Contract Complexity, Incentives, and the Value of Delegation

Transcription

1 Contract Complexity, Incentives, and the Vale of Delegation NAHUM MELUMAD Gradate School of Bsiness Colmbia University New York, NY 007 DILIP MOOKHERJEE Boston University Boston, MA 05 STEFAN REICHELSTEIN Haas School of Bsiness University of California, Berkeley Berkeley, CA 9470 and BWZ University of Vienna Vienna, Astria In settings where the revelation principle applies, delegation arrangements are freqently inferior to centralized decision making, and at best achieve the same level of performance. This paper stdies the vale of delegation when organizations are constrained by a bond on the nmber of contingencies in any contract. For a principal-agent setting with asymmetric information, we compare centralized mechanisms where the principal retains sole responsibility for contracting and coordinating prodction, with delegation mechanisms where one agent (a manager) is delegated athority to contract with other agents and coordinate prodction. Relative to centralization, delegation entails a control loss, bt allows decisions to be more sensitive to the manager s private information. We identify circmstances nder which the flexibility gain otweighs the control loss, so that delegation emerges sperior to centralized contracting. Mookherjee and Reichelstein grateflly acknowledge financial spport from National Science Fondation grant SES q 997 Massachsetts Institte of Technology. Jornal of Economics & Management Strategy, Volme 6, Nmber, Smmer 997, 57 89

2 58 Jornal of Economics & Management Strategy. Introdction The Revelation Principle has played a central role in the theory of incentives. It presmes that agents can costlessly enter into comprehensive contracts and process nlimited amonts of information. While the Revelation Principle has proven sefl in nderstanding the incentive constraints imposed by private information, it has also been an impediment to the theory of organization design. As Myerson (98) demonstrated, when this principle applies, a completely centralized organization performs at least as well as any other organizational arrangement. In particlar, any noncooperative eqilibrim otcome of a decentralized organization can be replicated by a centralized revelation mechanism where all agents commnicate their private information to a center and receive instrctions from it concerning actions to be taken. It ths precldes a theory sccessfl in explaining the widespread prevalence of decentralized decision making in organizations and contracting sitations. Optimal revelation mechanisms provide a sefl performance benchmark for evalating particlar organizational arrangements observed in practice. For instance, Myerson (98) and Myerson and Satterthwaite (983) have sed the characterization of optimal revelation mechanisms to examine the relative efficiency of specific action and bargaining procedres. Similarly Baron and Besanko (99), Gilbert and Riordan (995), McAfee and McMillan (995), and Melmad et al. (995) have examined the performance of delegated contracting arrangements, sing optimal revelation mechanisms as the reference point. Nevertheless, as long as the Revelation Principle applies, one cannot explain why centralized revelation mechanisms are not more widely sed in practice. Or approach in this paper is to dispense with one of the more qestionable assmptions nderlying the Revelation Principle. Specifically, we postlate that writing detailed incentive contracts is costly; in particlar, contracts corresponding to revelation mechanisms are prohibitively costly. Under this hypothesis, we compare centralized decision making with a decentralized scheme in which the principal commnicates and contracts with only one agent (the manager) and. Similar perspectives have been adopted in the literatre on reglation (e.g. Laffont and Tirole, 993) and intrafirm resorce allocation (e.g., Harris et al., 98; Kanodia, 993).

3 Contract Complexity, Incentives, and the Vale of Delegation 59 delegates athority to that agent to commnicate and contract with other agent(s). We view a contract as a collection of if..., then... statements that specify the parties obligations (the then... parts) for different contingencies (the if... parts). Depending on the organizational context, a contingency will be determined either by agents reports, or by their action choices, which are pblicly verifiable. Since each contingency has to be specified in advance as part of the formal contract, often with the aid of lawyers, it is plasible to sppose that, everything else remaining the same, contracts with more contingencies are more expensive to write and nderstand. 3 Moreover, in case of a contract dispte, it will take third parties (sch as corts) more time and effort to comprehend contracts involving more contingencies. 4 Or model does not specify an explicit cost for inclding contingencies. We only reqire that contracts corresponding to revelation mechanisms wold be prohibitively costly, since they typically involve an infinite nmber of contingencies. As a conseqence, contracts will be incomplete in the sense that they are not flly state-contingent. 5 To. This is a broader notion of delegation than one in which the principal retains athority over contracting with all agents bt allows the latter to decide their own prodction levels. Isses concerning sch narrower forms of delegation can be raised in a singleagent setting. It trns ot that limiting the nmber of contract contingencies does not serve to distingish sch forms of delegation from centralization, a point discssed in frther detail following Proposition. It is for this reason that we consider the broader notion of delegation in this paper. 3. Hart and Holmstrom (987) have arged that in some contexts the costs of incorporating a continm of contingencies into a contract may not be prohibitively large. For instance, if the parties can specify a simple mathematical fnction for the relationship between contingencies and decisions taken by the principal, the complexity of the contract will not vary significantly between contexts where the domain of this fnction is binary, has a finite nmber of elements, or forms a continm. In certain contexts this may indeed be the case, e.g., where contracts are indexed proportionally to the rate of inflation. However, in many other contexts contingencies are defined by a mltitde of variables (e.g. measres of qantity or qality of otpts delivered, cost conditions, or the state of technology), whose relationship to sbseqent prodction assignments and transfer payments cannot be represented by a simple mathematical formla. In sch sitations it seems plasible that the contract has to be phrased as a list of if..., then... statements, whence the length of the list will affect the costs of writing the contract. 4. For an analysis of costly contract contingencies in a general eqilibrim setting see Dye (985). 5. In the literatre on incomplete contracts the basic assmption is that the state of the world is not verifiable even thogh it is known to the contracting parties. Moreover, parties make specific investments that are sbject to a holdp problem. Parts of this literatre have considered the se of revelation mechanisms to provide the parties with appropriate incentives for specific investment; see, for example, Green and Laffont (988), Rogerson (99), and Che and Hasch (996). While or analysis does not inclde specific investments, it emphasizes asymmetric information and imposes contractal incompleteness for complexity reasons, mch in the spirit of Williamson (975, 985).

4 60 Jornal of Economics & Management Strategy keep the analysis tractable, or model assmes that each agent s private information is real-valed, and that contracts can involve only a finite nmber of contingencies. As a conseqence the information held privately by the agents is rich in comparison with what they can report to others, i.e., the set of contingencies that can be formally incorporated in the contract. 6 In addition, if payments are conditioned on actions chosen by agents, then only a finite nmber of action choices are permitted, in order to limit the complexity of the contract. When contracts cannot be flly state-contingent, delegated contracting gains a potential advantage over centralization. Under delegation, the principal empowers one agent (the manager) to decide the agents prodction assignments. These decisions are made on the basis of better information than the principal cold possibly obtain nder centralization, owing to the limited natre of reports sbmitted by agents. We shall refer to this effect as the flexibility gain inherent in delegated contracting. On the other hand, earlier work on contractal hierarchies has shown that delegated contracting is prone to experience a control loss: the manager (respectively, the prime contractor) has an inherent tendency to exploit his monopsony power to procre too little from sbordinates (the sbcontractors). This is also known in the vertical-integration literatre as the distortion arising from the doble marginalization of rents. Or earlier work [Melmad, Mookherjee, and Reichelstein (MMR hereafter), 995] has shown that absent any restriction on contracts, the principal can overcome the control loss by monitoring external procrement and sbsidizing it at a sitable rate. Under certain additional assmptions regarding the seqence of commnication and contracts, delegation can then be shown to replicate the performance of the optimal revelation mechanism. With limitations on the nmber of contract contingencies it will be impossible for the principal to completely eliminate the control loss that accompanies delegated contracting. As a conseqence, the performance comparison between centralized and delegated contracting depends on the relative magnitde of the flexibility gain and the control loss. Or main reslt in this paper is that, irrespective of the nmber of admissible contract contingencies, the flexibility gain always ot- 6. Ths, a more realistic formlation wold represent the private information of each agent as involving a large nmber of dimensions, all of which cannot be formally represented in the contract. We shall not prse this approach, becase the analysis of optimal contracts wold become significantly more difficlt.

5 Contract Complexity, Incentives, and the Vale of Delegation 6 weighs the control loss, ths implying that delegated contracting is the preferred organizational mode. 7 This reslt reqires a nmber of conditions that inclde risk netrality, an appropriate seqencing of contracts, the absence of collsion between the manager and his sbordinates, and, perhaps most important, the principal s ability to monitor the manager s prodction contribtion. 8 By example we demonstrate that if the principal were to observe only the aggregate team otpt, the reslting control loss nder delegation cold become sfficiently severe so as to reverse or ranking and render centralization sperior. The importance of the other reqirements in determining the performance of delegation was explained in MMR (995) in the context of costless contracting. In the crrent context it is evident that in the absence of one or several of the above reqirements, delegation may perform worse than centralization. 9 In earlier work (MMR, 99) we have also investigated the costs and benefits of delegated contracting, bt in a setting of costly commnication. Specifically, agents were constrained to report messages from a finite set, while contracts cold still be based on a continm of possible action contingencies. Since action choices can serve as a partial sbstitte for commnication of reports, delegated contracting was less constrained in MMR (99). In the setting of this paper, the limit on contract contingencies implies restrictions on both reports and action choices, which additionally constrains delegated contracting compared to the setting where only commnication is restricted. Hence the reslts obtained in this paper are stronger than those obtained previosly. In addition, in this paper we obtain sfficient conditions for delegated contracting to strictly dominate centralization, whereas the earlier paper only established conditions for weak dominance. The paper is organized as follows. The model and the corresponding optimal revelation mechanism are presented in Section. The optimal centralized contract sbject to limited contingencies is developed in the first part of Section 3. In Proposition, we show that becase of 7. Other agency models with limited commnication inclde Green and Laffont (986, 987) and Laffont and Martimort (996). 8. Ability to monitor financial payments to sbordinates (respectively, sbcontractors) wold also sffice, if monitoring of prodction contribtions were not possible, for the reasons explained in MMR (99). 9. For instance, with a sfficiently large nmber of contingencies, the performance levels of either contracting mode will be close to those obtained in a context with an nlimited nmber of contingencies. Since centralized contracting achieves sperior performance when contracting is costless and any of the above reqirements is not met, it follows from a continity argment that centralization will also be sperior when the nmber of contingencies allowed in the contract is large enogh.

6 6 Jornal of Economics & Management Strategy the constraint on contracts the principal prefers that the agents report seqentially rather than simltaneosly (as in a revelation mechanism). Proposition then demonstrates that given the optimal centralized contracting arrangement with seqential reporting (corresponding to any given nmber of contingencies), there exists a delegated contracting scheme involving fewer contingencies, which generates a level of expected profit for the principal that is at least as high. Under some additional assmptions, Proposition 3 shows that profits generated by delegation are strictly higher. Conclding remarks are contained in Section 4.. The Model We consider a team prodction problem involving a principal and two agents. 0 Agent i can take some prodctive action a i from a set A i. The set of jointly feasible actions is the prodct A 4 A A. For every action profile a [ (a, a ), the principal receives a monetary benefit B(a). If agent i takes action a i, he bears the cost C i (a i, i), where i [ Q i [ [ i, i] is a parameter representing the agent s private information (his type). As a conseqence, the cost C i (a i, i) is not observable to anyone except agent i. The cost parameters i are drawn independently from commonly known prior distribtions F i ( i) with positive densities ƒ i ( i). Each agent has a reservation tility level normalized to zero. Agents are compensated for the costs they bear by transfer payments x i made either by the principal or by another agent. All parties are assmed to be risk-netral. Hence, each agent s tility is the (expected) difference between the transfer payment he receives and his cost. Similarly, the principal seeks to maximize the difference between the benefit B(a) and the sm of the (expected) payments made to agents. We make the following assmptions regarding the strctre of prodction and information: (A) C i (a i, i) 4 b i ( i)c i (a i ), where b i ( ) is twice differentiable, strictly positive, strictly increasing, and convex. (A) Each agent has the option of not prodcing at all at zero cost, i.e., there exists 0 [ A i with c i (0) 4 0 and c i (a i ). 0 for all other a i [ A i. 0. The analysis easily extends to the case of n agents, where athority over contracting and coordination with n agents is delegated to the remaining agent. The case n 4 simplifies the notation considerably, so we adopt this version throghot.

7 e Contract Complexity, Incentives, and the Vale of Delegation 63 (A3) b i ( i ) b i ( i ) F i ( i ) ƒ i ( i ) is increasing in i. Assmption (A) postlates that the cost fnction is mltiplicatively separable in the state variable i and the prodction level. The role of this assmption will become clear sbseqently. We do not impose a specific strctre for the prodction sets, except for (A), which allows each agent to be inactive at zero cost. Finally, assmption (A3) is a variant of the sal reqirement that the inverse hazard rate F i ( i)/ƒ i ( i) be increasing in i. Basically, (A3) enables s to solve for optimal contracts on the basis of the local conditions for incentive compatibility only. It also ensres that mens of linear contracts are optimal. It will be sefl to recall the natre of optimal mechanisms when the organization faces no contracting constraints. Then the Revelation Principle applies, implying that the principal can withot loss of generality restrict himself to centralized revelation mechanisms, where agent i reports his entire private information i to the principal, who sbseqently decides transfers x i (, ) and prodction assignments a i (, ). The optimal revelation mechanism solves the following problem: 3 O max E B(a( )) a( ),x( ) i 4 x i ( ) 4 sbject to: for all i [ Q i, i [ $, }: (i) i [ arg max E [x j i( i, j ) C i (a i ( i, j ), i )], i (ii) E j [x i( i, j ) C i (a i ( i, j ), i )] $ 0. The first constraint (i) is the standard incentive compatibility condition, reqiring that trthtelling be a Bayesian-Nash eqilibrim. Constraint (ii) is a participation constraint that reflects the fact that each agent knows his cost prior to contracting. 3 The soltion to the above problem is well known from the literatre on adverse selection. For any given prodction assignments, a( ). See MMR (99) for frther discssion of this assmption.. We se the notation 4 (, ) and E [ ] 4 e [ ] df ( ) df ( ). Similarly, E [ ] 4 e i i i [ ] df i( i). 3. Alternatively, agents receive their private information after contracting, and they can costlessly qit after receiving their information.

8 64 Jornal of Economics & Management Strategy [ (a (, ), a (, )), the payments to the agents are niqely determined by the incentive and participation constraints. Frthermore, the agents will earn informational rents becase of their private information; i.e., their expected payoffs will generally exceed their reservation tility levels. The principal needs to trade off prodction efficiency against the informational rents earned by the agents. The optimal balance between these conflicting objectives is smmarized by the following reslt 4 : Lemma 0 (MMR, 995): Given assmptions (A) (A3), the prodction assignments a*( ) in the optimal revelation mechanism satisfy (a* (, ), a* (, )) [ where h i ( i) [ b i ( i) ` Fi( i) b i (. ƒ i ( i) b i ( i) arg max$ B(a, a ) h ( )c (a ) h ( )c (a )}, (a, a ) Expression () implies that the principal effectively marks p each agent s nit cost by a factor exactly eqal to the modified inverse hazard rate [as defined in assmption (A3)]. The principal calclates optimal prodction assignments with this measre of virtal cost, i.e., h i ( i)c i (a i ), rather than with the tre cost, b i ( i)c i (a i ). In expectation, the markp represents the informational rent that agent i earns de to his private information. () 3. Limited Contract Contingencies In this section, we introdce restrictions on the complexity of contracts. We identify the notion of complexity primarily with the nmber of contingencies in a contract, and secondarily with the nmber of decisions stiplated in each contingency. These collectively define the length of the contract. Or basic notion of complexity is that contracts that involve more contingencies and stiplate more decisions in each contingency are costlier to write and enforce. To illstrate the notion of limited contract contingencies in the context of a revelation mechanism, note that agent i s contract consists 4. The proof of this reslt can be fond in MMR (995).All other proofs are contained in the Appendix.

9 Contract Complexity, Incentives, and the Vale of Delegation 65 of the two fnctions $ a i (, ), x i (, )} (, )[ Q Q. We interpret the pairs (, ) as the contingencies of the contract, since the agent s action choice and payment mst be specified for all possible combinations of reports. A revelation mechanism therefore involves a continm of contingencies, and for each contingency the contract specifies two variables. In the sbseqent analysis we impose the exogenos restriction that the nmber of contract contingencies is finite. This restriction reflects the notion that the cost of preparing and enforcing a contract is increasing in the nmber of contingencies. While it will not be necessary for or prposes to specify an explicit cost fnction, we are de facto rling ot fll revelation mechanisms as prohibitively costly. Or approach is consistent with the notion that in many contracting sitations the set of possible contingencies constittes a large mltidimensional set, yet contracts are written on jst a few smmary variables Centralized Contracting With limitations on the nmber of contract contingencies the Revelation Principle no longer applies, and the search for optimal mechanisms becomes sbstantially more complicated. For instance, it is no longer obvios that both agents shold send their reports simltaneosly to the principal. Indeed, we shall show below that it is typically better from the principal s perspective for them to report seqentially. Frthermore, mechanisms with seqential reporting cold conceivably be dominated by ones with even more complicated message-sending rles, e.g., where agents send reports iteratively. The goal of this paper, thogh, is to compare the performance of specific organizational arrangements that differ with respect to the allocation of decision rights. In that sense, we are not aiming for a theory of optimal mechanisms when contracts are limited in complexity. We begin with the natral extension of a revelation mechanism to a setting with finitely many contingencies. Both agents simltaneosly send a message to the principal, bt each agent is restricted to select messages from a finite set M i, with M i [ k i. The contract stiplates action choices a i (m, m ) and payments x i (m, m ) for all possible reports (m, m ). The reslting contract then involves k k contingencies, 5. Related ideas have been explored in varios strands of the decentralization literatre; for instance, Hrwicz (977, 987), Mont and Reiter (974), Jordan (989) and Baiman (99).

10 66 Jornal of Economics & Management Strategy Principal Agent i Agent i Prodction offers observes i, reports m i and contracts participation transfers decision determined FIGURE. TIME LINE : Simltaneos Centralized Contracting with two variables specified for each contingency. The seqence of moves is represented in Figre. We shall refer to these mechanisms as simltaneos centralized mechanisms; they stiplate for each agent a reporting (message-sending) rle l i : Q i M i and a contract a i : M M A i, x i : M M R. The principal seeks to maximize his expected profit: max E a i (z),x i 3 B(a(l (, ))) (z) i O 4 l i(z) x i ( l (, )) 4 sbject to: for all i, # i #, (i) l i( i) [ arg max m i [ M i E j [x i(m i, l j( j)) b i ( i)c i (a i (m i, l j( j)))], (ii) E j [x i(l i( i), l j( j)) b i ( i)c i (a i (l i( i), l j( j)))] $ 0. () The notation a(l (, )) is shorthand for (a (l ( ), l ( )), a (l ( ), l ( ))), and l denotes (l, l ). The two constraints represent the reqirement that participating in the mechanism and reporting according to the sggested rles form a Bayes-Nash eqilibrim. The main difference from a revelation mechanism is that agents have to select reports from a finite message set that indces partial pooling for sbsets of types. In this sense the complexity constraint restricts the extent to which prodction assignments and payments can be fine-tned to variations in the tre state of the world. As we show below, the mltiplicative separability assmption (A) ensres that any sch mechanism is eqivalent to one where the pattern of pooling represents an interval partition of the type space. In other words, if any two types report the same message, then so do all intermediate types. This greatly simplifies or analysis, since we can confine attention to

11 Contract Complexity, Incentives, and the Vale of Delegation 67 FIGURE. SIMULTANEOUS CENTRALIZED CONTRACTING reporting rles that indce a rectanglar partition of the type space, i.e., a grid. Formally, the principal selects for each agent a partition of 0 the type space Q i into k i intervals ( i, i ], 4,..., k i, with i k 4 i, i i 4 i. All types in the same interval ( i, i ] then report the same message m i [ M i, so that the principal can no longer distingish among them. See Figre for an illstration of the case of for contingencies, with two possible messages per agent. The commnication can be interpreted as each agent reporting that costs are either high or low ; more detail cannot be incorporated, owing to the need to limit the complexity of the contract. In terms of the prodction assignments, the only difference from a revelation mechanism is that the principal is constrained to select the same assignments for all types in the same interval. Given any assignment, the sal argments nderlying the revene eqivalence theorem (see, for example, Myerson, 98) apply, and therefore an agent s expected payment has to be eqal to the virtal cost of his prodction assignment. The principal s optimization problem ths redces to max $ },$ v } $ a v i } k O 4 O k v 4 E v E v B(a v, a v ) O i 4 h i ( i)c i ( a v i ) df ( )df ( ), where the variable a v i denotes agent i s action choice following reports (m, m v ). To smmarize, the problem of selecting an optimal centralized mechanism with simltaneos reporting and k k contract contingencies can be represented as follows: the principal chooses a grid for Q (3)

12 68 Jornal of Economics & Management Strategy Principal Agent i Agent : Agent : Prodction offers observes i participation participation and contracts decision, decision, transfers reports m reports m determined FIGURE 3. TIME LINE : Seqential Centralized Contracting Q consisting of k k rectangles. For each rectangle (contingency) an agent s contract specifies a pair (a v i, x v i ). The action choices a v i can be chosen optimally, while the corresponding payments, by virte of the revene eqivalence theorem, are determined niqely by the incentive and participation constraints. Lemma : The vale of the centralized contracting problem with simltaneos reporting as stated in () is eqal to the vale of the program in (3). With limited contract contingencies it is conceivable that the principal can gain from asking the agents to make their reports in seqence rather than simltaneosly. The second agent cold be instrcted to send his message in response to the first agent s report. The exact seqence of events is described in Figre 3. The advantage of seqential reporting is that the second agent can condition his report on the information revealed by the first agent. This additional flexibility will generally indce better coordination of the agents decisions. The principal s optimization problem with seqential reporting is similar to that with simltaneos reporting. However, agent s reporting rle now takes the form l (, l ( )). Given assmption (A), the principal can again restrict attention to reporting rles that form an interval partition of the type space of each agent. Seqential reporting allows for the partition of Q to depend on the report sent by the first,v v agent. Specifically, types in (, ) can send the report m v, following the report m of the first agent. Figre 4 illstrates the additional flexibility of seqential mechanisms when each agent can send one of two messages. Contrasted with the increased flexibility is the fact that incentive and participation constraints for agent are strengthened, since agent has the advantage of knowing agent s report before responding to

13 Contract Complexity, Incentives, and the Vale of Delegation 69 FIGURE 4. SEQUENTIAL CENTRALIZED CONTRACTING the principal. Specifically, agent s incentive compatibility constraint takes the following form: m v [ arg max m [ M [x (m, m ) b ( )c (a (m, m ))] for all # # k and [ (,v, v ). Frthermore, the participation constraint has to hold ex post, i.e., x (m, m v ) b ( )c (a (m, m v )) $ 0. It trns ot, however, that these seemingly stronger constraints impose no additional cost on the principal. The reason is essentially the same as that described in Mookherjee and Reichelstein (99); an optimal mechanism in which trthfl reporting is a Bayes-Nash eqilibrim and the participation constraints apply in an interim sense can be replaced by another mechanism in which trthtelling is a dominant strategy and the participation constraints hold ex post. Frthermore the new mechanism yields the same expected payoff to the principal. We are now in a position to characterize the principal s expected payoff nder centralized contracting with seqential reporting. Proposition : The vale of centralized contracting with seqential reporting of k and k messages, respectively, is eqal to the vale of the following optimization problem: max $ },$ v } $ a v i } k O 4 O k v4 E E,v,v B(a v, a v ) O i 4 h i ( i)c i (a v i ) df ( )df ( ). (4)

14 70 Jornal of Economics & Management Strategy Comparison of (3) and (4) immediately shows that the principal is better off with seqential than with simltaneos reporting for any given message sets M and M. Problem (4) redces to (3) if one additionally imposes the constraint that the ctoff vales for agent s report v be independent of Agent s report. As mentioned before, thogh, there may well exist other centralized contracting arrangements which dominate even (4). Sch mechanisms may involve iterative message sending and/or delegation of the action choices once agents have sent their messages. The centralized contracting scenario we consider is nonetheless of considerable interest, since it is the direct analoge of a revelation mechanism in a setting with limited contract contingencies. 3. Delegated Contracting Consider now a contracting arrangement wherein the principal only contracts with agent, who in trn is athorized to sbcontract with agent. This arrangement amonts to a three-tier hierarchy in which agent acts as an intermediate principal (a manager or prime contractor ). As in many contexts of procrement contracting, agent (the prime contractor) pays agent (the sbcontractor) ot of her own pocket, and this payment cannot be monitored by the principal. However, as explained above, we assme that the principal does monitor the allocation of prodction assignments between the two agents. When contracting is costly and necessarily incomplete, a threetier hierarchy possesses one potential advantage. Agent can design the sbcontract for agent on the basis of fll information abot her own cost. This may reslt in improved decision making when compared to a centralized arrangement in which decisions can only be based on agent s limited commnication. On the other hand, a potential drawback of delegated contracting is that the principal may experience a control loss, owing to the monopsony power granted to agent. This problem has been identified in varios models of hierarchical contracting, inclding McAfee and McMillan (995) and Qian (994). Or earlier work (MMR, 995) has shown that in the absence of contracting constraints, the principal can alleviate the control loss completely by constrcting a sitable sbsidy for otsorcing to agent. To calibrate the sbsidy correctly, however, the principal has to know agent s tre cost state, as this determines the magnitde of the monopsony distortion. To elicit this information, however, the principal has to offer agent a contract with a continm of contingencies, corresponding to different possible tre vales of. This is of corse not feasible in the present setting. Hence the control loss cannot be eliminated with a limit on the nmber of contract contingencies.

15 Contract Complexity, Incentives, and the Vale of Delegation 7 Principal Agent Agent Agent Agent Transfers offers observes, selects observes, decides a determined Agent participation m and a participation a contract decision sbcontract decision, for Agent reports m ; a determined FIGURE 5. TIME LINE 3: Delegated Contracting The seqence of events nder delegation is described in Figre 5. In particlar, we assme that agent can commit to the prime contract before entering into a sbcontract with agent. 6 The principal designs a contract for agent that stiplates the payment x as a fnction of the benefit level B delivered, the manager s own contribtion a, and m, a message that agent sends to the principal at the time of contracting. The message m is selected from a finite set M, ths allowing the manager to self-select into different incentive contracts depending on his private information. Following the selection of a contract for himself, the manager selects a pair of fnctions $ x (m ), a (m )} m [ M specifying the payment that the manager will make to agent and the associated action choice. The nmber of contingencies in the sbcontract clearly eqals k 4 M, and for each contingency the sbcontract specifies two variables, jst as in the centralized arrangement. The contract for agent is, however, different in that it mst be conditioned on action choices made by her. Hence limits mst be imposed on the range of actions that agent is permitted to select from. For each message m, the principal can specify a control set S(m ) sch that S(m ), A $ B(a, a ) a [ A, a [ A }. (5) The interpretation of this control set is that, having selected m, agent is contractally obligated to deliver a combination of a and B that belongs to the finite set S(m ). The nmber of contingencies in the prime 6. MMR (995) demonstrated that this seqence of contracting is the most beneficial to delegation.

16 7 Jornal of Economics & Management Strategy contract is then given by å m [ M S(m ). 7 In particlar, if M 4 k, and S(m ) 4 k for all m, then the prime contract involves k k contingencies. In this manner, restricting the nmber of contingencies in the delegation contract limits the range of prodction assignments as well as reports that agent can select from. In contrast, the setting of limited commnication stdied in MMR (995) did not impose any limits on the range of prodction assignments. Hence delegation is more constrained when it is contractal complexity rather than commnication capacity which is restricted. Given the prime contract, agent s type and his message m, we denote agent s payoff by p (a, a, x x ( ), m, ) [ x (B(a, a ), a, m ) x b ( )c (a ). l Q The manager s sbseqent problem of designing a sbcontract is to select for agent a reporting rle ( ) : M and fnctions a (m ), a (m ), x (m ) to maximize E [p (a (l ( )), a (l ( )), x (l ( )) x ( ), m, )] (6) sbject to the constraints that for all [ Q, l ( ) [ arg max m [ M [x (m ) b ( )c (a (m ))], x (l ( )) b ( )c (a (l ( ))) $ 0, (a (m ), B(a (m ), a (m ))) [ S(m ) for all m [ M. The choice of sbcontract will generally depend on the prime contract and agent s type. In the terminology of Maskin and Tirole (990), the sbcontracting problem involves an informed principal problem with private vales: i.e., agent s valation of the otpt delivered by agent depends on, regarding which agent is privately informed. Nevertheless, this variable exercises no direct effect on agent s tility. De to the assmed risk netrality, however, the informedprincipal problem has essentially no effect, and we can solve for the optimal sbcontract as if agent s type were common knowledge between the two agents We are implicitly assming here that the contract states that agent will be paid nothing, nless one of the prodction combinations in S(m ) is delivered, and this exclsion clase is costless to write into the contract, or alternatively is eqivalent to the cost of writing a single contingency. 8. As Maskin and Tirole demonstrate in their paper, this is generally tre when both parties have tility that is linear in money. The only advantage that agent cold conceivably extract from his private information is to delay revelation of his own type ntil after agent responds to the offered contract, thereby imposing risk on agent associated with the realization of. With risk netrality this serves no prpose at all.

17 Contract Complexity, Incentives, and the Vale of Delegation 73 We denote the principal s maximm payoff nder centralized contracting with seqential reporting and k [ k k contingencies for both agents by p c (k k, k k ). Formally, p c (k k, k k ) is the optimal vale of the objective fnction in (4). The principal s maximm payoff nder delegated contracting is denoted by p d (k, k*) when the prime contract and sbcontract involve k and k* contingencies, respectively. The main reslt of this paper is the following. p # p Proposition : Centralized contracting with seqential reporting is dominated by delegated contracting in the sense that c (k k, k k ) d (k k, k ) for all k and k. Note that in comparison with the corresponding centralized contract, delegation involves fewer contingencies in the contract for agent, while the nmber of contingencies in agent s contract remains the same. The message sets of all agents have the same size: hence the reslt here implies also the speriority of delegation when only commnication is limited, as in MMR (99). Moreover, the sbcontract specifies the same variables per contingency, i.e., the prodction assignment and the payment for agent. For agent, a contingency is defined by the realization of the three variables B, a, m. For each contingency the prime contract stiplates jst one variable, i.e., x, rather than the two variables (x, a ) nder centralization. Hence the delegation mechanism is less complex than a centralized mechanism, jdging not only by the total nmber of contingencies, bt also by the nmber of decisions the principal takes in each contingency. The proof of Proposition is constrctive. Given a centralized contract, the principal can design the prime contract nder delegation so as to leave agent s message space M and the domain of possible prodction decisions $ a v, a v } nchanged. Given that agent sends message m, his discretion over prodction decisions is restricted exactly to the set of possible prodction assignments reslting nder centralization following the message m. In other words, the control set S(m ) is the set of k possible prodction assignments $ a v, a v } k v 4 in the centralized contract. Despite this restriction, thogh, agent now decides on the prodction assignments with fll knowledge of his own type, while nder centralization the principal only knows that agent s cost belongs to the interval (, ]. This featre entails a flexibility gain for delegation Note that this advantage wold contine to exist even if the centralized regime were permitted to se iterative reporting schemes. While the latter increase the flexibility of the centralized arrangement by narrowing down the ncertainty faced by the principal concerning agent s type, they cannot eliminate it entirely, owing to the finite restriction on message spaces.

18 74 Jornal of Economics & Management Strategy FIGURE 6. SUPERIORITY OF DELEGATION OVER CENTRALIZATION On the other hand, recall that the control loss problem inherent in delegation is that agent has a tendency to se his monopsony power to bias prodction in his favor when allocating tasks between agent and himself. The principal can conteract this distortion by sbsidizing procrement from agent. The limitation on the nmber of contract contingencies however restricts the set of reports that agent can send regarding his own type, ths preventing the principal from calibrating the otsorcing sbsidy correctly. Nevertheless, we find that the control loss can be ameliorated sfficiently so as to indce agent to make niformly better decisions than wold have reslted nder centralized contracts with the same nmber of contingencies. To frther illstrate the reasoning nderlying Proposition, consider an example in which there are two competing sppliers with cost fnctions C i (a i, i) 4 ia i, respectively. Sppose both i s are distribted according to the same density ƒ( i), reslting in common virtal cost fnctions h( i). The benefit fnction B(, ) takes the form B(a, a ) 4 max $ a, a }, and the benefit level is exogenosly fixed at B 4. The optimal revelation mechanism then calls for agent to deliver a 4 if and only if h( ) # h( ), which is eqivalent to #. Otherwise agent is asked to deliver a 4, so the low-cost prodcer is given the entire contract. 0 The optimal prodction assignments are ths described by the diagonal in Figre 6: this necessitates 0. It can be shown that in this setting the optimal revelation mechanism can be implemented by a second price action.

19 g Contract Complexity, Incentives, and the Vale of Delegation 75 each agent reporting his entire private information, i.e., the realization of i. In trn this cases the corresponding contract to contain a continm of contingencies. Consider now a scenario in which there are for contingencies in the centralized contract, with two messages from each agent. It is then no longer possible to implement second-best prodction assignments, as the principal cannot identify which prodcer has the lower cost in all states of the world. We know from Proposition that the optimal centralized contract with seqential reporting will take the form of a rectanglar partition of the cost parameter space, rather than the diagonal partition. Heristically, it will involve choosing the rectanglar partition that is the closest approximation to the diagonal partition. In particlar, if agent has cost #, he reports m. Hearing this, agent reports m if # and m otherwise. In the former case agent prodces the entire qantity, and in the latter case agent does. If., agent reports m. Sbseqently agent reports m if # and m otherwise. The vales of and are also the prices at which agent is offered the contract depending on agent s message. The ctoff levels and are sch that for some type ( * or ** ) of agent in the corresponding interval, prodction decisions are exactly second-best (see Figre 6). In delegated contracting, the principal can se the following mechanism to achieve higher profits. Agent is asked to se the same reporting rle as in the two-tier mechanism. Following message m, he is offered an incentive scheme of the form x 4 b a g, where b is a fixed payment, and g is a tax parameter chosen as follows: g 4 h( * ) * and g 4 h( ** ) **. Conseqently, agent will make a take-it-or-leave-it offer to agent at the price h ( ` g ) if #, or, at the price h ( ` g ) if.. Hence, the nit tax i mitigates the control loss, since in the absence of monitoring agent wold have chosen the price h ( ). By constrction, the coefficients g and g are sch that the types * and ** contine to implement the second-best decisions. Figre 6 shows that the prodction assignments corresponding to the h ( ` g i ) rle are niformly closer to the diagonal than the prodction assignments in the optimal centralized mechanism as represented by the flat lines corresponding to the constant prices and. As a conseqence, the prodction decisions are niformly better from the principal s standpoint. When commnication rather than the nmber of contract contingencies is limited, as in or previos paper MMR (99), the only constraint pertains to the size of each agent s message set; prodction decisions are not restricted at all. Agent can then select arbitrary

20 76 Jornal of Economics & Management Strategy prodction assignments nder delegation, based on his own private information, besides reports from agent. The flexibility advantage of delegation is then frther boosted: a continm of different assignments can be selected by agent, corresponding to different vales of, given any report sent by agent. In the preceding example, however, this additional flexibility was not valable, since the aggregate benefit level desired by the principal was fixed exogenosly. More generally, if the principal s aggregate benefit were a continos fnction of team otpt, the added flexibility in prodction assignments wold enable achievement of a higher expected payoff. The contrast between the context of limited commnication and limited contract contingencies can be illstrated more sharply in a single-agent setting. Sppose the only restriction is on the size of the agent s message space; then delegation of decisions concerning prodction to the agent wold generally generate sperior performance to centralization. This is becase delegation permits a continm of possible prodction decisions that are fine-tned to the agent s private information, nlike centralization, where the principal decides on prodction based on reports from the agent. On the other hand, when the only constraint is on the nmber of contract contingencies, centralized and delegated decision making are eqivalent. The reason is that delegation also mst be characterized by a finite range of possible prodction levels that can be selected by the agent, owing to the need to limit the nmber of contingencies in the delegation contract. In the example sed above to illstrate the reasoning of Proposition, delegation achieved a strictly higher expected payoff for the principal than did centralization. The following reslt provides sfficient conditions for this to hold in more general settings. In stating the reslt, we shall need the following definitions. First, say that commnication with agent i is valable nder centralization if the principal s payoff in (4) increases as one moves from k i 4 to some k i. (holding k j, j ¹ i, fixed). Second, given a vector of nit prices p 4 ( p, p ) for the two inpts, let a( p) denote the vector of inpt levels (a, a ) that maximizes B(a) p c (a ) p c (a ). We define the prodction strctre to be nonseparable if for any pair of inpt prices p, p both lying in [h ( ), h ( )] [h ( ), h ( )], and any corresponding pair of optimal prodction assignments a( p), a( p ), if a ( p) ¹ a ( p ), then also a ( p) ¹ a ( p ). In words, when the prodction contribtion procred from one agent changes, so mst the contribtion procred from the other agent. This condition ths reqireness jointness in the prodction of the

21 Contract Complexity, Incentives, and the Vale of Delegation 77 two agents, necessitating coordination of their respective prodction assignments. Proposition 3: Sppose that commnication with agent is valable nder centralization, and the prodction strctre is nonseparable. Then the dominance of delegated contracting over centralization in Proposition is strict, i.e., p c (k k, k k ), p d (k k, k ) for all k, k. To conclde this section, we illstrate the importance of the assmption that the principal can monitor the prodction assignments selected by agent. If the principal were to consider only the aggregate benefit level delivered, bt not agent s contribtion a, the control loss might otweigh the flexibility gain inherent in delegation. To demonstrate this point, consider again the above procrement example where the two agents are ex ante identical and the desired benefit level is fixed exogenosly at B 4. The optimal delegation mechanism will then be of the following simple form: the principal offers agent a fixed payment in the amont of x 4 h ( )F(h ( )) ` [ F(h ( ))]. (7) The amont x is calclated to exactly compensate the highest cost type ( ) of agent for delivery cost incrred. This prime contract indces agent to make agent a take-it-orleave-it offer to prodce the entire assignment at the price h ( ). Hence agent ends p as the exclsive prodcer if and only if # h ( ). The flexibility of the three-tier mechanism is embodied in this decision rle, since the allocation of prodction is decided on the basis of agent s exact information abot his own cost. This flexibility gain, however, is not necessarily desirable, since the reslting decision rle is distorted relative to the second-best rle. The following seqential centralized mechanism performs better than the above delegation mechanism. Agent is asked to report m if #. The principal awards the entire contract to agent if, h ( ) [, and to agent otherwise. If agent reports m (i.e.,. ), the contract is awarded to agent if and only if # h ( ). Note that if the optimal prodction assignments of the two agents were completely separable, e.g., if the principal s benefit fnction B were additively separable in a and a, then delegation cold not lead to any flexibility gain at all: better information abot the cost conditions in the prodction of one agent wold not generate any improvements in the contract designed for the other agent. Hence some degree of nonseparability is necessary for delegation to achieve more flexible prodction assignments than centralization.

22 78 Jornal of Economics & Management Strategy (i.e., agent reports m ). The principal s expected cost from this centralized mechanism eqals K( )F( ) ` K( ) [ F( )], (8) where K( ) [ h ( )F(h ( )) ` [ F(h ( ))]. The expected cost in (8) is clearly less than in (7), i.e., with 4 h ( ) and x 4 K( ); therefore the principal is better off nder centralization. By adopting a centralized mechanism, the principal loses some prodction efficiency associated with delegation. At the same time, thogh, types, earn lower rents in the three-tier hierarchy, and the reslting expected cost is lower nder centralization. Pt differently, the advantage of flexibility nder delegation is appropriated by agent rather than by the principal. 4. Conclding Remarks Or analysis has shown that with limited contract contingencies a principal will generally benefit from delegating athority to coordinate prodction and to contract with other agents. By designing a sitable prime contract the principal can align her own preferences with those of the manager sfficiently so as to take advantage of the flexibility inherent in delegation. These reslts are consistent with the widespread prevalence of managerial hierarchies and with the practice of sbcontracting. At the same time, or reslts point to several featres that appear essential in order for a sbcontracting arrangement to indeed be sperior. Or model ignored a nmber of factors that may favor centralization over delegation. These inclde the possibility of collsion between agents; the presence of limited-liability constraints on agents, which prevent intermediate contractors from bearing too mch risk; and the inability of the principal to ensre the appropriate seqencing of contracts nder delegation. In a setting where contract complexity restrictions are absent and the Revelation Principle applies, the importance of these factors for the performance of delegation arrangements has been established in MMR (995). When there are restrictions on contract complexity and some of the above conditions are not satisfied, the organization designer is likely to face the following trade-off: delegation is beneficial in that it enables prodction decisions to be more flexible,. Collsion between agents or inappropriate seqencing may involve the agents entering into a side contract before agent responds to the contract offered by the principal. This expands the extent of private information of agent, and therefore also the rents that the latter can captre. In a similar vein, ex post limited liability constraints on agent wold also enable her to captre larger rents. See MMR (995) for frther details.