2009/3 ■ The principal's dilemma Dunia López-Pintado and Juan D. Moreno-Ternero CORE Voie du Roman Pays 34 B-1348 Louvain-la-Neuve, Belgium. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 E-mail: corestat-library@uclouvain.be http://www.uclouvain.be/en-44508.html CORE DISCUSSION PAPER 2009/3 The principal's dilemma Dunia LÓPEZ-PINTADO1 and Juan D. MORENO-TERNERO2 January 2009 Abstract A recurrent dilemma in team management is to select between a team-based and an individual- based wage scheme. We explore such a dilemma in a simple model of production in teams, in which the team members may differ in their effort choices and qualification. We show that, in spite of enhancing output as the basis for payment, a team-based wage scheme might be less profitable for the principal than an individual-based wage scheme. We also highlight a deep misalignment between designing optimal (output-based) incentives for a team and treating its members impartially. Finally, upon introducing the possibility of liquidity constraints in our model, we provide rationale for the so-called “rich get richer” hypothesis. Keywords: team production, management, incentives, effort. JEL Classification: C70, D23, D78 1 Universidad Pablo de Olavide, Spain. 2 Universidad de Malaga and Universidad Pablo de Olavide, Spain; Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium. We are grateful to Ignacio Ortuño-Ortin for insightful discussion and detailed comments. We also thank Miguel A. Ballester, Francis Bloch, Luis Corchón, François Maniquet, Jordi Massó, David Pérez-Castrillo, Pedro Rey-Biel, and the participants of seminars and conferences at Barcelona, Vigo, Louvain-la-Neuve, Evanston, Málaga, Copenhagen and Zaragoza for helpful comments and suggestions. All remaining errors are ours. Financial support from the Spanish Ministry of Science and Innovation (ECO2008-03883/ECON) and Junta de Andalucıa (P06-SEJ-01645; P08-SEJ-04154) is gratefully acknowledged. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors. 1 Introduction A large theoretical literature has emphasized how employers design compen- sation contracts to induce employees to operate in their interests (see, for instance, Prendergast (1999) and the references cited therein). It is typically assumed that workers respond to incentives and that, in particular, paying on the basis of output induces workers to supply more output (e.g., Lazear, 2000). When firms can accurately measure the contribution of individual workers simple piece-rate incentive plans have proven to be effective, pro- vided there is careful measurement of output (e.g., Baker, 1992; Nagin et al., 2002). There are some features of employment relationships, though, that limit the effectiveness of simple piece-rate incentive pay plans and that force managers to consider other forms of incentive pay (e.g., Ichniowski and Shaw, 2003). For instance, in many production processes, output is a function not of the effort of a single worker, but of the combined effort of many workers, and the reward for participation in such teams is likely to be some form of group-based pay (e.g., Holmstrom and Milgrom, 1991). In this paper, we present a simple model of production in teams to ex- plore the dilemma for a principal between guaranteeing (deserving) workers a fixed wage, or making wages contingent on collective (besides individual) performance. The dilemma can be interpreted as a choice between a team- based or an individual-based wage scheme for the management of teams. More precisely, imagine a project that has to be managed by a team of agents each of whom is responsible for a different task. Agents (who might differ in their skills) decide whether to exert effort or not in order to perform their tasks. Exerting effort is a costly action and the higher the skill, the lower the cost. The overall project succeeds with a probability which is an increasing function of the number of agents exerting effort. The principal (who knows each agent’s skill and observes each agent’s effort) chooses between two management scenarios. In the first one, the principal designs a mechanism rewarding agents exerting effort only if the project ends successfully. In the second one, the scheme is not contingent and therefore the principal rewards (deserving) agents independently of the success of the overall project. We show in this paper that the dilemma is tilted in favor of the second management scenario, which is typically more profitable for the principal. More precisely, we show that, under general conditions, the expected benefits of a principal are lower under the contingent scenario than under the non- contingent one. As a byproduct of our analysis, we will also show that agents have the dual preferences over the two scenarios. That is, they prefer 1 the contingent scenario, as their expected benefits will be lower under the contingent scenario than under the non-contingent one. These results have implications that might even be seen as counterintu- itive at first sight. On the one hand, we obtain that, when it comes to the management of teams, naive stimulus measures, such as making payments contingent to the overall success of the team project, might not necessarily be a good option for the principal. On the other hand, agents would typ- ically prefer a scheme that, albeit risky, would enhance incentives further, rather than a scheme guaranteeing them a secure wage. Our analysis will also provide rationale for the so-called “rich get richer” hypothesis. In a market economy, there is no clear implication as to whether economic activities will tend to reduce or else to widen initial wealth dis- parities (e.g., Durham et al., 1998). The so-called Paradox of Power (e.g., Hirshleifer, 1991) is the observation that poorer or weaker contestants im- prove their position relative to richer or stronger opponents. Nevertheless, in some social and economic contexts the reverse occurs, i.e., initially richer and/or more powerful contestants do exploit weaker rivals and thus the rich get richer.1 Our model and results take a side on this debate upon endorsing the latter instance. To elaborate on this, one just has to assume the exis- tence of liquidity constraints in our model. For instance, think of the case of start-up companies without enough stock resources to face wages if there is a team failure (i.e., rewards could not exceed the revenues of the team). In such a case, the principal would be forced to the contingent management option described above. Our results would therefore tell us that a princi- pal without liquidity constraints is likely to obtain higher expected benefits than a principal with liquidity constraints, which is to be interpreted as an instance of the “rich get richer” hypothesis. As it can be inferred from the above, this paper deals with team manage- ment, a topic that has been the object of intense study in economics since Marschak (1955).2 Holmstrom (1982) initiated the interest on moral-hazard problems within teams, an aspect that has received considerable attention 1For instance, the accruing of greater increments of recognition for particular scientific contributions to scientists of considerable repute and the withholding of such recognition from scientists who have not yet made their mark, was already reported in the sociology of science long time ago (e.g., Merton, 1968). Similarly, in the literature on networks, the counterpart to this hypothesis refers to the idea that nodes gain new links with probabili- ties that are proportional to the number of links that they already have. This hypothesis is widely accepted as the explanation for the occurrence of node connectivities following a power-law distribution in systems as diverse as genetic networks, citation networks or the World Wide Web (e.g., Barabàsi and Albert, 1999). 2See Marschak and Radner (1972) for a comprehensive survey. 2 ever since (e.g., McAfee and McMillan, 1991; Itoh, 1993; Che and Yoo, 2001; Winter, 2004). In our case, the moral-hazard problem is absent as agents’ effort is observable. This feature allows us to scrutinize the robustness of some of the results obtained in the mentioned literature. For instance, Win- ter (2004) argues that even when agents are identical and act simultaneously (i.e., with no information among peers) the principal may gain by discrimi- nating among them. Nevertheless, this feature happens in Winter’s model if and only if technology functions exhibit decreasing returns of scale, whereas, as we shall show later, in our model this feature occurs without imposing additional conditions whatsoever on the technology functions. Our benchmark model relies on some key assumptions, such as agents’ neutrality to risk; a flat (as opposed to hierarchical) organization of the team; the viability of wage schemes, or the use of the Nash equilibrium concept to design them. We, nonetheless, explore the extensions of our benchmark model in each of the corresponding directions that arise when relaxing each of these assumptions. In doing so, we test the robustness of our results. The rest of the paper is organized as follows. In Section 2, we set up the benchmark model. In Section 3, we obtain the main results of the paper. In Section 4 we address some extensions of the benchmark model and their corresponding results. We conclude in Section 5. 2 The benchmark model There is a project involving n activities performed by n agents of a team. Each agent decides simultaneously whether to exert effort (invest) or not towards the performance of her activity.3 We denote by δi ∈ {0, 1} the effort decision of agent i, where δi = 1 (0) if player i does (not) exert effort. The cost of exerting effort of agent i is ci. This parameter is to be interpreted as a sign of the agent’s skill (i.e., the lower the cost of exerting effort, the higher the skill). We assume, without loss of generality, that c1 ≤ c2 ≤ · · · ≤ cn. An agent will invest if and only if her expected benefits (i.e., her expected wage minus her cost) are non-negative.4 The project’s technology is a strictly increasing function p : {0, 1, . . . , n} → [0, 1] specifying the probability of success for any given number of agents exerting effort. In doing so, we are 3This is a way of modeling the fact that the team has a flat (rather than hierarchical) organization. In doing so, we are implicitly assuming that communication among agents does not necessarily exist (perhaps reflecting geographical constraints) and that individual effort choices might not be observed by the other agents. 4We assume all agents are risk neutral. We will elaborate on this assumption later in the text. 3 implicitly assuming that agents’ efforts are equally valuable for the success of the project. A principal observes agents’ skills and effort decisions, and designs the wage scheme for the team with the aim of maximizing her benefits. Let β > 0 denote the proceeds for the principal if the project is successful and assume that an unsuccessful project yields 0. Agents are subject to limited liability, which means that the principal cannot impose negative wages on them.5 Let ωi ≥ 0 denote agent i’s wage, which will obviously depend on i’s effort decision.6 The principal will have two options to design the scheme {ωi}i∈N . One in which wages are conditional on the success of the project, which can therefore be considered as a team-based scheme, and another in which they are not, which can therefore be considered as an individual- based scheme. Under each option, the principal will have an optimal group of agents K ⊆ N she would like to see exerting effort. If the principal obeys the vNM axioms, K is obtained, respectively, by solving the programs max k=0,1,...,n p(k) · ( β − ∑ i∈K ωi ) , (1) and max k=0,1,...,n p(k) · β − ∑ i∈K ωi, (2) where, in each case, k denotes the cardinality of K.7 The value of k solving a program of this sort will be referred to as the optimal size of the team. In order to solve the above optimization problems, we shall need to con- struct the mechanism that induces agents within a given group (and only them) to exert effort at the minimum possible cost. A mechanism achiev- ing such an aim will be called an optimal investment-inducing mechanism. Note that any given wage scheme defines a game. An investment-inducing mechanism for a group of agents K would be a wage scheme for which its cor- responding game would have a unique Nash equilibrium in which all agents in K, and only them, exert effort. The optimal investment-inducing mech- 5Limited liability of the agents may arise from workers’ having the freedom to quit or from institutional constraints such as laws banning firms’ exacting payments from workers. In any case, dropping this assumption would not alter the message of our results. 6Note that, given the assumptions, it is natural to assume that ωi = 0 for each i such that δi = 0. 7At the risk of stressing the obvious, note that the values of ωi in (1) and (2) will not coincide and that, therefore, the corresponding optimal sets (and their cardinalities) in each program need not be the same. 4 anism for K would be the least expensive investment-inducing mechanism for K. Once the above optimization problems are solved, the corresponding wage schemes are easily described. More precisely, let K1 and K2 denote the optimal groups obtained from (1) and (2) respectively, and {ω1 i }i∈N and {ω2 i }i∈N denote the corresponding optimal investment-inducing mech- anism for these groups. Then, the wage schemes for each option would be, respectively, ωi = { ω1 i if δi = 1, i ∈ K1, and the project is successful, 0 otherwise, and ωi = { ω1 2 if δi = 1 and i ∈ K2, 0 otherwise. 3 The main results We start this section exploring the principal’s behavior under both scenarios. Proposition 1 The following statements hold: 1. If wages are contingent on the project’s success, the principal solves max k=0,1,...,n p(k) · ( β − k∑ i=1 ci p(i) ) . (3) 2. If wages are not contingent on the project’s success the principal solves max k=0,1,...,n p(k) · β − k∑ i=1 ci. (4) Proof. Since the second statement of the proposition is straightforward, we focus on the first one. Let (ω1, ω2, . . . , ωn) be a scheme inducing a game whose unique Nash equilibrium is (δ1, δ2, . . . , δn) = (1, 1, . . . , 1). Then, in particular, (δ1, δ2, . . . , δn) = (0, 0, . . . , 0) is not a Nash equilibrium for that game, which implies that there exists, at least, an agent i1 ∈ N wanting 5 to deviate by investing. In other words, there exists i1 ∈ N for which p(1)ωi1 ≥ ci1 .8 Equivalently, ωi1 ≥ ci1 p(1) .9 (5) Let us consider now the profile (δ1, δ2, . . . , δn) in which δi1 = 1 and δi = 0 otherwise.10 Since this profile cannot be an equilibrium either, it follows that there exists an agent i2 ∈ N \ {i1} wanting to deviate by investing. In other words, there exists i2 ∈ N \ {i1} for which p(2)ωi2 ≥ ci2 .11 Equivalently, ωi2 ≥ ci2 p(2) . This argument can be subsequently repeated for the remaining profiles to show, in the end, that there exists some permutation π of the set {1, . . . , n}, for which (ω1, ω2, . . . , ωn) ≥ (ωπ1 , . . . , ω π n) = ( c1 p(π(1)) , . . . , cn p(π(n)) ) . Now, since p is increasing, and c1 ≤ c2 ≤ · · · ≤ cn, it is straightforward to show that n∑ i=1 ci p(i) ≤ n∑ i=1 ωπi , for any permutation π, which shows that (ω1, ω2, . . . , ωn) = ( c1 p(1) , . . . , cn p(n) ) , is the optimal investment-inducing mechanism for N (together with the schemes obtained by permuting indices corresponding to agents with a same cost, which would also generate the same overall wage). 8Note that if i1 deviates there would only be one agent exerting effort, which would make p(1) the probability of success and therefore the probability for agent i1 of getting a positive wage ωi1 . 9Note that this condition guarantees that exerting effort is a dominant strategy for agent i1. 10Note that (5) not only guarantees that the profile in which no agent exerts effort is not a Nash equilibrium, but also that no profile in which only one agent (different from i1) exerts effort constitutes a Nash equilibrium either. 11Note that if i2 deviates there would be only two agents exerting effort, which would make p(2) the probability of success and therefore the probability for agent i2 of getting a positive wage ωi2 . 6 Thus, it follows from there that the optimal wage scheme to guarantee that agents in K (and only them) exert effort is given by ωi = ci p(σ(i)) , for all i ∈ K, where σ(i) denotes the rank of i in K, and ωi = 0 for all i /∈ K. Therefore, among the sets with the same cardinality of K, the optimal one for the principal would be {1, 2, . . . , k}. Consequently, the objective program (1) translates into p(k) · ( β − k∑ i=1 ci p(i) ) , as desired. We now compare the objective functions in the statement of Proposition 1. It is straightforward to show that p(k) · ( β − k∑ i=1 ci p(i) ) ≤ p(k) · β − k∑ i=1 ci, with a strict inequality for any k > 1. This proves the following corollary: Corollary 1 The principal gets higher expected benefits when wages are not contingent on the project’s success. Corollary 1 shows the superiority of what we called the non-contingent option. In other words, the principal would prefer guaranteeing workers a salary with no risk whatsoever rather than enhancing incentives further upon linking wages to (collective) performance. Corollary 1 also says, in particular, that when the principal faces liquidity constraints in the design of the wage scheme (i.e., she is forced to make wages contingent on the project’s success) then she typically obtains lower benefits than without liquidity constraints (and therefore avoiding contingency on the project’s success). Thus, Corollary 1 can provide rationale for the so-called “rich get richer” hypothesis, as a “rich” principal (e.g., a company with enough stock resources) increases her benefits with respect to a “poor” principal (e.g., a start-up company) without the option of offering a wage scheme deprived of contingencies. Proposition 1 also provides information regarding the preferences of the team members. More precisely, it shows that the expected benefits for each 7 agent are always 0 under the non-contingent management strategy, whereas under the contingent strategy this is only the case if the agent does not belong to the optimal set K = {1, 2, . . . , k}. Otherwise, agent i ∈ K gets p(k)ωi − ci = ci ( p(k) p(i) − 1 ) ≥ 0. Thus, we have the following corollary: Corollary 2 Agents get higher expected benefits when wages are contingent on the project’s success. Corollary 2 illustrates how principal and agents have opposite preferences regarding management strategies. It says that (risk-neutral) agents prefer a scheme that, albeit risky, would enhance incentives further, rather than a scheme guaranteeing them a secure wage. The information provided by Proposition 1 regarding agents’ (expected) wages shows another interesting feature. On the one hand, we observe that with the optimal non-contingent wage scheme agents would end up receiving the same (actually, zero) benefits despite having different skills (and there- fore receiving different wages). Thus, it would be, ex-post, an (extremely) egalitarian scheme. Things, however, differ with the optimal contingent wage scheme. In such a case, the expected benefits in equilibrium of an agent i exerting effort, would be p(k)ωi − ci = ci ( p(k) p(i) − 1 ) ≥ 0, where, recall that K = {1, 2, . . . , k} is the set of agents exerting effort, and therefore i ∈ K. This implies that agents would typically end up receiving not only different wages, but also different (and positive) benefits. In other words, the optimal contingent wage scheme is not only inegalitarian ex-ante, but also ex-post. As a matter of fact, the inegalitarian aspect of this scheme is obvious as it violates the most fundamental notion of horizontal equity: Corollary 3 The contingent wage scheme violates equal treatment of equals both from an ex-ante and an ex-post viewpoint. More precisely, Corollary 3 says that, under the optimal contingent wage scheme, equal agents (in terms of their skills) might not only receive different wages, but also enjoy different benefits. We conclude the analysis of the benchmark model by showing another difference between both management options. 8 Corollary 4 The optimal size of the team when the principal uses the con- tingent management option is never higher than when using the non-contingent management option. Proof. Let us denote by k (k̂) the optimal size of the team for a princi- pal using the contingent management (non-contingent) management option. Formally, k is the value where (3) is maximized, whereas k̂ is the value where (4) is maximized. We show that k ≤ k̂. By contradiction, assume that the opposite holds. Then, since k maximizes (3), it follows that p(k) · β − k∑ i=1 ci p(i)  ≥ p(k̂) · β − bk∑ i=1 ci p(i)  , and therefore, p(k)β ≥ p(k̂)β + k∑ i=1 p(k) p(i) ci − bk∑ i=1 p(k̂) p(i) ci. Thus, it follows from here that p(k)β − k∑ i=1 ci ≥ p(k̂)β + k∑ i=1 p(k) p(i) ci − bk∑ i=1 p(k̂) p(i) ci − k∑ i=1 ci = p(k̂)β − bk∑ i=1 ci + k∑ i=1 p(k) p(i) ci − bk∑ i=1 p(k̂) p(i) ci − k∑ i=bk+1 ci ≥ p(k̂)β − bk∑ i=1 ci + k∑ i=1 p(k) p(i) ci − bk∑ i=1 p(k̂) p(i) ci − k∑ i=bk+1 p(k) p(i) ci = p(k̂)β − bk∑ i=1 ci + bk∑ i=1 ( p(k) p(i) − p(k̂) p(i) ) ci > p(k̂)β − bk∑ i=1 ci. Thus, p(k)β − k∑ i=1 ci > p(k̂)β − bk∑ i=1 ci, 9 which represents a contradiction, as k̂ maximizes (4). Corollary 4 says that the non-contingent management option typically induces more agents to exert effort. Thus, if the principal values per se that members of the team exert effort, Corollary 4 shows an additional advantage of the non-contingent management strategy. 4 Further results In this section we explore several directions in which our benchmark model could be extended and obtain the corresponding results. 4.1 Intermediate management strategies Our benchmark model only considers two extreme cases of management strategies in which wages are either fully contingent on the project’s success or not contingent at all. A natural question arising from here (and our previous analysis) is whether our results would extend for intermediate cases in which wages are only partially contingent on the project’s success. More precisely, assume now that the principal faces some liquidity constraints and, as a result, only has some stock resources (S), although maybe not enough to face the salaries of all workers if the project is not successful. In this context, a semi-contingent management option consists of a scheme guaranteeing to each deserving agent a fraction si ≤ ci from the stock for sure, as well as a wage ωi contingent on the project’s success. That is, a principal using this option, and obeying the vNM axioms, would maximize the following function: S − ∑ i∈K si + p(k) · ( β − ∑ i∈K ωi ) , (6) where K is the group of agents exerting effort and k its cardinality. A similar argument to the one at the proof of Proposition 1, allows us to show that, provided (s1, . . . , sn) is the allocation of stock resources S, then the wage scheme to guarantee that agents in K (and only them) exert effort would be given by ωi = ci − si p(σ(i)) , (7) for all i ∈ K, where σ(i) denotes the rank of i in K, and ωi = 0 for all i /∈ K. 10 This implies that maximizing (6) is equivalent to maximizing S − k∑ i=1 si + p(k) · ( β − k∑ i=1 ci − si p(i) ) , (8) where (s1, . . . , sn) is such that S = ∑n i=1 si and the optimal set K is K = {1, 2, . . . , k}. Ultimately, this amounts to maximizing k∑ i=1 αi · si, (9) where αi = ( p(k) p(i) − 1 ) for all i = 1, . . . , k. It is straightforward to show that α1 ≥ α2 ≥ · · · ≥ αk. Thus, the optimal semi-contingent investment-inducing mechanism for K = {1, . . . , k} is given by (si, ωi) = (ci, 0) for all i ∈ {1, . . . i0}, (si, ωi) = ( r, ci0+1 − r p(i0 + 1) ) for i = i0 + 1, (si, ωi) = ( 0, ci p(i) ) for all i ∈ {i0 + 2, . . . k}, where i0 is such that 1 ≤ i0 ≤ k, ∑i0 i=1 ci ≤ S < ∑i0+1 i=1 ci and r = S −∑i0 i=1 ci. It is straightforward to show from here that, for S = 0, we obtain the (contingent) mechanism at statement 1 of Proposition 1, whereas for S > ∑k i=1 ci we obtain the (non-contingent) mechanism at statement 2 of Proposition 1. One can easily conclude that principals prefer non-contingent mechanisms to semi-contingent mechanisms and these ones to (fully) contin- gent mechanisms, hence supporting further the “rich get richer” hypothesis mentioned above. 4.2 Equity constraints We have shown that optimal (contingent) wage schemes typically sacrifice the idea of equal treatment of equals. This feature can be seen as another instance of the so-called equality-efficiency trade-off (e.g., Okun, 1975). Nev- ertheless, a wide number of advanced democracies have passed, in the last decades, bills promoting different forms of equality in wage schemes. Thus, 11 it might be reasonable to impose in our model some sort of equity constraints in the design of wage schemes. In this section, we compute the efficiency loss that would arise as a consequence of imposing these constraints. Let i1, · · · , ik be the agents for which the cost ranking has a discontinuity, i.e., i1, · · · , ik are such that ij ≤ ij + 1 for all j and c1 = c2 = · · · = ci1−1 < ci1 = ci1+1 = · · · = ci2−1 < ci2 ≤ · · · ≤ cn. Then, an analogous argument to the proof of Proposition 1 allows us to show that the following scheme constitutes the optimal contingent mechanism, out of those preserving equal treatment of equals. ωi = ci p(1) for all i ∈ {1, . . . i1 − 1}, ωi = ci p(i1) for all i ∈ {i1, . . . i2 − 1}, · · · ωi = ci p(ik) for all i ∈ {ik, . . . n}. Thus, the existence of equality constraints can exacerbate the effect of liq- uidity constraints widening the initial wealth disparities that might exist among principals. As one might expect, the more demanding is the notion of equality being considered, the higher the burden for a principal forced to adopt the con- tingent wage scheme. Instances of more demanding options would be the so-called weak equity axiom introduced by Sen (1973), which imposes a pos- itive discrimination towards disabled individuals (to be interpreted in this model as those with higher costs) or, more generally, the so-called priority axiom, introduced by Moreno-Ternero and Roemer (2006), which imposes a reasonable limit on the scope of Sen’s axiom, by endorsing the view that the discrimination in favor of the disabled should never be to the extent of making the disabled better-off than abler individuals, after the allocation takes place.12 12It is straightforward to show that the non-contingent wage scheme we have described would satisfy the priority axiom. As for the contingent wage scheme, it can actually be shown that, for a given technology function, it satisfies the priority axiom if and only if the distribution of skills in the team is sufficiently dispersed. Thus, in order to be prioritarian with teams having a low level of skill heterogeneity, the contingent wage scheme should be modified (at the expense of the principal) which, in other words, says that the imposition of a more demanding notion of equality (such as the priority axiom), at least under the presence of low skill heterogeneity, would intensify the effect of liquidity constraints. 12 4.3 Risk attitudes Our results are based on the implicit assumption that agents are all risk neutral. The next result summarizes our findings for other risk attitudes. Proposition 2 The following statements hold: 1. If agents are risk averse, the principal gets higher expected benefits when wages are not contingent on the project’s success. 2. There is a critical threshold of agents’ risk lovingness, above (below) which the principal gets lower (higher) expected benefits when wages are not contingent on the project’s success. Proof. We start focussing on the case in which agents are (equally) risk averse. Formally, let u(·) be each agent’s (strictly concave) utility function over sure prospects. Let (ω1, ω2, . . . , ωn) be the scheme inducing a game in this case, whose unique Nash equilibrium is (δ1, δ2, . . . , δn) = (1, 1, . . . , 1). In particular, (δ1, δ2, . . . , δn) = (0, 0, . . . , 0) is not a Nash equilibrium for that game, which implies that there exists, at least, an agent i1 ∈ N wanting to deviate by investing. In other words, there exists i1 ∈ N for which p(1)ui1(ωi1 − ci1) + (1− p(1))ui1(−ci1) ≥ ui1(0). Now, due to the strict concavity of ui1(·), we have that p(1)ui1(ωi1 − ci1) + (1− p(1))ui1(−ci1) < ui(p(1)ωi1 − ci1) Thus, ωi1 > ci1 p(1) . If we proceed iteratively, we show that, for any set of agents K, and i ∈ K, ωi > ci p(σ(i)) , where σ(i) denotes the rank of i in K. Thus, for a given size of the firm (k), the objective function for the principal in this case satisfies that p(k) · ( β − ∑ i∈K ωi ) ≤ p(k) · ( β − k∑ i=1 ci p(i) ) , (10) which shows that the wage scheme would be more expensive under the as- sumption of risk neutral agents, and which, combined with Corollary 1, 13 implies that the principal gets higher expected benefits when wages are not contingent on the project’s success. We now move to the case in which agents are (equally) risk loving. For- mally, let u(·) be each agent’s (strictly convex) utility function over sure prospects. Analogously to the previous case, there exists i1 ∈ N for which p(1)ui1(ωi1 − ci1) + (1 − p(1))ui1(−ci1) ≥ ui1(0). Now, due to the strict convexity of ui1(·), we have that p(1)ui1(ωi1 − ci1) + (1− p(1))ui1(−ci1) > ui1(p(1)ωi1 − ci1). Thus, by continuity, if follows that there exists εi1 > 0 for which, if ωi1 = ci1 p(1) − εi1 , then p(1)ui1(ωi1 − ci1) + (1− p(1))ui1(−ci1) = ui1(0). Obviously, εi1 is correlated to the agent’s degree of risk lovingness in a positive way, i.e., the higher the degree of risk lovingness, the higher εi1 . If we proceed iteratively, we obtain that the optimal wage scheme to guarantee that agents in K (and only them) exert effort is given by ωi = ci p(σ(i)) − εi, for all i ∈ K, where σ(i) denotes the rank of i in K, and ωi = 0 for all i /∈ K. This implies that, in this case, the principal solves max k=0,1,...,n p(k) · ( β − k∑ i=1 ci p(i) − εi ) , (11) which, combined with Corollary 1, shows that the resulting wage scheme would be more (less) expensive than the non-contingent wage scheme for low (high) values of εi, i.e., low (high) degrees of risk lovingness. Incidentally, the argument also shows that there is a sequence of thresholds {εi}, or, for that matter, a precise degree of agents’ risk lovingness, for which the principal would be indifferent between both management options. 4.4 Optimistic principals In our analysis, while designing (contingent) investment-inducing mecha- nisms, we have imposed that the profile in which all agents in a group (and 14 only them) exert effort be the only existing (Nash) equilibrium of the game induced by the corresponding wage scheme. Another (less demanding) op- tion would be to find investment-inducing mechanisms where the profile in which all agents in a group (and only them) exert effort is an equilibrium, but not necessarily the only one. An interpretation for this alternative option is that the principal is optimistic and believes that agents will coordinate on the right equilibrium and therefore does not need to worry about the other existing equilibria. In other words, the principal is not concerned with the strategic uncertainty induced by the presence of multiple equilibria and, more precisely, by the existence of an equilibrium in which no agent exerts effort. We show next that this change alters our results substantially. Let us start by noting that if a principal in this new setting commits to reward agents independently of the project’s success, nothing changes, i.e., the optimal way to do so is also by rewarding each (deserving) agent within the optimal set with her cost and giving nothing to all other agents. Formally, the principal would solve max k=0,1,...,n p(k) · β − k∑ i=1 ci, (12) and if k̂ denotes the solution to this program, the ensuing wage scheme would be ωi = { ci if δi = 1 and i ∈ {1, . . . , k̂}, 0 otherwise. However, if the principal makes wages contingent on the project’s suc- cess, then she would solve the following program: max k=0,1,...,n p(k) · ( β − k∑ i=1 ci p(k) ) . (13) If k̃ denotes the solution to this program, the ensuing wage scheme would be ωi = { ci p(ek) if δi = 1, i ∈ {1, . . . , k̃}, and the project is successful, 0 otherwise. It is not difficult to show that there is indeed strategic uncertainty for the above wage scheme as two equilibria arise; namely, the one in which all agents shirk and the one in which all agents receiving a positive wage exert 15 effort.13 It is also easy to show that this is indeed the cheapest wage scheme supporting the existence of an equilibrium in which all agents in a given group exert effort. It follows from the above that the objective function in programs (12) and (13) is the same, which implies that the expected benefits for an optimistic principal are the same no matter whether she makes wages contingent on the project’s success or not. In particular, this shows that the effect of liquidity constraints would be mitigated at the cost of assuming strategic uncertainty in the design of contingent wage schemes. 4.5 Flat Vs. Hierarchical structures Our benchmark model has assumed a flat organization for the team. As mentioned above, this could be interpreted as a way of assuming that com- munication among agents does not necessarily exist (perhaps reflecting ge- ographical constraints) and that, therefore, individual effort choices might not be observed by the other agents. Nevertheless, an alternative option would be to assume a hierarchical organization in which agents instead of performing their tasks simultaneously do so sequentially. That is, agents would decide sequentially (instead of simultaneously) whether to exert ef- fort or not towards the performance of their activity. In such a case, the natural equilibrium notion to be used, while designing mechanisms, would be the so-called subgame perfect Nash equilibrium. As before, if the principal aims to induce agents exert effort (at the minimum possible cost) commiting to reward them independently of the project’s success, then the optimal way to do so would be by rewarding each (deserving) agent within the optimal set with her cost and giving nothing to all other agents. However, if the principal makes wages contingent on the project’s suc- cess, then the optimal wage scheme to guarantee that agents in K (and only them) exert effort is given by ωi = ci p(k) , (14) for all i ∈ K, where k denotes the cardinality of K, and ωi = 0 for all i /∈ K.14 13Note that an agent i would deviate from the profile in which all agents shirk only if p(1)ωi ≥ ci or, equivalently, ωi ≥ ci p(1) . Now, if ek ≥ 2 (otherwise the scheme is identical to the one in the benchmark model) and since p is strictly increasing, it follows that ci p(1) > ci p(ek) , which shows that no deviation would occur. 14Note that the location of agents in the hierarchy does not affect this result. 16 Thus, an analogous argument to the one in the previous section would allow us to show that the objective function the principal faces to determine the optimal set is the same for both cases, which therefore implies that the expected benefits for a principal of a hierarchical organization would be independent of the fact that wages are contingent on the project’s success. Again, as before, this shows that the effect of liquidity constraints in our model of team production would be alleviated were principals allowed to freely design the architecture of their firms. 4.6 More flexible management strategies Our analysis of contingent wage schemes has imposed that individual con- tracts depend only on the individual effort decision and the success (or failure) of the joint venture, which could be considered as a public signal. This might partially be justified on the grounds that agents (ex post) neither observe their peers’ decisions nor the realization of their wages, and there- fore would not find credible contingent contracts depending on additional aspects to the ones just mentioned. Nevertheless, one might think of alternative contexts of team production in which more flexibility is allowed while designing wage schemes, and ad- ditional information (e.g., private signals that only the principal observes) might be considered. If so, as we show next, and similarly to what we ob- tain in the previous sections, the principal would be indifferent between a contingent scheme and a non-contingent scheme and, therefore, the effect of liquidity constraints would also be mitigated. More precisely, consider the following (contingent) scheme. Let k̃ denote the solution of the following program: max k=0,1,...,n p(k) · ( β − k∑ i=1 ci p(k) ) , and let k̂ denote the total number of agents exerting effort within the team (i.e., k̂ = ∑ i∈N δi). Then, consider the following wage scheme, described by a menu of options depending on k̂, For each k̂ = 1, 2, . . . , n, ωi = { ci p(bk) if δi = 1, i ∈ {1, . . . , k̃}, and the project is successful, 0 otherwise. Note that, with this scheme, individual contracts not only depend on the individual effort decision and the success (or failure) of the team, but 17 also on the number of agents in the team who actually exerted effort, as described in the above menu. It is clear that, on the equilibrium path, i.e., when the agents exerting effort are those in {1, . . . , k̃}, and therefore, k̂ = k̃, the scheme coincides with the one described in Section 4.4. However, the strategic uncertainty inherent there does not occur here, thanks to the off- equilibrium path. More precisely, an agent i would deviate from the profile in which all agents shirk only if p(1)ωi ≥ ci or, equivalently, ωi ≥ ci p(1) , which is precisely the amount that this scheme guarantees to agent i, provided she is the only one exerting effort. A similar argument allows to show that no other profile with some agents in {1, . . . , k̃} shirking is an equilibrium, and that the profile in which all agents in {1, . . . , k̃} exert effort is indeed the unique equilibrium (being the scheme described above the cheapest one to achieve that goal). It then follows that the overall wage the principal faces turns out to be the same than the one under the non-contingent scenario, which shows that allowing more flexibility to design contingent contracts (in particular, to consider the menu described above) also vanishes the comparative advantage of the individual-based wage scheme we have considered throughout this paper. 5 Discussion We have analyzed in this paper a simple model of organization and shown that an optimal management strategy for team production may involve guar- anteeing workers a fixed wage, rather than linking wages to collective (be- sides individual) performance.15 This implies, in particular, that when the principal faces liquidity constraints (and therefore is forced to link wages to the team’s performance) then she is expected to obtain lower benefits than without liquidity constraints. Thus, we provide rationale for the so- called “rich get richer” hypothesis. Our finding can also be interpreted as an argument to endorse individual-based wage schemes rather than team- based wage schemes for the management of teams in which agents differ in their qualification and effort choices. This is in line with some related lit- erature in which it has been suggested that team-based compensation gives rise to problems when workers vary in their ability (e.g., Prendergast, 1999; Meidinger et al., 2003). Furthermore, we show a deep misalignment between optimal team-based 15Our dichotomy is also reminiscent of the so-called make-or-buy decision in the theory of the firm (e.g., Milgrom and Roberts, 1994). 18 compensation schemes and an impartial treatment of its members. More pre- cisely, in our model, an optimal team-based compensation scheme violates equal treatment of equals. In other words, equally talented (and deserving) agents might well receive different wages and end up with different benefits. Discriminating among equals in a team production model is a feature already obtained by Winter (2004) in the case in which principals only care about making all agents exert effort. Nevertheless, Winter (2004) only obtains this feature for the case in which production functions exhibit increasing returns of scale, whereas we obtain it here without imposing additional conditions on the production function.16 We have also shown that if the team is managed under a profit-sharing plan, less agents are expected to exert effort, which might be considered as an additional advantage of individual-based wage schemes for team pro- duction. The lack of success of profit-sharing plans in fostering individual effort, within a context of teams, has been observed, for instance, in medical practices (e.g., Newhouse, 1973) or partnerships in law firms (e.g., Leibowitz and Tollison, 1980). Our main finding is robust to the extension of our benchmark model in three directions; namely, the existence of intermediate management options (with semi-contingent mechanisms), agents’ aversion to risk, and equity con- straints. On the other hand, it is not robust to the extension in three other directions. More precisely, one amounts to assume a hierarchical (rather than a flat) organization for the team. This, however, seems to be an un- realistic assumption nowadays, where hierarchies are being challenged from below or are transforming themselves from top-down structures into more horizontal and collaborative ones (e.g., Friedman, 2007). Another amounts to assume the existence of optimistic principals, who might be satisfied with guaranteeing that all agents exerting effort is an equilibrium, but not the only one. The flaw of this option is its inherent strategic uncertainty induced by the existence of multiple equilibria (and, in particular, the equilibrium in which no agent exerts effort).17 A third one amounts to assume more flex- ibility in the design of contingent wage schemes upon allowing contracts to offer a menu of options depending on the number of agents exerting effort, 16It is worth remarking that Winter (2004) analyzes a different model in which the principal does not observe agents’ effort decisions and hence the moral-hazard problem becomes the priority of the analysis. 17This feature is somehow reminiscent of the literature on public goods (see, for instance, Ledyard (1995) for an excellent survey) in which voluntary contributions mechanisms, that might yield strategic uncertainty, are considered as a way of combating the free-riding equilibrium. 19 which might be plausible in some instances, although not always. Alto- gether, it can be safely argued that even though liquidity constraints can have an important effect in the rise of initial wealth disparities, the more freedom we provide a principal with (either by allowing different team archi- tectures, strategic uncertainty, or more flexibility in the design of contracts) the more she can mitigate their effect. References [1] Baker, G., (1992) Incentive Contracts and Performance Measurement, Journal of Political Economy 100, 598-614 [2] Barabàsi, A-L., Albert, R., (1999) Emergence of Scaling in Random Networks, Science 286, 509-512. [3] Che, Y-K., Yoo, S-W., (2001) Optimal Incentives for Teams, Ameri- can Economic Review 91, 525-541 [4] Durham, Y., Hirshleiffer, J., Smith, V., (1998) Do the Rich Get Richer and the Poor Poorer? Experimental Tests of a Model of Power, Amer- ican Economic Review 88, 979-983 [5] Friedman, T.L., (2007) The World Is Flat: A Brief History of the Twenty-first Century. Picador. New York. [6] Hirshleiffer, J., (1991) The Paradox of Power, Economics and Poli- tics 3, 177-200 [7] Holmstrom, B., (1982) Moral Hazard in Teams, Bell Journal of Eco- nomics 13, 324-340 [8] Holmstrom, B., Milgrom, P., (1991) Multitask Principal-Agent Anal- yses: Incentive Contracts, Asset Ownership and Job Design, Journal of Law, Economics and Organization 7, 24-52 [9] Holmstrom, B., Milgrom, P., (1994) The Firm as an Incentive System, American Economic Review 84, 972-991 [10] Ichniowski, C., Shaw, K., (2003) Beyond Incentive Pay: Insiders Esti- mates of the Value of Complementary Human Resource Management Practices, Journal of Economic Perspectives 17, 155-180. 20 [11] Itoh, H., (1991) Incentives to Help Multi-Agent Situations, Economet- rica 59, 611-636 [12] Lazear E.P., (2000) Performance Pay and Productivity, American Economic Review 90, 1346-1361. [13] Leibowitz, A., Tollison R., (1980) Free Riding, Shirking and Team Pro- duction in Legal Partnerships, Economic Inquiry 18, 380-94. [14] Marschak, J., (1955) Elements for a Theory of Teams, Management Science 1, 127-137. [15] Marschak, J., Radner, R., (1972) Economic Theory of Teams. New Haven: Yale University Press [16] McAfee, P., McMillan, J., (1991) Optimal Contracts for Teams, Inter- national Economic Review 32, 561-577. [17] Meidinger, C., Rullire, J.L., Villeval, M.C., (2003) Does Team-Based Compensation Give Rise to Problems When Agents Vary in Their Abil- ity? Experimental Economics 6, 253-272. [18] Merton, R.K., (1968) The Matthew Effect in Science, Science 159, 56-63. [19] Moreno-Ternero, J.D., Roemer, J.E. (2006), Impartiality, priority and solidarity in the theory of justice. Econometrica 74, 1419-1427. [20] Nagin, D., Rebitzer, J., Sanders, S., Taylor, L., (2002) Monitoring, Mo- tivation and Management: The Determinants of Opportunistic Behav- ior in a Field Experiment, American Economic Review 84, 850-873 [21] Newhouse, J. (1973) The Economics of Group Practice, Journal of Human Resources 8, 37-56. [22] Okun, A., (1975). Equality and Efficiency: The Big Trade-off. Brook- ings Institution Press [23] Prendergast C., (1999) The Provision of Incentives in Firms, Journal of Economic Literature 37, 7-63. [24] Sen, A. (1973), On Economic Inequality, Clarendon Press, Oxford. [25] Winter, E., (2004) Incentives and Discrimination, American Eco- nomic Review 94, 764-773 21 Recent titles CORE Discussion Papers 2008/47. Shin-Huei WANG and Cheng HSIAO. An easy test for two stationary long processes being uncorrelated via AR approximations. 2008/48. David DE LA CROIX. Adult longevity and economic take-off: from Malthus to Ben-Porath. 2008/49. David DE LA CROIX and Gregory PONTHIERE. On the Golden Rule of capital accumulation under endogenous longevity. 2008/50. Jean J. GABSZEWICZ and Skerdilajda ZANAJ. Successive oligopolies and decreasing returns. 2008/51. Marie-Louise LEROUX, Pierre PESTIEAU and Grégory PONTHIERE. Optimal linear taxation under endogenous longevity. 2008/52. Yuri YATSENKO, Raouf BOUCEKKINE and Natali HRITONENKO. Estimating the dynamics of R&D-based growth models. 2008/53. Roland Iwan LUTTENS and Marie-Anne VALFORT. Voting for redistribution under desert- sensitive altruism. 2008/54. Sergei PEKARSKI. Budget deficits and inflation feedback. 2008/55. Raouf BOUCEKKINE, Jacek B. KRAWCZYK and Thomas VALLEE. Towards an understanding of tradeoffs between regional wealth, tightness of a common environmental constraint and the sharing rules. 2008/56. Santanu S. DEY. A note on the split rank of intersection cuts. 2008/57. Yu. NESTEROV. Primal-dual interior-point methods with asymmetric barriers. 2008/58. Marie-Louise LEROUX, Pierre PESTIEAU and Gregory PONTHIERE. Should we subsidize longevity? 2008/59. J. Roderick McCRORIE. The role of Skorokhod space in the development of the econometric analysis of time series. 2008/60. Yu. NESTEROV. Barrier subgradient method. 2008/61. Thierry BRECHET, Johan EYCKMANS, François GERARD, Philippe MARBAIX, Henry TULKENS and Jean-Pascal VAN YPERSELE. The impact of the unilateral EU commitment on the stability of international climate agreements. 2008/62. Giorgia OGGIONI and Yves SMEERS. Average power contracts can mitigate carbon leakage. 2008/63. Jean-Sébastien TANCREZ, Philippe CHEVALIER and Pierre SEMAL. A tight bound on the throughput of queueing networks with blocking. 2008/64. Nicolas GILLIS and François GLINEUR. Nonnegative factorization and the maximum edge biclique problem. 2008/65. Geir B. ASHEIM, Claude D'ASPREMONT and Kuntal BANERJEE. Generalized time- invariant overtaking. 2008/66. Jean-François CAULIER, Ana MAULEON and Vincent VANNETELBOSCH. Contractually stable networks. 2008/67. Jean J. GABSZEWICZ, Filomena GARCIA, Joana PAIS and Joana RESENDE. On Gale and Shapley 'College admissions and stability of marriage'. 2008/68. Axel GAUTIER and Anne YVRANDE-BILLON. Contract renewal as an incentive device. An application to the French urban public transport sector. 2008/69. Yuri YATSENKO and Natali HRITONENKO. Discrete-continuous analysis of optimal equipment replacement. 2008/70. Michel JOURNÉE, Yurii NESTEROV, Peter RICHTÁRIK and Rodolphe SEPULCHRE. Generalized power method for sparse principal component analysis. 2008/71. Toshihiro OKUBO and Pierre M. PICARD. Firms' location under taste and demand heterogeneity. 2008/72. Iwan MEIER and Jeroen V.K. ROMBOUTS. Style rotation and performance persistence of mutual funds. 2008/73. Shin-Huei WANG and Christian M. HAFNER. Estimating autocorrelations in the presence of deterministic trends. 2008/74. Yuri YATSENKO and Natali HRITONENKO. Technological breakthroughs and asset replacement. Recent titles CORE Discussion Papers - continued 2008/75. Julio DÁVILA. The taxation of capital returns in overlapping generations economies without financial assets. 2008/76. Giorgia OGGIONI and Yves SMEERS. Equilibrium models for the carbon leakage problem. 2008/77. Jean-François MERTENS and Anna RUBINCHIK. Intergenerational equity and the discount rate for cost-benefit analysis. 2008/78. Claire DUJARDIN and Florence GOFFETTE-NAGOT. Does public housing occupancy increase unemployment? 2008/79. Sandra PONCET, Walter STEINGRESS and Hylke VANDENBUSSCHE. Financial constraints in China: firm-level evidence. 2008/80. Jean GABSZEWICZ, Salome GVETADZE, Didier LAUSSEL and Patrice PIERETTI. Pubic goods' attractiveness and migrations. 2008/81. Karen CRABBE and Hylke VANDENBUSSCHE. Are your firm's taxes set in Warsaw? Spatial tax competition in Europe. 2008/82. Jean-Sébastien TANCREZ, Benoît ROLAND, Jean-Philippe CORDIER and Fouad RIANE. How stochasticity and emergencies disrupt the surgical schedule. 2008/83. Peter RICHTÁRIK. Approximate level method. 2008/84. Çağatay KAYI and Eve RAMAEKERS. Characterizations of Pareto-efficient, fair, and strategy- proof allocation rules in queueing problems. 2009/1. Carlo ROSA. Forecasting the direction of policy rate changes: The importance of ECB words. 2009/2. Sébastien LAURENT, Jeroen V.K. ROMBOUTS and Francesco VIOLANTE. Consistent ranking of multivariate volatility models. 2009/3. Dunia LÓPEZ-PINTADO and Juan D. MORENO-TERNERO. The principal's dilemma. Books Y. POCHET and L. WOLSEY (eds.) (2006), Production planning by mixed integer programming. New York, Springer-Verlag. P. PESTIEAU (ed.) (2006), The welfare state in the European Union: economic and social perspectives. Oxford, Oxford University Press. H. TULKENS (ed.) (2006), Public goods, environmental externalities and fiscal competition. New York, Springer-Verlag. V. GINSBURGH and D. THROSBY (eds.) (2006), Handbook of the economics of art and culture. Amsterdam, Elsevier. J. GABSZEWICZ (ed.) (2006), La différenciation des produits. Paris, La découverte. L. BAUWENS, W. POHLMEIER and D. VEREDAS (eds.) (2008), High frequency financial econometrics: recent developments. Heidelberg, Physica-Verlag. P. VAN HENTENRYCKE and L. WOLSEY (eds.) (2007), Integration of AI and OR techniques in constraint programming for combinatorial optimization problems. Berlin, Springer. CORE Lecture Series C. GOURIÉROUX and A. MONFORT (1995), Simulation Based Econometric Methods. A. RUBINSTEIN (1996), Lectures on Modeling Bounded Rationality. J. RENEGAR (1999), A Mathematical View of Interior-Point Methods in Convex Optimization. B.D. BERNHEIM and M.D. WHINSTON (1999), Anticompetitive Exclusion and Foreclosure Through Vertical Agreements. D. BIENSTOCK (2001), Potential function methods for approximately solving linear programming problems: theory and practice. R. AMIR (2002), Supermodularity and complementarity in economics. R. WEISMANTEL (2006), Lectures on mixed nonlinear programming.