to show that under both allocation schemes, the first- best aggregate effort increases in the team size at every q. This difference is a consequence of the free-riding effect being absent in this case, so that the encouragement effect alone leads a larger team to always work on aggregate harder than a smaller team.

It is noteworthy that the thresholds of Theorem 2 need not always be interior. Under budget allocation, it is possible that �n,m =−∞, which would imply that each member of the smaller

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Figure 2

An example with quartic effort costs (p=3). The upper panels illustrate that under both allocation schemes, �n,m is interior, whereas the lower panels illustrate that

�n,m =0, in which case the aggregate effort in the larger team always exceeds that of the smaller team.

team always works harder than each member of the larger team. However, numerical analysis indicates that�n,m is always interior under both allocation schemes. Turning to�n,m, the proof of Theorem 2 ensures that it is interior only under budget allocation if effort costs are quadratic, while one can find examples in which�n,m is interior as well as examples in which�n,m =0 otherwise. Numerical analysis indicates that the most important parameter that determines whether�n,m is interior is the convexity of the effort cost function, and it is interior as long as c(·) is not too convex (i.e. p is sufficiently small). This is intuitive, as more convex effort costs favour the larger team more.17 In addition, under public good allocation, for �n,m to be interior, it is also necessary that n and m are sufficiently small. Intuitively, this is because the size of the pie increases in the team size under this scheme, which (again) favours the larger team. Figure 2 illustrates an example with quartic effort costs (i.e. p=3) in which case �n,m is interior but �n,m =0 under both allocation schemes.

4.1. Partnership formation

In this section, I examine the problem faced by a group of agents who seek to organize into a partnership. Proposition 3 characterizes the optimal partnership size.

17. This finding is consistent with the results of Esteban and Ray (2001), who show that in a static setting, the aggregate effort increases in the team size if effort costs are sufficiently convex. In their setting, however, individual effort always decreases in the team size irrespective of the convexity of the effort costs. To further examine the impact of the convexity of the agents’ effort costs, in Appendix A.5, I consider the case in which effort costs are linear, and I establish an analogous result to Theorem 2: members an (n+1)-person team have stronger incentives relative to those of an n-person team as long as n is sufficiently small.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 199

Proposition 3. Suppose that the partnership composition is finalized before the agents begin to work, so that the optimal partnership size satisfies argmaxn{Jn(q0)}.

(i) Under public good allocation, the optimal partnership size n=∞ independent of the project length |q0|.

(ii) Under budget allocation, the optimal partnership size n increases in the project length |q0|.

Increasing the size of the partnership has two effects. First, the expected completion time of the project changes; from Theorem 2 it follows that it decreases, thus increasing each agent’s expected discounted reward, if the project is sufficiently long. Secondly, in equilibrium, each agent will exert less effort to complete the project, which implies that his total expected discounted cost of effort decreases. Proposition 3 shows that if each agent’s reward does not depend on the partnership size (i.e. under public good allocation), then the latter effect always dominates the former, and hence agents are better off the bigger the partnership. Under budget allocation, however, these effects outweigh the decrease in each agent’s reward caused by the increase in the partnership size only if the project is sufficiently long, and consequently, the optimal partnership size increases in the length of the project.

An important assumption underlying Proposition 3 is that the partnership composition is finalized before the agents begin to work. Under public good allocation, this assumption is without loss of generality, because the optimal partnership size is equal to ∞ irrespective of the length of the project. However, it may not be innocuous under budget allocation, where the optimal partnership size does depend on the project length. If the partnership size is allowed to vary with progress, an important modelling assumption is how the rewards of new and exiting members will be determined. While a formal analysis is beyond the scope of this article, abstracting from the above modelling issue and based on Theorem 2, it is reasonable to conjecture that the agents will have incentives to expand the partnership after setbacks, and to decrease its size as the project nears completion.

5. MANAGER’S PROBLEM

Most projects require substantial capital to cover infrastructure and operating costs. For example, the design of a new pharmaceutical drug, in addition to the scientists responsible for the drug design (i.e. the project team), necessitates a laboratory, expensive and maintenance-intensive machinery, as well as support staff. Because individuals are often unable to cover these costs, projects are often run by corporations instead of the project team, which raises the questions of: (i) how to determine the optimal team size; and (ii) how to best incentivize the agents. These questions are addressed in this section, wherein I consider the case in which a third party (to be referred to as a manager) is the residual claimant of the project, and he/she hires a group of agents to undertake it on his/her behalf. Section 5.1 describes the model, Section 5.2 establishes some of the properties of the manager’s problem, and Section 5.3 studies his/her contracting problem.

5.1. The model with a manager

The manager is the residual claimant of the project, he/she is risk neutral, and he/she discounts time at the same rate r>0 as the agents. The project has (expected) length |q0|, and it generates a pay-off U>0 upon completion. To incentivize the agents, at time 0, the manager commits to an incentive contract that specifies the size of the team, denoted by n, a set of milestones q0<Q1<...<QK =0

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(where K ∈N), and for every k ∈{1,...,K}, allocates non-negative payments {Vi,k}ni=1 that are due upon reaching milestone Qk for the first time.18

5.2. The manager’s profit function

I begin by considering the case in which the manager compensates the agents only upon completing the project, and I show in Theorem 3 that his/her problem is well-defined and it satisfies some desirable properties. Then I explain how this result extends to the case in which the manager also rewards the agents for reaching intermediate milestones.

Given the team size n and the agents’ rewards {Vi}ni=1 that are due upon completion of the project (where I can assume without loss of generality that

∑n i=1Vi ≤U), the manager’s expected

discounted profit function can be written as

F (q)= (

U − n∑

i=1 Vi

) Eτ

[ e−rτ |q] ,

where the expectation is taken with respect to the project’s completion time τ , which depends on the agents’ strategies and the stochastic evolution of the project.19 By using the first-order condition for each agent’s equilibrium effort as determined in Section 3, the manager’s expected discounted profit at any given state of the project satisfies

rF (q)= [

n∑ i=1

f ( J ′i (q)

)] F′(q)+ σ

2

2 F′′(q) (5)

defined on (−∞,0] subject to the boundary conditions

lim q→−∞F (q)=0 and F (0)=U −

n∑ i=1

Vi , (6)

where Ji (q) satisfies equation (2) subject to equation (3). The interpretation of these conditions is similar to equation (3). As the state of the project diverges to −∞, its expected completion time diverges to ∞, and because r>0, the manager’s expected discounted profit diminishes to 0. The second condition asserts that the manager’s profit is realized when the project is completed, and it equals her pay-off U less the payments

∑n i=1Vi disbursed to the agents.

Theorem 3. Given ( n, {Vi}ni=1

) , a solution to the manager’s problem defined by equation (5)

subject to the boundary conditions (6) and the agents’ problem as defined in Theorem 1 exists, and it has the following properties:

(i) F (q)>0 and F′(q)>0 for all q. (ii) F (·) is unique if the agents’ rewards are symmetric (i.e. if Vi =Vj for i �= j).

18. The manager’s contracting space is restricted. In principle, the optimal contract should condition each agent’s pay-off on the path of qt (and hence on the completion time of the project). Unfortunately, however, this problem is not tractable; for example, the contracting approach developed in Sannikov (2008) boils down a partial differential equation with n+1 variables (i.e. the state of the project q and the continuation value of each agent), which is intractable even for the case with a single agent. As such, this analysis is left for future research.

19. The subscript k is dropped when K =1 (in which case Q1 =0).

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 201

Now let us discuss how Theorems 1 and 3 extend to the case in which the manager rewards the agents upon reaching intermediate milestones. Recall that he/she can designate a set of milestones, and attach rewards to each milestone that are due as soon as the project reaches the respective milestone for the first time. Let Ji,k (·) denote agent i’s expected discounted pay-off given that the project has reached k−1 milestones, which is defined on (−∞,Qk], and note that it satisfies equation (4) subject to limq→−∞Ji,k (q)=0 and Ji,k (Qk)=Vi,k +Ji,k+1(Qk), where Ji,K+1(0)= 0.The second boundary condition states that upon reaching milestone k, agent i receives the reward attached to that milestone, plus the continuation value from future rewards. Starting with Ji,K (·), it is straightforward that it satisfies the properties of Theorem 1, and in particular, that Ji,K

( Qk−1

) is unique (as long as rewards are symmetric) so that the boundary condition of Ji,K−1(·) at QK−1 is well defined. Proceeding backwards, it follows that for every k, Ji,k (·) satisfies the properties of Theorem 1.

To examine the manager’s problem, let Fk (·) denote his/her expected discounted profit given that the project has reached k−1 milestones, which is defined on (−∞,Qk], and note that it satisfies equation (5) subject to limq→−∞Fk (q)=0 and Fk (Qk)=Fk+1(Qk)−

∑n i=1Vi,k ,

where FK+1(Qk)=U. The second boundary condition states that upon reaching milestone k, the manager receives the continuation value of the project, less the payments that he/she disburses to the agents for reaching this milestone. Again starting with k =K and proceeding backwards, it is straightforward that for all k, Fk (·) satisfies the properties established in Theorem 3.

5.3. Contracting problem

The manager’s problem entails choosing the team size and the agents’ incentive contracts to maximize his/her ex ante expected discounted profit subject to the agents’ incentive compatibility constraints.20 I begin by analysing symmetric contracts. Then I examine how the manager can increase his/her expected discounted profit with asymmetric contracts.

5.3.1. Symmetric contracts. Theorem 4 shows that within the class of symmetric contracts, one can without loss of generality restrict attention to those that compensate the agents only upon completion of the project.

Theorem 4. The optimal symmetric contract compensates the agents only upon completion of the project.

To prove this result, I consider an arbitrary set of milestones and arbitrary rewards attached to each milestone, and I construct an alternative contract that rewards the agents only upon completing the project and renders the manager better off. Intuitively, because rewards are sunk in terms of incentivizing the agents after they are disbursed, and all parties are risk-neutral and they discount time at the same rate, by backloading payments, the manager can provide the same incentives at the early stages of the project, while providing stronger incentives when it is close to completion.21

20. While it is possible to choose the team size directly via the incentive contract (e.g. by setting the reward of n̄<n agents to 0, the manager can effectively decrease the team size to n− n̄), it is analytically more convenient to analyse the two “levers” (for controlling incentives) separately.

21. As shown in part II of the proof of Theorem 4, the agents are also better off if their rewards are backloaded. In other words, each agent could strengthen his/her incentives and increase his/her expected discounted pay-off by depositing any rewards from reaching intermediate milestones in an account with interest rate r, and closing the account upon completion of the project.

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This result is consistent with practice, as evidenced for example by Lewis and Bajari (2014), who study incentive contracts in highway construction projects. Moreover, it is valuable from an analytical perspective, because it reduces the infinite-dimensional problem of determining the team size, the number of milestones, the set of milestones, and the rewards attached to each milestone into a two-dimensional problem, in which the manager only needs to determine his/her budget B=∑ni=1Vi for compensating the agents and the team size. Propositions 4–6 characterize the manager’s optimal budget and his/her optimal team size.

Proposition 4. Suppose that the manager employs n agents whom she compensates symmetri- cally. Then her optimal budget B increases in the length of the project |q0|.

Contemplating an increase in his/her budget, the manager trades off a decrease in her net profit U −B and an increase in the project’s expected present discounted value Eτ

[ e−rτ |q0

] . Because

a longer project takes (on average) a larger amount of time to be completed, a decrease in his/her net profit has a smaller effect on his/her ex ante expected discounted profit the longer the project. Therefore, the benefit from raising the agents’ rewards outweighs the decrease in his/her net profit if and only if the project is sufficiently long, which in turn implies that the manager’s optimal budget increases in the length of the project.

Lemma 1. Suppose that the manager has a fixed budget B and he/she compensates the agents symmetrically. For any m>n, there exists a threshold Tn,m such that he/she prefers employing an m-member team instead of an n-member team if and only if |q0|≥Tn,m.

Given a fixed budget, the manager’s objective is to choose the team size to minimize the expected completion time of the project. This is equivalent to maximizing the aggregate effort of the team along the evolution path of the project. Hence, the intuition behind this result follows from statement (B) of Theorem 2. If the project is short, then on expectation, the aggregate effort of the smaller team will be greater than that of the larger team due to the free-riding effect (on average) dominating the encouragement effect. The opposite is true if the project is long. Figure 3 illustrates an example.

Applying the Monotonicity Theorem of Milgrom and Shannon (1994) leads one to the following Proposition.

Proposition 5. Given a fixed budget to (symmetrically) compensate a group of agents, the manager’s optimal team size n increases in the length of the project |q0|.

Proposition 5 suggests that a larger team is more desirable while the project is far from completion, whereas a smaller team becomes preferable when the project gets close to completion. Therefore, it seems desirable to construct a scheme that dynamically decreases the team size as the project progresses. Suppose that the manager employs two identical agents on a fixed budget, and he/she designates a retirement state R, such that one of the agents is permanently retired (i.e. he/she stops exerting effort) at the first time that the state of the project hits R. From that point onwards, the other agent continues to work alone. Both agents are compensated only upon completion of the project, and the payments (say V1 and V2) are chosen such that the agents are indifferent with respect to who will retire at R; i.e. their expected discounted pay-offs are equal at qt =R.22

22. Note that this is one of many possible retirement schemes. A complete characterization of the optimal dynamic team size management scheme is beyond the scope of this article, and is left for future research.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 203

Figure 3

Illustration of Lemma 1.

Given a fixed budget, the manager’s expected discounted profit is higher if she recruits a 5-member team relative to a

3-member team if and only if the initial state of the project q0 is to the left of the threshold −T3,5; or equivalently, if and only if |q0|≥T3,5.

Proposition 6. Suppose the manager employs two agents with quadratic effort costs. Consider the retirement scheme described above, where the retirement state R>max

{ q0,−T1,2

} and T1,2

is taken from Lemma 1. There exists a threshold �R> |R| such that the manager is better off implementing this retirement scheme relative to allowing both agents to work together until the project is completed if and only if its length |q0|<�R.

First, note that after one agent retires, the other will exert first-best effort until the project is completed. Because the manager’s budget is fixed, this retirement scheme is preferable only if it increases the aggregate effort of the team along the evolution path of the project. A key part of the proof involves showing that agents have weaker incentives before one of them is retired as compared to the case in which they always work together (i.e. when a retirement scheme is not used). Therefore, the benefit from having one agent exert first-best effort after one of them retires outweighs the loss from the two agents exerting less effort before one of them retires (relative to the case in which they always work together) only if the project is sufficiently short. Hence, this retirement scheme is preferable if and only if |q0|<�R.

From an applied perspective, this result should be approached with caution. In this environment, the agents are (effectively) restricted to playing the MPE, whereas in practice, groups are often able to coordinate to a more efficient equilibrium, for example, by monitoring each other’s efforts, thus mitigating the free-rider problem (and hence weakening this result). Moreover, Weber (2006) shows that while efficient coordination does not occur in groups that start off large, it is possible to create efficiently coordinated large groups by starting with small groups that find it easier to coordinate, and adding new members gradually who are aware of the group’s history. Therefore, one should be aware of the tension between the free-riding effect becoming stronger with progress, and the force identified by Weber.

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204 REVIEW OF ECONOMIC STUDIES

5.3.2. Asymmetric contracts. Insofar, I have restricted attention to contracts that compensate the agents symmetrically. However, Proposition 6 suggests that an asymmetric contract that rewards the agents upon reaching intermediate milestones can do better than the best symmetric one if the project is sufficiently short. Indeed, the retirement scheme proposed above can be implemented using the following asymmetric rewards-for-milestones contract.

Remark 6. Let Q1 =R, and suppose that agent 1 receives V as soon as the project is completed, while he/she receives no intermediate rewards. However, agent 2 receives the equilibrium present discounted value of B−V upon hitting R for the first time (i.e. (B−V)Eτ

[ e−rτ |R]),

and he/she receives no further compensation, so that he/she effectively retires at that point. From Proposition 6 we know that there exists a V ∈(0,B) and a threshold�R such that this asymmetric contract is preferable to a symmetric one if and only if |q0|<�R.

It is important to note that while the expected cost of compensating the agents in the above asymmetric contract is equal to B, the actual cost is stochastic, and in fact, it can exceed the project’s pay-off U. As a result, unless the manager is sufficiently solvent, there is a positive probability that he/she will not be able to honour the contract, which will negatively impact the agents’ incentives.

The following result shows that an asymmetric contract may be preferable even if the manager compensates the (identical) agents upon reaching the same milestone; namely, upon completing the project.

Proposition 7. Suppose that the manager has a fixed budget B>0, and he/she employs two agents with quadratic effort costs whom he/she compensates upon completion of the project. Then

for all �∈ (

0, B2

] , there exists a threshold T� such that the manager is better off compensating the

two agents asymmetrically such that V1 = B2 +� and V2 = B2 −� instead of symmetrically, if and only if the length of the project |q0|≤T� .23

Intuitively, asymmetric compensation has two effects: first, it causes an efficiency gain in that the agent who receives the smaller share of the payment has weak incentives to exert effort, and hence the other agent cannot free-ride as much.At the same time however, because effort costs are convex, it causes an efficiency loss, as the total costs to complete the project are minimized when the agents work symmetrically; which occurs in equilibrium only when they are compensated symmetrically. By noting that the efficiency loss is increasing in the length of the project, and that the manager’s objective is to allocate his/her budget so as to maximize the agents’ expected aggregate effort along the evolution path of the project, it follows that the manager prefers to compensate the agents asymmetrically if the project is sufficiently short.

6. CONCLUDING REMARKS

To recap, I study a dynamic problem in which a group of agents collaborate over time to complete a project, which progresses at a rate that depends on the agents’ efforts, and it generates a pay- off upon completion. The analysis provides several testable implications. In the context of the MRF, for example, one should expect that principal investigators will allocate more resources to MRF activities as the goal comes closer into sight. Secondly, in a drug discovery venture for

23. Note that the solution to the agents’ problem need not be unique if the contract is asymmetric. However, this comparative static holds for every solution to equation (5) subject to equations (6), (4), and (3) (if more than one exists).

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 205

instance, the model predicts that the amount of time and resources (both individually and on aggregate) that the scientists allocate to the project will be positively related to the group size at the early stages of the project, and negatively related near completion. Moreover, this prediction is consistent with empirical studies of voluntary contributions by programmers to open-source software projects (Yildirim, 2006). These studies report an increase in the average contributions with the number of programmers, especially in the early stages of the projects, and a decline in the mature stages. Thirdly, the model prescribes that the members of a project team should be compensated asymmetrically if the project is sufficiently short.

In a related paper, Georgiadis et al. (2014) consider the case in which the project size is endogenous. Motivated by projects involving design or quality objectives that are often difficult to define in advance, they examine how the manager’s optimal project size depends on his/her ability to commit to a given project size in advance. In another related paper, Ederer et al. (2014) examine how the team size affects incentives in a discrete public good contribution game using laboratory experiments. Preliminary results support the predictions of Theorem 2.

This article opens several opportunities for future research. First, the optimal contracting problem is an issue that deserves further exploration. As discussed in Section 5, I have considered a restricted contracting space. Intuitively, the optimal contract will be asymmetric, and it will backload payments (i.e. each agent will be compensated only at the end of his/her involvement in the project). However, each agent’s reward should depend on the path of qt , and hence on the completion time of the project. Secondly, the model assumes that efforts are unobservable, and that at every moment, each agent chooses his/her effort level after observing the current state of the project. An interesting extension might consider the case in which the agents can obtain a noisy signal of each other’s effort (by incurring some cost) and the state of the project is observed imperfectly. The former should allow the agents to coordinate to a more efficient equilibrium, while the latter will force the agents to form beliefs about how close the project is to completion, and to choose their strategies based on those beliefs. Finally, from an applied perspective, it may be interesting to examine how a project can be split into subprojects that can be undertaken by separate teams.

APPENDIX A

A. ADDITIONAL RESULTS

A.1. Flow payoffs while the project is in progress

An important assumption of the base model is that the agents are compensated only upon completion of the project. In this section, I extend the model by considering the case in which during any small [t, t+dt) interval while the project is in progress, each agent receives h(qt)dt, in addition to the lump sum reward V upon completion. To make the problem tractable, I shall make the following assumptions about h(·):

Assumption 1. h(·) is thrice continuously differentiable on (−∞,0], it has positive first, second, and third derivatives, and it satisfies limq→−∞h(q)=0 and h(0)≤rV.

Using a similar approach as in Section 3, it follows that in an MPE, the expected discounted pay-off function of agent i satisfies

rJi (q)=max ai

⎧⎨ ⎩h(q)−c(ai)+

⎛ ⎝ n∑

j=1 aj

⎞ ⎠J ′i (q)+ σ 22 J ′′i (q)

⎫⎬ ⎭

subject to equation (3), and his optimal effort level satisfies ai (q)= f ( J ′i (q)

) , where f (·)=c′−1 (max{0, ·}).

Proposition 8 below characterizes the unique MPE of this game, and it shows: (i) that each agent’s effort level is either increasing, or hump-shaped in q; and (ii) the team size comparative static established in Theorem 2 continues to hold.

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206 REVIEW OF ECONOMIC STUDIES

Figure A1

An example in which agents receive flow pay-offs while the project is in progress with h(q)=10eq/2. Observe that effort strategies are hump shaped in q, and the predictions of Theorem 2 continue to hold under both

allocation schemes.

Proposition 8. Suppose that each agent receives a flow pay-off h(q) while the project is in progress, , and h(·) satisfies Assumption 1.

(i) A symmetric MPE for this game exists, it is unique, and it satisfies 0≤Jn (q)≤V and J ′n (q)≥0 for all q. (ii) There exists a threshold ω (not necessarily interior) such that each agent’s effort a′n (q)≥0 if and only if q≤ω.

(iii) Under both allocation schemes and for any m>n, there exists a threshold �n,m (�n,m) such that am (q)≥an (q) (mam (q)≥nan (q)) if and only if q≤�n,m (q≤�n,m).

The intuition why effort can be decreasing in q when the project is close to completion can be explained as follows: far from completion, the agents are incentivized by the future flow pay-offs and the lump sum V upon completion. As the project nears completion, the current flow pay-offs become larger, and hence the agents have less to gain by bringing the project closer to completion, and consequently, they decrease their effort. While establishing conditions under which ω is interior does not seem possible, numerical analysis indicates that this is the case if h(0)/r is sufficiently close to V .

Finally, statement (iii) follows by noting that J ′n (q) being unimodal in q is sufficient for the proof of Theorem 2. Figure A1 illustrates an example.

A.2. Cancellation states

In this section, I consider the case in which the project is cancelled at the first moment that qt hits some (exogenous) cancellation state QC>−∞ and the game ends with the agents receiving 0 pay-off. The expected discounted pay-off for each agent i satisfies equation (4) subject to the boundary conditions

Ji (QC)=0 and Ji (0)=V . In contrast to the model analysed in Section 3, with a finite cancellation state, it need not be the case that J ′i (QC)=0. It follows that all statements of Theorem 1 hold except for (iii) (which asserts that effort increases with progresses).24

Instead, there exists some threshold ω (not necessarily interior), such that a′n (q)≥0 if and only if q≥ω. Similarly, by noting that J ′n (q) being unimodal in q is sufficient for the proof of Theorem 2, it follows that even with

cancellation states, members of a larger team work harder than members of a smaller team, both individually and on aggregate, if and only if the project is sufficiently far from completion. These results are summarized in the Proposition 9 below.

24. This result requires that limq→−∞J ′i (q)=0.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 207

Figure A2

Illustration of the agents’ effort functions given three different cancellation states.

Observe that when QC is small (e.g. QC =−30), effort increases in q. When QC is in an intermediate range (e.g. QC =−10), then effort is U-shaped in q, while it decreases in q if QC is sufficiently large (e.g. QC =−4.5).

Proposition 9. Suppose that the project is cancelled at the first moment such that qt hits a given cancellation state QC>−∞ and the game ends with the agents receiving 0 pay-off.

(i) A symmetric MPE for this game exists, it is unique, and it satisfies 0≤Jn (q)≤V and J ′n (q)≥0 for all q. (ii) There exists a threshold ω (not necessarily interior) such that each agent’s effort a′n (q)≥0 if and only if q≥ω.

(iii) Under both allocation schemes and for any m>n, there exists a threshold �n,m (�n,m) such that am (q)≥an (q) (mam (q)≥nan (q)) if and only if q≤�n,m (q≤�n,m).

While a sharper characterization of the MPE is not possible, numerical analysis indicates that effort increases in q if QC is sufficiently small (i.e. ω=−∞), it is U-shaped in q if QC is in some intermediate range (i.e. ω is interior), while it decreases in q (i.e. ω=0) if QC is close to 0. An example is illustrated in Figure A2.

Intuitively, the agents have incentives to exert effort to: (i) complete the project; and (ii) avoid hitting the cancellation state QC . Moreover, observe that the incentives due to the former (latter) are stronger the closer the project is to completion (to QC ). Therefore, if QC is small, then the latter incentive is weak, so that the agents’ incentives are driven primarily by (i), and effort increases with progress. As QC increases, (ii) becomes stronger, so that effort becomes U-shaped in q, and if QC is sufficiently close to 0, then the incentives from (ii) dominate those from (i), and consequently, effort decreases in q.

A.3. Effort affects drift and variance of stochastic process

A simplifying assumption in the base model is that the variance of the process that governs the evolution of the project (i.e. σ ) does not depend on the agents’ effort levels. As a result, even if no agent ever exerts any effort, the project is completed in finite time with probability 1. To understand the impact of this assumption, in this section, I consider the case in which the project progresses according to

dqt = n∑

i=1 ai,tdt+

√√√√ n∑ i=1

ai,tσdWt .

25 The expected discounted pay-off function of agent i satisfies the HJB equation

rJi (q)=−c ( ai,t )+ ⎛ ⎝ n∑

j=1 aj,t

⎞ ⎠(J ′i (q)+ σ 22 J ′′i (q)

)

25. Note that the total effort of the team is instantly observable here. Therefore, there typically exist non-Markovian equilibria that are sustained via trigger strategies that revert to the MPE after observing a deviation. Moreover, provided that the state qt is verifiable, the team’s total effort becomes contractible.

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208 REVIEW OF ECONOMIC STUDIES

Figure A3

An example in which the agents’ effort influences both the drift and the variance of the stochastic process.

Observe that effort increases in q, and that the predictions of Theorem 2 continue to hold under both allocation schemes.

subject to equation (3). Restricting attention to symmetric MPE and guessing that each agent’s first order condition always

binds, it follows that his effort level satisfies a(q)= f (

J ′ (q)+ σ 22 J ′′ (q) )

. Using a similar approach to that used to prove

Theorem 1, one can show that a non-trivial solution to this ODE exists. However, the MPE need not be unique in this case: unless a single agent is willing to undertake the project single handedly, then there exists another equilibrium in which no agent ever exerts any effort, and the project is never completed.

Unfortunately, analysing how the agents’ effort levels change with progress and how individual and aggregate effort depends on the team size is analytically intractable. However, as illustrated in Figure A3, numerical examples indicate that the main results of the base model continue to hold: effort increases with progress (i.e. a′ (q)≥0 for all q) and the predictions of Theorem 2 continues to hold: under both allocation schemes and for any m>n, there exists a threshold �n,m (�n,m) such that am (q)≥an (q) (mam (q)≥nan (q)) if and only if q≤�n,m (q≤�n,m).

A.4. Equilibria with non-markovian strategies

Insofar, I have restricted attention to Markovian strategies, so that at every moment, each agent’s effort is a function of only the current state of the project qt . This raises the question whether agents can increase their expected discounted pay-off by adopting non-Markovian strategies that at time t depend on the entire evolution path of the project {qs}s≤t . Sannikov and Skrzypacz (2007) study a related model in which the agents can change their actions only at times t = 0, ,2 ,..., where >0 (but small), and the information structure is similar; i.e. the state variable evolves according to a diffusion process whose drift is influenced by the agents’ actions. They show that the pay-offs from the best symmetric PPE converge to the pay-offs corresponding to the MPE as →0 (see their Proposition 5).

A natural, discrete-time analogue of the model considered in this article is one in which at t ∈{0, ,2 ,...} each agent chooses his effort level ai,t at cost c

( ai,t ) , and at t+ the state of the project is equal to qt+ =

qt + (∑n

i=1 ai,t ) +�t+ , where �t+ ∼N

( 0,σ 2

) . In light of the similarities between this model and the model in

Section VI of Sannikov and Skrzypacz (2007), it is reasonable to conjecture that in the continuous-time limit (i.e. as →0), there does not exist a PPE in which agents can achieve a higher expected discounted pay-off than the MPE at any state of the project. However, because a rigorous proof is difficult for the continuous-time game and the focus of this article is on team formation and contracting, a formal analysis of non-Markovian PPE of this game is left for future work.

Nevertheless, it is useful to present some intuition. FollowingAbreu et al. (1986), an optimal PPE involves a collusive regime and a punishment regime, and in every period, the decision whether to remain in the collusive regime or to switch

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 209

is guided by the outcome in that period alone. In the context of this model, at t+ , each agent will base his decision on qt+ −qt

. As decreases, two forces influence the scope of cooperation. First, the gain from a deviation in a single

period decreases, which helps cooperation. On the other hand, because V (

qt+ −qt

) = σ 2

, the agents must decide whether

to switch to the punishment regime by observing noisier information, which increases the probability of type I errors (i.e. triggering a punishment when no deviation has occurred), thus hurting cooperation. As Sannikov and Skrzypacz (2007) show, the latter force becomes overwhelmingly stronger than the former as →0, thus eradicating any gains from cooperation.

A.5. Linear effort costs

The assumption that effort costs are convex affords tractability as it allows for comparative statics despite the fact that the underlying system of HJB equations does not admit a closed-form solution. However, convex effort costs also favour larger teams. Therefore, it is useful to examine how the comparative statics with respect to the team size extend to the case in which effort costs are linear; i.e. c(a)=a. In this case, the marginal value of effort is equal to J ′i (q)−1, so agent i finds it optimal to exert the largest possible effort level if J ′i (q)>1, he is indifferent across any effort level if J ′i (q)=1, and he exerts no effort if J ′i (q)<1. As a result, I shall impose a bound on the maximum effort that each agent can exert: a∈ [0,u]. Moreover, suppose that agents are symmetric, and σ =0 so that the project evolves deterministically.26 This game has multiple MPE: (i) a symmetric MPE with bang-bang strategies; (ii) a symmetric MPE with interior strategies; and (iii) asymmetric MPE. The reader is referred to Section 5.2 of Georgiadis et al. (2014) for details. Because (ii) is sensitive to the assumption that σ =0, I shall focus on the symmetric MPE with bang-bang strategies.27

By using equation (2) subject to equation (3) and the corresponding first-order condition, it follows that there exists a symmetric MPE in which each agent’s discounted pay-off and effort strategy satisfies

Jn (q)= [ − u

r + (

Vn + u r

) e

rq nu

] 1{q≥ψn} and an (q)=u1{q≥ψn} ,

where ψn = nur ln (

nu rVn+u

) . In this equilibrium, the project is completed only if q0 ≥ψn.28 Observe that agents have

stronger incentives the closer the project is to completion, as evidenced by the facts that J ′′n (q)≥0 for all q, and an (q)=1 if and only if q≥ψn. To investigate how the agents’ incentives depend on the team size, one needs to examine how ψn depends on n. This threshold decreases in the team size n under both allocation schemes (i.e. both if Vn =V and Vn =V/n for some V>0) if and only if n is sufficiently small. This implies that members of an (n+1)-member team have stronger incentives relative to those of an n-member team as long as n is sufficiently small.

If agents maximize the team’s rather than their individual discounted pay-off, then the first-best threshold ψ̂n = nu r ln

( u

rVn+u )

, and it is straightforward to show that it decreases in n under both allocation schemes. Therefore, similar

to the case in which effort costs are convex, members of a larger team always have stronger incentives than those of a smaller one.

B. PROOFS

This proof is organized in seven parts. I first show that an MPE for the game defined by equation (1) exists. Next I show that properties (i) through (iii) hold, and that the value functions are infinitely differentiable. Finally, I show that with symmetric agents, the equilibrium is symmetric and unique.

Part I: Existence of an MPE. To show that an MPE exists, it suffices to show that a solution satisfying the system of ordinary nonlinear differential

equations defined by equation (4) subject to the boundary conditions (3) for all i=1,...,n exists.

26. While the corresponding HJB equation can be solved analytically if effort costs are linear, the solution is too complex to obtain the desired comparative statics if σ >0.

27. In the MPE with interior strategies, J ′n (q)=1 for all q, and the equilibrium effort is chosen so as to satisfy this indifference condition. Together with the boundary condition Jn (0)=Vn, this implies that Jn (q)=0 and an (q)=0 for all q≤−Vn. However, such an equilibrium cannot exist if σ >0, because in this case, Jn (q)>0 for all q even if an (q)=0.

28. If q0 ∈ [ψn,ψ1) so that each agent is not willing to undertake the project single handedly, then there exists another equilibrium in which no agent exerts any effort and the project is never completed.

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210 REVIEW OF ECONOMIC STUDIES

To begin, fix some arbitrary N ∈N and rewrite equations (4) and (3) as

J ′′i,N (q) = 2

σ 2

⎡ ⎣rJi,N (q)+c(f (J ′i,N (q)))−

⎛ ⎝ n∑

j=1 f (

J ′j,N (q) )⎞⎠J ′i,N (q)

⎤ ⎦ (B.1)

subject to Ji,N (−N)=0 and Ji,N (0)=Vi for all i. Let gi

( JN ,J ′N

) denote the the RHS of equation (B.1), where JN and J ′N are vectors whose i-th row corresponds

to Ji,N (q) and J ′i,N (q), respectively , and note that gi (·,·) is continuous. Now fix some arbitrary K>0, and define a new function

gi,K ( JN ,J

′ N

)=max{min{gi (JN ,J ′N ),K},−K} . Note that gi,K (·,·) is continuous and bounded. Therefore, by Lemma 4 in Hartman (1960), there exists a solution to J ′′i,N,K =gi,K

( JN,K ,J ′N,K

) on [−N,0] subject to Ji,N,K (−N)=0 and Ji,N,K (0)=Vi for all i. This Lemma, which is due to

Scorza-Dragoni (1935), states:

Let g ( q,J,J ′

) be a continuous and bounded (vector-valued) function for α≤q≤β and arbitrary (J,J ′).

Then, for arbitrary qα and qβ , the system of differential equations J ′′ =g ( q,J,J ′

) has at least one

solution J =J (q) satisfying J (α)=qα and J (β)=qβ . The next part of the proof involves showing that there exists a K̄ such that gi,K

( Ji,N,K (q),J ′i,N,K (q)

)∈(−K̄,K̄) for all i, K , and q, which will imply that the solution Ji,N,K̄ (·) satisfies equation (B.1) for all i. The final step involves showing that a solution exists when N →∞, so that the first boundary condition in equation (B.1) is replaced by limq→−∞Ji (q)=0.

First, I show that 0≤Ji,N,K (q)≤Vi and J ′i,N,K (q)≥0 for all i and q. Because Ji,N,K (0)>Ji,N,K (−N)=0, either Ji,N,K (q)∈ [0,Vi] for all q, or it has an interior extreme point z∗ such that Ji,N,K (z∗) /∈ [0,Vi]. If the former is true, then the desired inequality holds. Suppose the latter is true. By noting that Ji,N,K (·) is at least twice differentiable, J ′i,N,K (z∗)=0, and hence J ′′i,N,K (z∗)=max

{ min

{ 2r σ 2

Ji,N,K (z∗),K } ,−K

} . Suppose z∗ is a global maximum. Then J ′′i,N,K (z∗)≤0�⇒

Ji,N,K (z∗)≤0, which contradicts the fact that Ji,N,K (0)>0. Now suppose that z∗ is a global minimum. Then, J ′′i,N,K (z∗)≥ 0�⇒Ji,N,K (z∗)≥0. Therefore, 0≤Ji,N,K (q)≤Vi for all i and q.

Next, let us focus on J ′i,N,K (·). Suppose that there exists a z∗∗ such that J ′i,N,K (z∗∗)<0. Because Ji,N,K (−N)=0, either Ji,N,K (·) is decreasing on [−N,z∗∗], or it has a local maximum z̄∈(−N,z∗∗). If the former is true, then J ′i,N,K (z∗∗)<0 implies that Ji,N,K (q)<0 for some q∈(−N,z∗∗], which is a contradiction because Ji,N,K (q)≥0 for all q. So the latter must be true. Then, J ′i,N,K (z̄)=0 implies that J ′′i,N,K (z̄)=max

{ min

{ 2r σ 2

Ji,N,K (z̄),K } ,−K

} . However, because z̄ is a

maximum, J ′′i,N,K (z̄)≤0, and together with the fact that Ji,N,K (q)≥0 for all q, this implies that Ji,N,K (q)=0 for all q∈ [−N,z∗∗). But since J ′i,N,K (z∗∗)<0, it follows that Ji,N,K (q)<0 for some q in the neighbourhood of z∗∗, which is a contradiction. Therefore, it must be the case that J ′i,N,K (q)≥0 for all i and q.

The next step involves establishing that there exists an Ā, independent of N and K , such that J ′i,N,K (q)< Ā for all i and q. First, let SN,K (q)=∑ni=1 ∣∣Ji,N,K (q)∣∣. By summing J ′′i,N,K =gi,K (Ji,N,K ,J ′i,N,K ) over i, using that (i) 0≤Ji,N,K (q)≤ Vi and 0≤J ′i,N,K (q)≤S′N,K (q) for all i and q, (ii) f (x)=x1/p, and (iii) c(x)≤xc′ (x) for all x≥0, and letting�=r

∑n i=1 Vi,

we have that for all q

∣∣S′′N,K (q)∣∣ ≤ 2σ 2 n∑

i=1

⎡ ⎣rJi,N,K (q)+c(f (J ′i,N,K (q)))+

⎡ ⎣ n∑

j=1 f (

J ′j,N,K (q) )⎤⎦J ′i,N,K (q)

⎤ ⎦

≤ 2 σ 2

⎡ ⎣�+ n∑

i=1 c′ (

c′−1 ( J ′i,N,K (q)

)) c′−1

( J ′i,N,K (q)

)+S′N,K (q) n∑

j=1 f (

J ′j,N,K (q) )⎤⎦

≤ 4 σ 2

[ �+nS′N,K (q)f

( S′N,K (q)

)]= 4 σ 2

[ �+n(S′N,K (q)) p+1p

] .

By noting that SN,K (0)=∑ni=1 Vi, SN,K (−N)=0, and applying the mean value theorem, it follows that there exists a z∗ ∈ [−N,0] such that S′N,K (z∗)=

∑n i=1 Vi N . It follows that for all z∈ [−N,0]

n∑ i=1

Vi> ∫ z

z∗ S′N,K (q)dq≥

σ 2

4

∫ z z∗

S′N,K (q) S′′N,K (q)

�+n(S′N,K (q)) p+1p dq≥ σ

2

4

∫ S′N (z) 0

s

�+ns p+1p ds,

where I let s=S′N,K (q) and used that S′N,K (q)S′′N,K (q)dq=S′N,K (q)dS′N,K (q). It suffices to show that there exists a Ā<∞ such that σ 24

∫ Ā 0

s

�+ns p+1

p ds=∑ni=1 Vi. This will imply that S′N,K (q)< Ā, and consequently J ′i,N,K (q)≤ Ā for all

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 211

q∈ [−N,0]. To show that such Ā exists, it suffices to show that ∫∞0 s �+ns

p+1 p

ds=∞. First, observe that if p=1, then∫∞ 0

s �+ns2 ds= 12n ln

( �+ns2)∣∣∞0 =∞. By noting that s�+ns2 is bounded for all s∈ [0,1], s

�+ns p+1

p > s �+ns2 for all s>1

and p>1, and ∫∞

0 s

�+ns2 ds=∞, integrating both sides over [0,∞] yields the desired inequality. Because Ā is independent of both N and K , this implies that J ′i,N,K (q)∈

[ 0,Ā ]

for all q∈ [−N,0], N ∈N and K>0. In addition, we know that Ji,N,K (q)∈ [0,Vi] for all q∈ [−N,0], N ∈N and K>0. Now let K̄ =maxi

{ 2 σ 2

[ rVi +c

( f ( Ā ))]}

,

and observe that a solution to J ′′ i,N,K̄

=gi,K̄ (

JN,K̄ ,J ′ N,K̄

) subject to Ji,N,K̄ (−N)=0 and Ji,N,K̄ (0)=Vi for all i exists, and

gi,K̄

( JN,K̄ (q),J

′ N,K̄

(q) ) =gi

( JN,K̄ (q),J

′ N,K̄

(q) )

for all i and q∈ [−N,0]. Therefore, Ji,N,K̄ (·) solves equation (B.1) for all i.

To show that a solution for equation (B.1) exists at the limit as N →∞, I use the Arzela–Ascoli theorem, which states that:

Consider a sequence of real-valued continuous functions (fn)n∈N defined on a closed and bounded interval [a,b] of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence

( fnk )

that converges uniformly.

Recall that 0≤Ji,N (q)≤Vi and that there exists a constant Ā such that 0≤J ′i,N (q)≤ Ā on [−N,0] for all i and N>0. Hence the sequences

{ Ji,N (·)

} and

{ J ′i,N (·)

} are uniformly bounded and equicontinuous on [−N,0]. By applying the

Arzela–Ascoli theorem to a sequence of intervals [−N,0] and letting N →∞, it follows that the system of ODE defined by equation (4) has at least one solution satisfying the boundary conditions (3) for all i.

Finally, note that (i) the RHS of (2) is strictly concave in ai so that the first-order condition is necessary and sufficient for a maximum and (ii) Ji (q)∈ [0,Vi] for all q and i so that the transversality condition limt→∞E

[ e−rtJi (qt)

]=0 is satisfied. Therefore, the verification theorem is satisfied (p. 123 in Chang, 2004), thus ensuring that a solution to the system given by equation (4) subject to equation (3) is indeed optimal for equation (1).

Part II: Ji (q)>0 for all q and i. By the boundary conditions we have that limq→−∞Ji (q)=0 and Ji (0)=Vi>0. Suppose that there exists an interior

z∗ that minimizes Ji (·) on (−∞,0]. Clearly z∗<0. Then J ′i (z∗)=0 and J ′′i (z∗)≥0, which by applying equation (4) imply that

rJi ( z∗ )= σ 2

2 J ′′i ( z∗ )≥0.

Because limq→−∞Ji (q)=0, it follows that Ji (z∗)=0. Next, let z∗∗ =argmaxq≤z∗ {Ji (q)}. If z∗∗ is on the boundary of the desired domain, then Ji (q)=0 for all q≤z∗. Suppose that z∗∗ is interior. Then J ′i (z∗∗)=0 and J ′′i (z∗∗)≤0 imply that Ji (z∗∗)≤0, so that Ji (q)=J ′i (q)=0 for all q<z∗. Using equation (4) we have that∣∣J ′′i (q)∣∣ ≤ 2rσ 2 |Ji (q)|+ 2σ 2 (n+1)f

( Ā )∣∣J ′i (q)∣∣ ,

where this bound follows from part I of the proof. Now let hi (q)=|Ji (q)|+ ∣∣J ′i (q)∣∣, and observe that hi (q)=0 for all

q<z∗, hi (q)≥0 for all q, and

h′i (q)≤ ∣∣J ′i (q)∣∣+∣∣J ′′i (q)∣∣≤ 2rσ 2 |Ji (q)|+ 2σ 2

[ (n+1)f (Ā)+ σ 2

2

]∣∣J ′i (q)∣∣≤Chi (q) , where C = 2

σ 2 max

{ r, (n+1)f (Ā)+ σ 22 }. Fix some ẑ<z∗, and applying the differential form of Grönwall’s inequality

yields hi (q)≤hi ( ẑ ) exp (∫ q

ẑ Cdx )

for all q. Because (i) hi ( ẑ )=0, (ii) exp(∫ qz∗ Cdx)<∞ for all q, and (iii) hi (q)≥0 for all

q, this inequality implies that Ji (q)=0 for all q. However this contradicts the fact that Ji (0)=Vi>0. As a result, Ji (·) cannot have an interior minimum, and there cannot exist a z∗>−∞ such that Ji (q)=0 for all q≤z∗. Hence Ji (q)>0 for all q.

Part III: J ′i (q)>0 for all q and i. Pick a K such that Ji (0)<Ji (K)<Vi. Such K is guaranteed to exist, because Ji (·) is continuous and Ji (0)>0=

limq→−∞Ji (q). Then by the mean-value theorem there exists a z∗ ∈(K,0) such that J ′i (z∗)= Ji(0)−Ji(K)−K = Vi−Ji(K)−K >0. Suppose that there exists a z∗∗ ≤0 such that J ′i (z∗∗)≤0. Then by the intermediate value theorem, there exists a z̄ between z∗ and z∗∗ such that J ′i (z̄)=0, which using equation (4) and part II implies that rJi (z̄)= σ

2

2 J ′′ i (z̄)>0 (i.e. z̄ is a local

minimum). Consider the interval (−∞,z̄]. Because limq→−∞Ji (q)=0, Ji (z̄)>0 and J ′′i (z̄)>0, there exists an interior local maximum ẑ< z̄. Since ẑ is interior, it must be the case that J ′i

( ẑ )=0 and J ′′i (ẑ)≤0, which using equation (4) implies

that Ji ( ẑ )≤0. However, this contradicts the fact that Ji (q)>0 for all q. As a result, there cannot exist a z̄≤0 such that

J ′i (z̄)≤0. Together with part II, this proves properties (i) and (ii).

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212 REVIEW OF ECONOMIC STUDIES

Part IV: Ji (q) is infinitely differentiable on (−∞,0] for all i. By noting that limq→−∞Ji (q)= limq→−∞J ′i (q)=0 for all i, and by twice integrating both sides of equation (B.1)

over the interval (−∞,q], we have that

Ji (q)= ∫ q

−∞

∫ y −∞

2r

σ 2 Ji (z)+ 2

σ 2

⎡ ⎣c(f (J ′i (z)))−

⎛ ⎝ n∑

j=1 f (

J ′j (z) )⎞⎠J ′i (z)

⎤ ⎦dzdy.

Recall that c(a)= ap+1p+1 , f (x)=x1/p, and J ′i (q)>0 for all q. Since Ji (q) and J ′i (q) satisfy equation (4) subject to the boundary conditions (3) for all i, Ji (q) and J ′i (q) are continuous for all i. As a result, the function under the integral is continuous and infinitely differentiable in Ji (z) and J ′i (z) for all i. Because Ji (q) is differentiable twice more than the function under the integral, the desired result follows by induction.

Part V: J ′′i (q)>0 and a′i (q)>0 for all q and i. I have thus far established that for all q, Ji (q)>0 and J ′i (q)>0. By applying the envelope theorem to equation (4)

we have that

rJ ′i (q)= [ f ( J ′i (q)

)+A−i (q)]J ′′i (q)+ σ 22 J ′′′i (q) , (B.2) where A−i (q)=∑nj �=i f (J ′j (q)). Choose some finite z≤0, and let z∗∗ =argmax{J ′i (q) : q≤z}. By part III, J ′i (z∗∗)>0 and because limq→−∞J ′i (q)=0, either z∗∗ =z, or z∗∗ is interior. Suppose z∗∗ is interior. Then J ′′i (z∗∗)=0 and J ′′′i (z∗∗)≤0, which using equation (B.2) implies that J ′i (z∗∗)≤0. However, this contradicts the fact that J ′i (z∗∗)>0, and therefore J ′i (·) does not have an interior maximum on (−∞,z] for any z≤0. Therefore, z∗∗ =z, and since z was chosen arbitrarily, J ′i (·) is strictly increasing; i.e. J ′′i (q)>0 for all q. By differentiating ai (q) and using that J ′i (q)>0 for all q, we have that

d

dq ai (q)= d

dq c′−1

( J ′i (q)

)= J ′′i (q) c′′ ( c′−1

( J ′i (q)

)) >0. Part VI: When the agents are symmetric, the MPE is also symmetric.

Suppose agents are symmetric; i.e. Vi =Vj for all i �= j. In any MPE, {Ji (·)}ni=1 must satisfy equation (4) subject to equation (3). Pick two arbitrary agents i and j, and let (q)=Ji (q)−Jj (q). Observe that (·) is smooth, and limq→−∞ (q)= (0)=0. Therefore, either (·)≡0 on (−∞,0], which implies that Ji (·)≡Jj (·) on (−∞,0] and hence the equilibrium is symmetric, or (·) has at least one interior global extreme point. Suppose the latter is true, and denote this extreme point by z∗. By using equation (4) and the fact that ′ (z∗)=0, we have r (z∗)= σ 22 ′′ (z∗). Suppose that z∗ is a global maximum. Then ′′ (z∗)≤0, which implies that (z∗)≤0. However, because (0)=0 and z∗ is assumed to be a maximum, (z∗)=0. Next, suppose that z∗ is a global minimum. Then ′′ (z∗)≥0, which implies that (z∗)≥0. However, because (0)=0 and z∗ is assumed to be a minimum, (z∗)=0. Therefore, it must be the case that (·)≡0 on (−∞,0]. Since i and j were chosen arbitrarily, Ji (·)≡Jj (·) on (−∞,0] for all i �= j, which implies that the equilibrium is symmetric.

Part VII: Suppose that Vi =Vj for all i �= j. Then the system of ordinary nonlinear differential equations defined by equation (4) subject to equation (3) has at most one solution.

From part VI of the proof, we know that if agents are symmetric, then the MPE is symmetric. Therefore to facilitate exposition, I drop the notation for the i-th agent. Any solution J (·) must satisfy

rJ (q)=−c(f (J ′ (q)))+nf (J ′ (q))J ′ (q)+ σ 2 2

J ′′ (q) subject to lim q→−∞J (q)=0 and J (0)=V .

Suppose that there exist two functions JA (q) ,JB (q) that satisfy the above boundary value problem. Then define D(q)= JA (q)−JB (q), and note that D(·) is smooth and limq→−∞D(q)=D(0)=0. Hence, either D(·)≡0 in which case the proof is complete, or D(·) has an interior global extreme point z∗. Suppose the latter is true. Then D′(z∗)=0, which implies that rD(z∗)= σ 22 D′′ (z∗). Suppose that z∗ is a global maximum. Then D′′ (z∗)≤0⇒D(z∗)≤0, and D(0)=0 implies that D(z∗)=0. Next, suppose that z∗ is a global minimum. Then D′′ (z∗)≥0⇒D(z∗)≥0, and D(0)=0 implies that D(z∗)=0 . Therefore, it must be the case that D(·)≡0 and the proof is complete.

In light of the fact that J ′i (q)>0 for all q, it follows that the first-order condition for each agent’s best response always binds. As a result, any MPE must satisfy the system of ODE defined by equation (4) subject to equation (3). Since this system of ODE has a unique solution with n symmetric, it follows that in this case, the dynamic game defined by equation (1) has a unique MPE. ‖ Proof of Proposition 1. See online Appendix. ‖ Proof of Proposition 2. See online Appendix. ‖

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 213

Proof of Theorem 2. This proof is organized in four parts.

Proof for (i) under public good allocation To begin, let us define Dn,m (q)=Jm (q)−Jn (q), and note that Dn,m (q) is smooth, and Dn,m (0)= limq→−∞Dn,m (q)=

0. Therefore, either Dn,m (·)≡0, or it has an interior extreme point. Suppose the former is true. Then Dn,m (·)≡D′n,m (·)≡ D′′n,m (·)≡0 together with equation (4) implies that f

( J ′n (q)

) J ′n (q)=0 for all q. However, this contradicts Theorem 1 (ii),

so that Dn,m (·) must have an interior extreme point, which I denote by z∗. Then D′n,m (z∗)=0⇒J ′m (z∗)=J ′n (z∗), and using equation (4) yields

rDn,m ( z∗ ) = σ 2

2 D′′n,m

( z∗ )+(m−n)f (J ′n (z∗))J ′n (z∗) .

By noting that any local interior minimum must satisfy D′′n,m (z∗)≥0 and hence Dn,m (z∗)>0, it follows that z∗ must satisfy Dn,m (z∗)≥0. Therefore, Jm (q)≥Jn (q) (i.e. Dn,m (q)≥0) for all q.

I now show that Dn,m (q) is single peaked. Suppose it is not. Then there must exist a local maximum z∗ followed by a local minimum z̄>z∗. Clearly, Dn,m (z̄)<Dn,m (z∗), D′n,m (z̄)=D′n,m (z∗)=0, D′′n,m (z̄)≥0≥D′′n,m (z∗), and by Theorem 1 (iii), J ′n (z̄)>J ′n (z∗). By using equation (4), at z̄ we have

rDn,m (z̄) = σ 2

2 D′′n,m (z̄)+(m−n)f

( J ′m (z̄)

) J ′m (z̄)

> σ 2

2 D′′n,m

( z∗ )+(m−n)f (J ′m (z∗))J ′m (z∗)=rDn,m (z∗) ,

which contradicts the assumption that z∗ is a local maximum and z̄ is a local minimum. Therefore, there exists a�n,m ≤0 such that J ′m (q)≥J ′n (q) (because D′n,m (q)≥0), and consequently am (q)>an (q), if and only if q≤�n,m. Proof for (i) under budget allocation

Recall that under the public good allocation scheme, we had the boundary condition Dn,m (0)=0. This condition is now replaced by Dn,m (0)= Vm − Vn <0. Therefore, Dn,m (·) is either decreasing, or it has at least one extreme point. Using similar arguments as above, it follows that any extreme point z∗ is a global maximum and Dn,m (·) may be at most single peaked. Hence either Dn,m (·) is decreasing in which case �n,m =−∞, or there exists an interior �n,m such that am (q)≥an (q) if and only if q≤�n,m. The details are omitted. Proof for (ii) under public good allocation

Note that c(a)= ap+1p+1 implies that f (x)=x1/p and c(f (x))= x p+1

p

p+1 . As a result, equation (4) can be written for an n-member team as

rJn (q)= (

n− 1 p+1

)( J ′n (q)

) p+1 p + σ

2

2 J ′′n (q) . (B.3)

To compare the total effort of the teams at every state of the project, we need to compare mf ( J ′m (q)

)=(mpJ ′m (q))1/p and nf

( J ′n (q)

)=(npJ ′n (q))1/p. Define D̄n,m (q)=mpJm (q)−npJn (q), and observe that D̄′n,m (q)≥0⇐⇒mam (q)≥nan (q). Note that D̄n,m (0)=(mp −np)V>0 and limq→−∞ D̄n,m (q)=0. As a result, either D̄n,m (q) is increasing for all q, which implies that mam (q)≥nan (q) for all q and hence�n,m =0, or D̄n,m (q) has an interior extreme point z∗. Suppose the latter is true. Then D̄′n,m (z∗)=0 implies that J ′m (z∗)=

( n m

)p J ′n (z∗). Multiplying both sides of equation (B.3) by mp and np for

Jm (·) and Jn (·), respectively, and subtracting the two quantities yields

rD̄n,m ( z∗ ) = np

p+1 (

m−n m

)( J ′n ( z∗ )) p+1

p + σ 2

2 D̄′′n,m

( z∗ ) ,

and observe that the first term in the RHS is strictly positive. Now suppose z∗ is a global minimum. Then D̄′′n,m (z∗)≥0, which implies that D̄n,m (z∗)>0, but this contradicts the facts that limq→−∞ D̄n,m (q)=0 and z∗ is interior. Hence, z∗ must be a global maximum or a local extreme point satisfying D̄n,m (z∗)≥0.

To complete the proof for this case, I now show that D̄n,m (·) can be at most single peaked. Suppose that the contrary is true. Then there exists a local maximum z∗ followed by a local minimum z̄>z∗. Because D̄′n,m (z∗)= D̄′n,m (z̄)= 0, D̄′′n,m (z̄)≥0≥ D̄′′n,m (z∗), and by Theorem 1 (iii) J ′n (z∗)<J ′n (z̄), it follows that D̄n,m (z∗)< D̄n,m (z̄). However, this contradicts the facts that z∗ is a local maximum and z̄ is a local minimum, which implies that D̄n,m (·) is either strictly increasing in which case �n,m =0, or it has a global interior maximum and no other local extreme points, in which case there exists an interior �n,m such that mam (q)≥nan (q) if and only if q≤�n,m. Proof for (ii) under budget allocation

The only difference compared to the proof under public good allocation is the boundary condition at 0; i.e. D̄n,m (0)= mpJm (0)−npJn (0)=

( mp−1 −np−1)V>0 (recall p≥1). As a result, the same proof applies. Note that if p=1 (i.e. effort

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214 REVIEW OF ECONOMIC STUDIES

costs are quadratic), then D̄n,m (0)=0 and hence �n,m must be interior (whereas otherwise D̄n,m (0)>0 and hence �n,m ≤0). ‖ Proof of Proposition 3. Let us first consider the statement under public good allocation. In the proof of Theorem 2 (i), I showed that Dn,n+1 (q)=Jn+1 (q)−Jn (q)≥0 for all q. This implies that Jn+1 (q0)≥Jn (q0) for all n and q0, and hence the optimal partnership size n=∞ for any project length |q0|.

Now consider the statement under budget allocation. In the proof of Theorem 2 (i), I showed that Dn,m (·)=Jm (·)− Jn (·) is either decreasing, or it has exactly one extreme point which must be a maximum. Because limq→−∞Dn,n+1 (q)=0 and Dn,m (0)<0, there exists a threshold Tn,m (may be −∞) such that Jm (q0)≥Jn (q0) if and only if q0 ≤−Tn,m, or equivalently if and only if |q0|≥Tn,m. By noting that the necessary conditions for the Monotonicity Theorem (i.e. Theorem 4) of Milgrom and Shannon (1994) to hold are satisfied, it follows that the optimal partnership size increases in the length of the project |q0|. ‖ Proof of Theorem 3. See online Appendix. ‖ Proof of Theorem 4. To prove this result, first fix a set of arbitrary milestones Q1<...<QK =0 where K is arbitrary but finite, and assume that the manager allocates budget wk>0 for compensating the agents upon reaching milestone k for the first time. Now consider the following two compensation schemes. Let B=∑Kk=1 wk . Under scheme (a), each agent is paid B/n upon completion of the project and receives no intermediate compensation while the project is in progress. Under scheme (b), each agent is paid wk/nEτk [e

rτk |Qk] when qt hits Qk for the first time, where τk denotes the random time to completion given that the current state of the project is Qk . I shall show that the manager is always better off using scheme (a) relative to scheme (b). Note that scheme (b) ensures that the expected total cost for compensating each agent equals B/n to facilitate comparison between the two schemes.

This proof is organized in three parts. In part I, I introduce the necessary functions (i.e. ODEs) that will be necessary for the proof. In part II, I show that each agent exerts higher effort under scheme (a) relative to scheme (b). Finally, in part III, I show that the manager’s expected discounted profit is higher under scheme (a) relative to scheme (b) for any choice of Qk’s and wk’s.

Part I: To begin, I introduce the expected discounted pay-off and discount rate functions that will be necessary for the proof. Under scheme (a), given the current state q, each agent’s expected discounted pay-off satisfies

rJ (q)=−c(f (J ′ (q)))+nf (J ′ (q))J ′ (q)+ σ 2 2

J ′′ (q) subject to lim q→−∞J (q)=0 and J (0)=

B

n .

However, under scheme (b), given the current state q and that k−1 milestones have been reached, each agent’s expected discounted pay-off, which is denoted by Jk (q), satisfies

rJk (q)=−c ( f ( J ′k (q)

))+nf (J ′k (q))J ′k (q)+ σ 22 J ′′k (q) on (−∞,Qk] subject to

lim q→−∞Jk (q)=0 and Jk (Qk)=

wk nEτk [e

rτk |Qk] +Jk+1 (Qk) ,

where JK+1 (QK )=0.29 The second boundary condition states that upon reaching milestone Qk for the first time, each agent is paid wk/nEτk [e

rτk |Qk], and he receives the continuation value Jk+1 (Qk) from future progress. Eventually upon reaching the K-th milestone, the project is completed so that each agent is paid wK/n, and receives no continuation value. Note that due to the stochastic evolution of the project, even after the k-th milestone has been reached for the fist time, the state of the project may drift below Qk . Therefore, the first boundary condition ensures that as q→−∞, the expected time until the project is completed so that each agent collects his/her reward diverges to ∞, which together with the fact that r>0, implies that his/her expected discounted pay-off asymptotes to 0. It follows from Theorem 1 that for each k, Jk (·) exists, it is unique, smooth, strictly positive, strictly increasing, and strictly convex on its domain.

Next, let us denote the expected present discounted value function under scheme (a), given the current state q, by T (q)=Eτ

[ e−rτ |q]. Using the same approach as used to derive the manager’s HJB equation, it follows that

rT (q)=nf (J ′ (q))T ′ (q)+ σ 2 2

T ′′ (q) subject to lim q→−∞T (q)=0 and T (0)=1.

The first boundary condition states that as q→−∞, the expected time until the project is completed diverges to ∞, so that limq→−∞T (q)=0. However, when the project is completed so that q=0, then τ=0 with probability 1, which implies that T (0)=1.

29. Since this proof considers a fixed team size n, we use to subscript k to denote that k−1 milestones have been reached.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 215

Next, let us consider scheme (b). Similarly, we denote the expected present discounted value function, given the current state q and that k−1 milestones have been reached, by Tk (q)=Eτk

[ e−rτk |q]. Then, it follows that

rTk (q)=nf ( J ′k (q)

) T ′k (q)+

σ 2

2 T ′′k (q) on (−∞,Qk]

subject to lim

q→−∞Tk (q)=0 , Tk (Qk)=Tk+1 (Qk) for all k ≤n, where TK+1 (QK )=1. The first boundary condition has the same interpretation as above. The second boundary condition ensures value matching; i.e. that upon reaching milestone k for the first time, Tk (Qk)=Tk+1 (Qk). Using the same approach as used in Theorem 3, it is straightforward to show that T (·) and for each k, Tk (·) exists, it is unique, smooth, strictly positive, and strictly increasing on its domain.

Note that by Jensen’s inequality, 1 Eτk [e

rτk ] ≤Eτk [ e−rτk

] . Therefore, using this inequality, and the second boundary

condition for Jk (·), it follows that Jk (Qk)≤ wkn Tk (Qk)+Jk+1 (Qk). Part II: The next step of the proof is to show that for any k, J (Qk)≥Jk (Qk), and as a consequence of Proposition

1 (i), J ′ (q)≥J ′k (q) for all q≤Qk . This will imply that agents exert higher effort under scheme (a) at every state of the project. To proceed, let us define k (q)=J (q)−Jk (q)− 1n

(∑k−1 i=1 wi

) Tk (q) on (−∞,Qk] for all k, and note that

limq→−∞ k (q)=0 and k (·) is smooth. First, I consider the case in which k =K , and then I proceed by backward induction. Noting that K (QK )=0 (where

QK =0), either K (·)≡0 on (−∞,QK ], or K (·) has some interior global extreme point z. If the former is true, then K (q)=0 for all q≤QK , so that J (QK )≥JK (QK ). Now suppose that the latter is true. Then ′K (z)=0 so that

r K (z) = −c ( f ( J ′ (z)

))+nf (J ′ (z))J ′ (z)+c(f (J ′K (z)))−nf (J ′K (z))J ′K (z) − (

m−1∑ i=1

wi

) f ( J ′K (z)

) T ′K (z)+

σ 2

2 ′′K (z) .

Because ′K (z)=0 implies that (∑k−1

i=1 wi )

T ′K (z)=n [ J ′ (z)−J ′K (z)

] , the above equation can be re-written as

r K (z) = c ( f ( J ′K (z)

))−c(f (J ′ (z)))+nf (J ′ (z))J ′ (z)−nf (J ′K (z))J ′ (z)+ σ 22 ′′K (z) = ⎧⎨ ⎩ [ J ′K (z)

] p+1 p −[J ′ (z)] p+1p p+1 +n

[ J ′ (z)

] p+1 p −n[J ′K (z)] 1p J ′ (z)

⎫⎬ ⎭+ σ

2

2 ′′K (z) .

To show that the term in brackets is strictly positive, note that J (QK )>JK (QK ) so that J ′ (z)>J ′K (z) by Proposition 1 (i), and J ′K (z)>0. Therefore, let x= J

′ K (z)

J ′(z) , where x<1, and observe that the term in brackets is non-negative if and only if

n(p+1)[J ′ (z)] p+1p −[J ′ (z)] p+1p ≥ n(p+1)[J ′K (z)] 1p J ′ (z)−[J ′K (z)] p+1p �⇒n(p+1)−1 ≥ n(p+1)x 1p −x p+1p .

Because the RHS is strictly increasing in x, and it converges to the LHS as x→1, it follows that the above inequality holds.

Suppose that z is a global minimum. Then, ′′K (z)≥0 together with the fact that the term in brackets is strictly positive implies that K (z)>0. Therefore, any interior global minimum must satisfy K (z)≥0, which in turn implies that K (q)≥0 for all q. As a result, K (QK−1)≥0 or equivalently J (QK−1)≥JK (QK−1)+ 1n

(∑K−1 i=1 wi

) TK (QK−1).

Now consider K−1 (·), and note that limq→−∞ K−1 (q)=0. By using the last inequality, that JK−1 (QK−1)≤ wK−1

n TK−1 (QK−1)+JK (QK−1), and TK−1 (QK−1)=TK (QK−1), it follows that

K−1 (QK−1)=J (QK−1)−JK−1 (QK−1)− 1 n

( K−2∑ i=1

wi

) TK−1 (QK−1)≥0.

Therefore, either K−1 (·) is increasing on (−∞,QK−1], or it has some interior global extreme point z<QK−1 such that ′K−1 (z)=0. If the former is true, then K−1 (QK−2)≥0. If the latter is true, then by applying the same technique as above we can again conclude that K−1 (QK−2)≥0.

Proceeding inductively, it follows that for all k ∈{2,...,K}, k (Qk−1)≥0 or equivalently J (Qk−1)≥Jk (Qk−1)+ 1 n

(∑k−1 i=1 wi

) Tk (Qk−1) and using that Jk−1 (Qk−1)≤ wk−1n Tk (Qk−1)+Jk (Qk−1), it follows that J (Qk−1)≥Jk−1 (Qk−1).

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216 REVIEW OF ECONOMIC STUDIES

Finally, by using Proposition 1 (i), it follows that for all k, J ′ (q)≥J ′k (q) for all q≤Qk . In addition, it follows that for all k, J (q)≥Jk (q) for all q≤Qk , which implies that given a fixed expected budget, the agents are better off if their rewards are backloaded.

Part III: Given a fixed expected budget B, the manager’s objective is to maximize Eτ [ e−rτ |q0

] or equivalently T (q0),

where τ denotes the completion time of the project, and it depends on the agents’ strategies, which themselves depend on the set of milestones {Qk}Kk=1 and payments {wk}Kk=1. Since q0<Q1<...<QK , it suffices to show that T (q0)≥T1 (q0) to conclude that given any arbitrary choice of {Qk,wk}Kk=1, the manager is better off compensating the agents only upon completing the project relative to also rewarding them for reaching intermediate milestones.

Define Dk (q)=T (q)−Tk (q)on (−∞,Qk] for all k ∈{1,...,K}, and note that Dk (·) is smooth and limq→−∞Dk (q)=0. Let us begin with the case in which k =K . Note that DK (QK )=0 (where QK =0). So either DK (·)≡0 on (−∞,QK ], or DK (·) has an interior global extreme point z̄<QK . Suppose that z̄ is a global minimum. Then D′K (z̄)=0 so that

rDK (z̄)=n [ J ′ (z̄)−J ′K (z̄)

] T ′ (z̄)+ σ

2

2 D′′K (z̄) .

Recall that J ′ (q)≥J ′k (q) for all q≤Qk from part II. Since z̄ is assumed to be a minimum, it must be true that D′′K (z̄)≥ 0, which implies that DK (z̄)≥0. Therefore, any interior global minimum must satisfy DK (z̄)≥0, which implies that DK (q)≥0 for all q≤QK . As a result, T (QK−1)≥TK (QK−1)=TK−1 (QK−1).

Next, consider DK−1 (·), recall that limq→−∞DK−1 (q)=0, and note that the above inequality implies that DK−1 (QK−1)≥0. By using the same technique as above, it follows that T (QK−2)≥TK−1 (QK−2)=TK−2 (QK−2), and proceeding inductively we obtain that D1 (q)≥0 for all q≤Q1 so that T (q0)≥T1 (q0). ‖ Proof of Proposition 4. See online Appendix. ‖ Proof of Lemma 1. Let us denote the manager’s expected discounted profit when he/she employs n (symmetric) agents by Fn (·), and note that limq→−∞Fn (q)=0 and Fn (0)=U −V>0 for all n. Now let us define n,m (·)=Fm (·)−Fn (·) and note that n,m (·) is smooth and limq→−∞ n,m (q)= n,m (0)=0. Note that either n,m (·)≡0, or n,m (·) has at least one global extreme point. Suppose that the former is true. Then, n,m (q)= ′n,m (q)= ′′n,m (q)=0 for all q, which together with equation (5) implies that [Am (q)−An (q)]F ′n (q)=0 for all q, where An (·)≡nan (·). However, this is a contradiction, because Am (q)>An (q) for at least some q by Theorem 2 (ii), and F ′n (q)>0 for all q by Theorem 3 (i). Therefore, n,m (·) has at least one global extreme point, which I denote by z̄. By using that ′n,m (z̄)=0 and (5), we have that

r n,m (z̄)= [Am (z̄)−An (z̄)]F ′n (z̄)+ σ 2

2 ′′n,m (z̄) .

Recall that F ′n (z̄)>0, and from Theorem 2 (ii) that for each n and m there exists an (interior) threshold �n,m such that Am (q)≥An (q) if and only if q≤�n,m. It follows that z̄ is a global maximum if z̄≤�n,m, while it is a global minimum if z̄≥�n,m. Next observe that if z̄≤�n,m then any local minimum must satisfy n,m (z̄)≥0, while if z̄≥�n,m then any local maximum must satisfy n,m (z̄)≤0. Therefore, either one of the following three cases must be true: (i) n,m (·)≥0 on (−∞,0], or (ii) n,m (·)≤0 on (−∞,0], or (iii) n,m (·) crosses 0 exactly once from above. Therefore, there exists a Tn,m such that n,m (q0)≥0 if and only if q0 ≤−Tn,m, or equivalently the manager is better off employing m>n rather than n agents if and only if |q0|≥Tn,m. By noting that Tn,m =0 under case (i), and Tn,m =∞ under case (ii), the proof is complete. ‖ Proof of Proposition 5. All other parameters held constant, the manager chooses the team size n∈N to maximize his/her expected discounted profit at q0; i.e. he/she chooses n(|q0|)=argmaxn∈N {Fn (q0)}. By noting that the necessary conditions for the Monotonicity Theorem (i.e. Theorem 4) of Milgrom and Shannon (1994) to hold are satisfied, it follows that the optimal team size n(|q0|) is (weakly) increasing in the project length |q0|. ‖ Proof of Propositions 6–9. See online Appendix. ‖

Acknowledgments. I am grateful to the co-editor, Marco Ottaviani, and to three anonymous referees whose comments have immeasurably improved this paper. I am indebted to Simon Board and Chris Tang for their guidance, suggestions and criticisms. I also thank Andy Atkeson, Sushil Bikhchandani, Andrea Bertozzi, Miaomiao Dong, Florian Ederer, Hugo Hopenhayn, Johannes Hörner, Moritz Meyer-Ter-Vehn, Kenny Mirkin, James Mirrlees, Salvatore Nunnari, Ichiro Obara, Tom Palfrey, Gabriela Rubio, Tomasz Sadzik, Yuliy Sannikov, Pierre-Olivier Weill, Bill Zame, Joe Zipkin, as well as seminar participants at Bocconi, BU, Caltech, Northwestern University, NYU, TSE, UCLA, UCSD, the University of Chicago, the University of Michigan, USC, UT Austin, UT Dallas, the Washington University in St. Louis, the 2012 Southwest Economic Theory conference, the 2012 North American Summer Meetings of the Econometric Society, GAMES 2012, and the SITE 2013 Summer Workshop for many insightful comments and suggestions.

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GEORGIADIS PROJECTS AND TEAM DYNAMICS 217

Supplementary Data

Supplementary materials are available at Review of Economic Studies online.

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• Issue Table of Contents
• American Sociological Review, Vol. 78, No. 4 (August 2013) pp. 521-726
• Front Matter
• The Historical Nature of Cities: A Study of Urbanization and Hazardous Waste Accumulation [pp. 521-543]
• Leading Associations: How Individual Characteristics and Team Dynamics Generate Committed Leaders [pp. 544-573]
• Deciding to Cross: Norms and Economics of Unauthorized Migration [pp. 574-603]
• Neighborhood Immigration, Violence, and City-Level Immigrant Political Opportunities [pp. 604-632]
• The Association of Social Class and Lifestyles: Persistence in American Sociability, 1974 to 2010 [pp. 633-661]
• The Grandparents Effect in Social Mobility: Evidence from British Birth Cohort Studies [pp. 662-678]
• Pathways to Empowerment: Repertoires of Women's Activism and Gender Earnings Equality [pp. 679-701]
• Conditional Decoupling: Assessing the Impact of National Human Rights Institutions, 1981 to 2004 [pp. 702-725]
• Back Matter