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Contract delegation with bargaining.

I. INTRODUCTION

Public procurement is a very important part of economic activity. It accounts for about 20% of gross domestic product (GDP) in the United States and 16% in the European Union (EU) countries. (1) Therefore, it is not surprising that many interest groups are at the stake whenever aspects of public procurement procedures are discussed. To limit the influence of these groups, the purpose of legislation in the United States and the European Community has been to unify public procurement procedures and restrict the discretionary power of contracting officers and agencies at lower administrative levels. The Federal Acquisition Regulation (FAR) in 2005 in the United States and directive 2004/18/EC of the European Parliament in 2004 are examples of this.

One of the most important aspects in public procurement is the extent to which subcontracting is allowed. Traditionally, subcontracting has been judged negatively. For example, in construction, which amounts to a substantial part of public procurement, the FAR prevents contracts from being awarded to a single firm that supplies both design and engineering services (i.e., it requires contract centralization). However, recently most states have legislations that allow for exceptions like the use of the design-build method (i.e., they allow contract decentralization). Lam, Chan, and Chan (2006) report that nowadays more than one-third of the construction projects in the United States and up to 50% in Japan use the design-build approach. Economic theory should explain whether this is because of the increased influence of pressure groups or if there exists an economic rational for this tendency.

However, economic theory has had difficulty in explaining the superiority of subcontracting. To see this, consider a general contractor (or principal) who needs two suppliers (or agents) to realize a project. Either, he/she can contract directly both of them, or contract only one supplier and let him/her subcontract the other. Subcontracting obviously implies a loss of control over the subcontractee's contract for the general contractor. However, because he/she can always replicate the same contracts if he/she contracts directly with both suppliers, it is not easy to see the advantages of subcontracting. A common assumption in this literature is that the principal has all the bargaining power and can make a "take it or leave it" offer to the agent. Regarding public procurement this is a realistic assumption in cases in which there are many suppliers, but it fails to apply in cases in which the principal only can choose among a small number of suppliers. For example, this is the case in public procurement of defense systems, aeronautic and space equipment, or specific construction facilities such as water depuration plants, airports, or electric power plants.

Although procurement in these cases is also based on fixed-price, cost-reimbursement, and incentive contracts, usually it also allows for negotiations. So, section 15.3 of FAR states that "negotiations are exchanges, in either a competitive or sole source environment, between the Government and offerors, that are undertaken with the intent of allowing the offeror to revise its proposal. These negotiations may include bargaining. Bargaining includes persuasion, alteration of assumptions and positions, give-and-take, and may apply to price, schedule, technical requirements, type of contract, or other terms of a proposed contract." This means that economic theory should take into account that contract design includes bargaining between parties at some stage of the process when contract centralization and decentralization are compared.

To analyze this issue, a very stylized model is used to examine the advantages of centralized and decentralized organizations. Two agents work jointly on a project for a principal. Usually, in public procurement this is a contracting officer who represents a public agency. The agents that differ in terms of productivity choose their inputs simultaneously. The principal faces moral hazard problems because efforts are nonobservable and nonverifiable. Because of limited liability and moral hazard, agents obtain informational rents under efficient contract design. The contracts are negotiated between the contracting parties (the principal and the agents). The negotiation process is modeled following the offer procedure analyzed by Suh and Wen (2006). They show that the offer procedure provides a noncooperative foundation for the Nash (1950) cooperative bargaining solution in a multiplayer bargaining context and thereby generalize the result of Binmore, Rubinstein, and Wolinsky (1986) for the two-player case. A legislator chooses between three organizations: a centralized organization in which the principal controls and negotiates the contracts with both agents and two decentralized organizations in which he/she contracts only one agent and lets him/her subcontract the other agent. The legislator's problem is to choose the organization which gives him/her the highest expected utility.

Standard theory suggests that in the absence of communication costs, collusion among agents and renegotiation, the principal always prefers a centralized structure to a decentralized one. This is because in a decentralized contracting structure the general contractor distorts the subcontractor's inputs to increase his/her rent. Compared to a centralized contracting structure, these distortions mean an efficiency loss for the principal. In the context of bargaining, however, contract decentralization also has an advantage: it allows the principal to bargain with fewer partners in the first round. That this can be potentially advantageous for the principal can be seen easily from the following example. (2)

EXAMPLE 1. Consider a three-person pure bargaining situation with a principal P and two agents, 1 and 2. The feasible set is defined by x + y + z [less than or equal to] 1 and x, y,z [greater than or equal to] 0. The breakdown point is (0, 0, 0). In a centralized structure, the principal bargains with both agents in the first round. This is the traditional bargaining problem in which each individual forms one group. So, the group structure is {{1}, {2}, {P}} with outcome (1/3, 1/3, 1/3). In a decentralized structure, the principal bargains in the first round with only one of the two agents who then, in the second round, bargains with the other agent. Consequently, the group structure is {{1, 2}, {P}} and the outcome is (1/4, 1/4, 1/2).

The example shows that the principal gains when he/she bargains with fewer agents in the first round. This is because when all three individuals act independently, they split the payoff equally. However, if agents 1 and 2 act as a group, the game works as a two-person bargaining game between this group and the principal and each group gets 1/2. This means that the decentralization of contracts has an important advantage for the principal. The organizational choice for the principal is shown to depend on the trade-off between the advantage of negotiating with fewer parties in a decentralized structure and the disadvantage of greater effort distortions when one party has control over the other party's contract as under decentralization.

This trade-off is not trivial. For example, in light of Example 1, it can be argued that the prediction that a centralized contracting structure is preferable for the principal is correct whenever the principal's bargaining power is much greater than that of the agents. This paper shows that this is not always true. In particular, it shows that a decentralized contracting structure might be preferable to a centralized structure from the principal's point of view even for infinitely small bargaining powers of the agents. Current theory predicts an almost unique use of centralized contracting. The results in this paper derive the factors that make either of the two structures the better choice and bring us closer to the evidence that centralized and decentralized contracting structures coexist.

The literature has tried to explain contract delegation by relaxing the assumptions underlying the Revelation Principle. Some authors have shown that under certain conditions contract centralization and decentralization can be equivalent. (3) However, only few authors have found reasons for contract decentralization to be superior to centralization. In adverse selection environments, Melumad, Mookherjee, and Reichelstein (1995, 1997) show that costly communication can explain the superiority of contract decentralization, while Laffont and Martimort (1998) relate its advantages to the possibility of collusion and (Poitevin 2000) to contract renegotiation. Baliga and Sjostrom (1998) show the superiority of contract decentralization in a moral hazard environment with sequential production for the case in which agents have limited liability, share hard information not available to the principal, and collude. Finally, in such a setting, Jelovac and Macho-Stadler (2002) find delegation to be superior to centralization when contracts are signed sequentially and firms inputs are complements. The contribution of this paper is that it gives a completely new argument for the advantage of contract delegation. Decentralization allows a better bargaining position for the principal. This might even be true if agents have almost no bargaining power.

The paper is organized as follows. Section II presents the model and the timing of the three stage-game. In Section III, the equilibrium effort values at stage 3 are derived. Section IV solves the contract negotiation process at stage 2 and provides the optimal contracts under contract centralization and contract decentralization. In Section V, the principal's organizational decision at stage 1 is analyzed. This section includes the main results of the paper. Section VI concludes. All proofs are included in the Appendix.

II. THE MODEL

Production. Suppose a principal hires two agents, 1 and 2, who perform two different tasks in a project. The monetary outcome of the project is represented by a random variable x [member of] X. It can take two possible values, X = {0, [bar.x]} which represent the joint outcome of the agents' performance. Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the probability of obtaining outcome [bar.x] when efforts ([e.sub.1], [e.sub.2]) [subset or equal to] [R.sup.2] are chosen. (4) It is assumed that [[alpha].sub.i] > 0, i = 1, 2 and [alpha] [equivalent to] [[alpha].sub.1] + [[alpha].sub.2] < 1. Both agents choose their efforts simultaneously.

Contracts. The monetary outcome is the only publicly observable and contractible variable. Hence agents are compensated with a payment schedule contingent on output, [w.sub.i] (0) = [[w.bar].sub.i] and [w.sub.i] ([bar.x]) = [[bar.w].sub.i], i = 1, 2. Agents are protected by limited liability or legal restrictions such that [[w.bar].sub.i], [[bar.w].sub.i] [greater than or equal to] 0.

Utilities. Agent i's and the principal's expected utility from a single project are given by

[EU.sub.i] = p([e.sub.1], [e.sub.2])[DELTA][w.sub.i] + [[bar.w].sub.i] - [c.sub.i][e.sb.i, and

[EU.sub.P] = [([e.sub.i], [e.sub.2]) (bar.x] - [DELTA][w.sub.i] - [DELTA][w.sub.2]) - [[bar.w].sub.1] - [[bar.w].sub.2],

respectively, where [DELTA][w.sub.i] = [[bar.w].sub.i] = [[w.bar.].sub.i] and [c.sub.i] > [bar.x] denotes the marginal cost of effort. The reservation utility of the principal and the agents is assumed to be equal to zero.

Organizational structure. The contracting process can be organized in three alternative ways: (1) As a centralized structure C in which the principal negotiates the contracts with the two agents. (2) As a decentralized structure in which he/she negotiates only with agent 1, who in the next round negotiates with agent 2. This structure is denoted as [D.sub.1]. (3) Finally, he/she can negotiate only with agent 2, who in the next round negotiates with agent 1. This structure is denoted as [D.sub.2]. The contracting structure is chosen by a legislator to maximize the expected net surplus of the project. (5)

Bargaining process. Contract negotiation is modeled following the offer procedure by Suh and Wen (2006). In a centralized structure, the multiplayer bargaining process is modeled via sequential bilateral bargaining sessions each of which proceeds as under the alternative-proposal framework of Rubinstein (1982). In session 1, the principal offers agent 1 a share of outcome x. If agent 1 accepts the offer the current session ends with the accepted proposal as a partial agreement contingent of final acceptation by agents 1 and 2 and the principal. In this case, the principal moves to the next session to bargain with agent 2. If agent 1 rejects the principal's offer in session 1, then the bargaining will proceed to the next period in the same session where agent 1 makes a proposal to the principal. Session 1 ends when the principal and agent 1 accept a partial agreement. Then, the principal moves to session 2 and makes a proposal to agent 2. If the principal and agent 2 reach an agreement, they also agree on the previous partial agreements because in order to obtain what they have agreed on, agent 1 has to agree. When the principal and agent 2 reach an agreement, all agreements will be implemented and the principal and the agents get their shares of outcome x. (6)

In a decentralized structure, the offer procedure of Sub and Wen (2006) is identical to Rubinstein's (1982) alternating-offer bargaining model. In structure [D.sub.1], the principal and agent 1 bargain to reach an agreement on how to share outcome x. When they reach a final agreement, agents 1 and 2 negotiate how to share what has been left by the principal. Similarly, bargaining proceeds in structure [D.sub.2]. So, concerning the bargaining process, the main difference between the centralized and a decentralized structure is that in the former, agreements between two parties are only partial, while in the latter, agreements between two parties are final. (7)

Suh and Wen (2006) show that under the offer procedure exists a unique subgame perfect equilibrium that is efficient. Throughout the paper it is assumed that the time interval between offers and counteroffers in all bargaining session is short and shrinks to zero. Then, Binmore, Rubinstein, and Wolinsky (1986) and Sub and Wen (2006) have shown that Rubinstein's (1982) alternating-offer model and the offer procedure, respectively, approximate the Nash bargaining solution. (8) According to Binmore, Rubinstein, and Wolinsky (1986) the principal's and the agents' bargaining powers, [[gamma].sub.p], [[gamma].sub.1], and [[gamma].sub.2], respectively, can be related, for example, to their discount rate. Without loss of generality, the normalization [[gamma].sub.p] = 1 - [[gamma].sub.1] - [[gamma].sub.2] is used.

Timing. The timing of the game is as follows: (1) The legislator determines the contracting structure k [member of] {C, [D.sub.1], [D.sub.2]} for the principal and the two agents. (2) In structure C, the principal negotiates contracts [w.sub.1] and [w.sub.2] following the offer procedure. In structure [D.sub.i], the principal negotiates in a first round the contract with agent i, who then negotiates the contract with agent j [not equal to] i in a second round. (3) Agents simultaneously make their effort choices. (4) The outcome is observed and the agents are paid contingent on their contracts.

III. EFFORT CHOICE

The game is solved by backward induction. Denote [[??].sub.i] agent i's net payment. In a centralized structure, this is just the wage he/she receives from the principal. In a decentralized structure in which agent i is the general contractor, this is the wage he/she receives from the principal minus the wage he/she pays to his/her subcontractee, agent j. At stage 3 agent i chooses his effort according to:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In a Nash equilibrium of this subgame efforts are:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equation (2) is the incentive compatibility constraints the contracts must satisfy. (9) Taking into account the equilibrium values of efforts, the expected utilities of the agents and the principal are:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The following lemma simplifies the form of the contracts that need to be considered to obtain the bargaining solution. It is stated without proof, as it is a straightforward extension of lemma 1 of Pitchford (1998).

LEMMA 1. Any Pareto efficient contract that satisfies [EU.sub.i] > 0 for i = 1, 2, P, must have [w.sub.i] = 0 for i = 1,2. Furthermore, if [[w.bar].sub.i] = 0 incentive compatibility guarantees that [EU.sub.i] [greater than or equal to] 0 for i = 1, 2 if [DELTA] [[??].sub.i] [greater than or equal to] .

The intuition behind this result is that if [[w.bar].sub.i] were strictly positive, the principal could reduce [[w.bar].sub.i] and increase [[bar.w].sub.i] in a way that would not change the agents' expected utility. However, this change would increase the principal's expected utility because an increase in [[bar.].sub.i] would incite higher efforts and thereby increase the likelihood of a high outcome.

Lemma 1 makes it possible to write the principal's expected utility as [EU.sub.P] - [p.sup.*] ([bar.x] - [DELTA][??]), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the equilibrium success probability and {DELTA][??] = [DELTA][[??].sub.1] + [DELTA][[??].sub.2] the total wage paid by the principal to the agents. There are two types of misallocation from the principal's point of view: among agents' wage shares and in total wage per outcome. The first effect influences the probability of success of the project. For a given total wage [DELTA][??], the probability reaches its maximum when the agents' wage ratio is [DELTA][[??].sub.1]/ [DELTA][[??].sub.2] = [[alpha].sub.1]/[[alpha].sub.2]. The second effect means that the principal must counterbalance the advantage of a lower total wage with the disadvantage of a lower probability of success of the project. The principal's expected utility reaches its maximum when the total wage outcome ratio is [DELTA][??]/[bar.x] = [alpha]. The principal's decision to decentralize or centralize contracting takes into account these two effects. The principal chooses the organizational structure in which the wage allocation between agents and the total wage are closest to her preferred combination.

IV. BARGAINING ON CONTRACTS

A. Centralization

In a centralized structure, the principal contracts both agents directly. Therefore [DELTA][[??].sub.i] = [DELTA][w.sub.1] for i = 1, 2. As mentioned before, with no delay between offers and counteroffers the offer procedure at stage 2 approximates the Nash bargaining solution where the principal's and agent 1's and 2's bargaining powers are [[gamma].sub.p], [[gamma].sub.1], and [[gamma].sub.2], respectively. Using the result from Lemma 1, the outcome of this game is given by the solution to program [[p.sup.C]]:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From elementary calculations we find that the optimal payments are:

[DELTA][w.sup.C.sub.i] = ([[alpha].sub.i] + (1 - [alpha]) [[gamma].sub.1]) [bar.x], i = 1, 2.

Because gross and net wages coincide in a centralized structure, the following result is obtained immediately.

LEMMA 2. In a centralized structure, the agents' net wages are [DELTA] [[??].sup.C.sub.i] = ([[alpha].sub.i] + (1 - [alpha]) [[gamma].sub.i]) [bar.x], i = 1, 2. Expected utilities are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice that [DELTA][[??].sup.C.sub.i]. Then, as stated in Lemma 1, incentive compatibility and participation for both agents are guaranteed. The result in Lemma 2 indicates that an agent's wage increases with both the importance of his/her effort for the success of the project and his/her bargaining power. His/her wage decreases with the importance of the other agent's effort for the success of the project and the bargaining power of the other agent and the bargaining power of the principal.

B. Decentralization

Consider a decentralized structure in which the principal bargains with agent 1 at stage 2 in a first round and agent 1 subcontracts agent 2 in a second round. In this case, the offer procedure corresponds to two bilateral bargaining sessions of the Rubinstein (1982) type. In the first round, the principal and agent 1 bargain until they reach a final agreement on how to share the outcome x. As shown by Binmore, Rubinstein, and Wolinsky (1986), without delay between offers and counteroffers the solution approximates the Nash bargaining solution. Similarly, in the second round agents 1 and 2 bargain until they reach a final agreement on how to share the part of outcome x that has been left by the principal. (10)

The game is solved by backward induction. Consider first the bargaining game at round 2 between agents 1 and 2. Under contract delegation to agent 1 we have [DELTA][[??].sub.1] = [DELTA][W.sub.1] - [DELTA][w.sub.2] and [DELTA][W.sub.2] = [DELTA][W.sub.2]. With bargaining powers [[gamma].sub.i] of agent i and with no delay, a solution to the bargaining game is given by the solution to program [[P.sup.1]]:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From simple calculations we find that the wage negotiated between agents 1 and 2 is given by:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This wage increases with agent 2's bargaining power and the importance of his/her effort for the success of the project. The wage decreases when agent 1's bargaining power increases and when agent 1's effort becomes more important for the success of the project.

At stage 2 in round 1 of the game, the principal negotiates agent 1's wage contract. Given agent 1's and the principal's bargaining power, [[gamma].sub.1] and [[gamma].sub.p] , respectively, the solution to the bargaining game is given by the solution to program [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following lemma resumes the equilibrium values in a decentralized contracting structure in which the principal negotiates with agent 1 who then subcontracts agent 2.

LEMMA 3. In a decentralized structure in which contracting is delegated to agent 1, who subcontracts agent 2, the agents' net wages are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Expected utilities are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Again, notice that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 0 such that incentive compatibility and participation for both agents is guaranteed (see Lemma 1). In a decentralized structure, an agent's net wage not only depends on his/her own bargaining power but also on the bargaining power of his/her colleague. Because the case in which the principal negotiates first with agent 2 is similar, we obtain:

LEMMA 4. In a decentralized structure in which contracting is delegated to agent 2, who subcontracts agent 1, the agents' net wages are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Expected utilities are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

V. CHOICE OF CONTRACTING STRUCTURES

At stage 1, the legislator must decide whether to choose the centralized structure in which the principal bargains with both agents or to decentralize contracts and let the principal bargain only with one agent. First, to obtain a benchmark we can use the previous results to analyze the effect of contract delegation under the assumption of standard principal-agent theory that the party that proposes the contract has all the bargaining power. Then, the general case is analyzed.

A. The Standard Case: [[gamma].sub.p] = 1

First, consider the two decentralized structures. We derive the optimal contracts for this case by assuming that the subcontracted agent has no bargaining power ([[gamma].sub.j] = 0) and that the bargaining power of the agent directly contracted by the principal shrinks to zero ([[gamma].sub.i] [right arrow] 0). A comparison of the principal's and the agents' expected utilities in Lemmas 3 and 4 yields the following result:

PROPOSITION 1. If [[gamma].sub.p] = 1, the legislator and the principal prefer the decentralized structure in which the general contractor is more important for the success of the project [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. An agent always prefers a decentralized structure in which he/she is the general contractor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, let us compare contract centralization with decentralization. Regarding centralized contracting from Lemma 2, we find that when agents have no bargaining power ([[gamma].sub.1] = [[gamma].sub.2] = 0), as is assumed in traditional principal-agent theory, agents wage ratio is [DELTA] [[??].sup.C.sub.1]/[DELTA] [[??].sup.C.sub.2] = [[alpha].sub.1]/[[alpha].sub.2] and the total wage outcome ratio is [DELTA][[??].sup.C]/[bar.x] = [alpha]. This means that there is neither a misallocation in the agents' wage shares nor in the total wage per outcome from the principal's point of view. Therefore, in this case, the centralized structure is at least as good as any other contracting structure for the principal.

Now consider a decentralized structure in which agent 1 is the general contractor who subcontracts agent 2. From Lemma 3 we find that the agents' wage ratio is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the total wage outcome ratio is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11) This means that there is a misallocation in the agents' wage shares while the total wage per outcome is not misallocated from the principal's point of view. Similarly, if agent 2 is the general contractor who subcontracts agent 1, from Lemma 4 we find that the agents' wage ratio is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the total wage outcome ratio is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Again, we have a misallocation in the agents' wage shares while the total wage per outcome is not misallocated from the principal's point of view. Together, these findings imply that

the principal is strictly better off in a centralized structure than in a decentralized structure, because in the former contracting structure there is no misallocation from the principal's point of view, while in a decentralized structure from his/her point of view the agents' wage shares are misallocated.

PROPOSITION 2. If [[gamma].sub.P] = 1, the legislator and the principal always prefer contract centralization to contract decentralization.

B. The General Case: [[gamma].sub.P] < 1

To see how the assumption that the principal has all the bargaining power influences this result from the classical principal-agent literature, let us now consider the case that [[gamma].sub.P] < 1. First, let us compare the two decentralized structures. We find:

PROPOSITION 3. If the legislator chooses a decentralized structure, the principal prefers to delegate contracting to the agent with the least bargaining power ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). An agent prefers the decentralized structure in which the agent with the highest bargaining power is the general contractor ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Proposition 3 has some interesting implications. First, as expected from Example 1, when agents bargain their contracts with the principal, the principal prefers to delegate contracting to the agent with the least bargaining power. This allows her to obtain a higher share of the residual output in the first round. For agents, however, it is irrelevant who subcontracts whom because wages are determined by bargaining. Therefore, they prefer that the agent who bargains with the principal is the one with the most bargaining power. Second, notice that this result is different to the case in which the general contractor has all the bargaining power. In that case, we have seen that what is important is which agent is the most important for the success of the project. With bargaining this plays no role because the agents take this into account when they bargain their wages and the result is independent from which of them is the subcontractor and which is the subcontractee.

Now consider the legislator's choice of the optimal contracting structure in general. We compare net expected surplus (the principal's utility) in a centralized structure with that in a decentralized structure in which the agent with most bargaining power is the subcontractor. (12) The following proposition shows that, depending on the principal's and the agent's bargaining power, both contracting structures can be optimal.

PROPOSITION 4. For any [[gamma].sub.P] < 1, [There exists] [[gamma].sub.1], [[gamma].sub.2] such that the legislator and the principal prefer contract decentralization to contract centralization.

Proposition 4 is the main result of the paper. It states that with strictly positive bargaining power of the agents, both centralized and decentralized contracting structures can be preferable for the legislator (and the principal). Figure 1 displays the regimes for which C, [D.sub.1] and [D.sub.2] are his/her optimal choice for different parameter values. This depends on the relationship between the relative bargaining strengths of the agents, [[gamma].sub.1]/[[gamma].sub.2], and their relative importance for the success of the project, [[alpha].sub.1]/[[alpha].sub.2]. We find that for [[gamma].sub.1]/[[gamma].sub.2] = [[alpha].sub.1]/[[alpha].sub.2] the legislator always prefers a decentralized structure to a centralized one. This result is opposed to the result from traditional principal-agent theory. To interpret it, first notice that in this case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So there is no inefficiency in the wage allocation between agents in centralized and decentralized structures. As far as the total wage the principal must pay is concerned, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A decentralized structure has lower total wage payments. Therefore, for [[gamma].sub.1]/[[gamma].sub.2] = [[alpha].sub.1]/[[alpha].sub.2] a decentralized structure has the advantage of being cheaper to implement without having the disadvantage of yielding a worse wage and effort allocation between agents. Consequently, the principal prefers decentralization. For [[gamma].sub.1]/[[gamma].sub.2] [not equal to] [[alpha].sub.1]/[[alpha].sub.2] the wage allocation between agents will be less favorable in a decentralized structure than in the centralized structure. So the legislator's decision must counterbalance both effects. If '[[gamma].sub.1]/[[gamma]sub.2] is far from [[alpha].sub.1]/[[alpha].sub.2] which means that an agent with low (high) relative importance for the project's success has high (low) relative bargaining power, the wage allocation is much worse in a decentralized structure than in a centralized one. So centralization is more likely to be better for the legislator and the principal. On the other hand, if [[gamma].sub.1] + [[gamma]sub.2] is large, the advantage of having a more favorable total wage in a decentralized structure than in a centralized one becomes more important. As seen from Example 1, this is because of the advantage of bargaining with fewer parties in round one, whose importance increases with the bargaining powers of the two agents. So, if [[gamma].sub.1] + [[gamma].sub.2] is large, decentralization is more likely to be the legislator's choice.

VI. CONCLUSIONS

The purpose of this paper is to analyze the consequences of the assumption made in the traditional principal-agent literature that the contractor has all the bargaining power and that the contractee and subcontractee have none. It is shown that when this assumption is relaxed, the standard result in the context of contract delegation that a centralized structure is always preferable to a decentralized structure does not hold any longer. Moreover, the standard result may even fail to hold if contractors have nearly all the bargaining power and subcontractees have nearly none. Because bargaining is present in many contracting processes, this result helps to understand why contract decentralization is observed much more than expected from previous results. This explains, for example, the coexistence of different contracting structures in public procurement regulations such as the FAR or to the directive 2004/18/EC of the European Parliament.

ABBREVIATIONS

EU: European Union

FAR: Federal Acquisition Regulation

GDP: Gross Domestic Product

doi: 10.1111/j.1465-7295.2011.00404.x

APPENDIX

Proof of Lemma 2

Consider a logarithmic transformation of Equation (5). Problem [[P.sup.C]] is equivalent to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first-order conditions are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second derivatives are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the first-order conditions are necessary and sufficient for a maximum. Using [[gamma].sub.P] = 1 - [[gamma].sub.1] - [[gamma].sub.2], from the first-order conditions we get: [DELTA][w.sub.1] = ([[alpha].sub.1] + (1 - [alpha])[[gamma].sub.1]) [bar.x] and [DELTA][w.sub.2] = ([[alpha].sub.2] + (1 - [alpha])[[gamma].sub.2])[bar.x]. Notice, that [DELTA][w.sub.i] < [bar.x] guarantees that the equilibrium success probability is less than unity. Finally, substitution into Equations (3) and (4) yields expected utilities in equilibrium.

Proof of Lemma 3

Consider a logarithmic transformation of equation (8). Problem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equivalent to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order condition is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the second derivative is

[[partial derivative].sup.2] L/[partial derivative][DELTA][w.sup.2.sub.1] = [[gamma].sub.P] (-[alpha]/1 - [alpha] 1/[DELTA][w.sup.2.sub.1] - 1/([bar.x] - [DELTA][w.sub.1]).sup.2]) - [[gamma].sub.1] 1/1 - [alpha] 1/[DELTA][w.sup.2.sub.1] < 0

So, the first-order condition is necessary and sufficient for a maximum. Using [[gamma].sub.P] = 1 - [[gamma].sub.1] - [[gamma].sub.2], from the first-order condition we obtain the optimal contract between the principal and agent 1: [DELTA][w.sub.1] = (1-[[gamma].sub.1] - [[gamma].sub.2]) [alpha] + [[gamma].sub.1/1 - [[gamma].sub.2] [bar.x]. Using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] yields the agents' equilibrium net wages which are less than [bar.x] such that the equilibrium success probability is less than unity. Finally, substitution into Equations (3) and (4) yields expected utilities in equilibrium.

Proof of Proposition 1

The legislator and the principal prefer a decentralized structure in which agent 1 is the general contractor to a decentralized structure in which agent 2 is the general contractor if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This means

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking logarithms on both sides we find that this is equivalent to

[f([[alpha].sub.1]) = [[alpha].sub.1] ln (1 - [[alpha].sub.2]) + [[alpha].sub.2] ln [[alpha].sub.2] - [[alpha].sub.1] ln [[alpha].sub.1] - [[alpha].sub.2] ln (1 - [[alpha].sub.1]) > 0.

This is a function with one turning point (f" = -[[alpha].sup.-1.sub.1] + [[alpha].sub.2](1 - [[alpha].sub.1]).sup.-2], [[alpha].sub.1] = 2 + [[alpha].sub.2]/2 - [square root of (2 + [[alpha].sub.2]/2).sup.2] - 1)] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Furthermore, f([[alpha].sub.2]) = 0, f(1 - [[alpha].sub.2]) = 0 and f'(1 - [[alpha].sub.2]) = 0. For [[alpha].sub.2] < 1 - ([[alpha].sub.2] the function f has a minimum at 1 -[[alpha].sub.2]. Thus, f([[alpha].sub.1]) < 0 for [[alpha].sub.1] < [[alpha].sub.2]), and f([[alpha].sub.1]) > 0 for [[alpha].sub.2] < [[alpha].sub.1] < 1 - [[alpha].sub.2]. For [[alpha].sub.2] > 1 - [[alpha].sub.2] the function f has a maximum at 1 - [[alpha].sub.2]. Thus, f([[alpha].sub.1]) < 0 for [[alpha].sub.1] < 1 - [[alpha].sub.2] < [[alpha].sub.2]. Therefore for [[alpha].sub.1] + [[alpha].sub.2] < 1, f([[alpha].sub.1]) < 0 for [[alpha].sub.1] < [[alpha].sub.2], f([[alpha].sub.1]) = 0 for [[alpha].sub.1] = [[alpha].sub.2] and f([[alpha].sub.1]) > 0 for [[alpha].sub.1] > [[alpha].sub.2].

Agent 1 prefers structure [D.sub.1] to structure [D.sub.2] iff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This means

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking logarithms on both sides we find that this is equivalent to

g([[alpha].sub.1]) = (1 - [[alpha].sub.1]) ln (1 - [[alpha].sub.2]) + [[alpha].sub.2] ln [[alpha].sub.2] - (1 - [[alpha].sub.2]) ln [[alpha].sub.1] - [[alpha].sub.2] ln (1 - [[alpha].sub.1) > 0.

This is a decreasing function (g' = 1 (1 - [alpha])/([[alpha].sub.1] (1 - [[alpha].sub.1]) < 0) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, g([[alpha].sub.1]) > 0 for 0 < [[alpha].sub.1] < - [[alpha].sub.2].

Proof of Proposition 3

From Lemma 3 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f ([[gamma].sub.i], [[gamma].sub.j], [alpha]) = (1 - [[gamma].sub.i]/1 - [[gamma].sub.j]) ((1 - [[gamma].sub.1] - [[gamma].sub.2]) [alpha] + [[gamma].sub.i]/(1 - [[gamma].sub.1] - [[gamma].sub.2]) [alpha]+[[gamma].sub.j]). First notice that f ([[gamma].sub.i], [[gamma].sub.i], [alpha]) = f ([[gamma].sub.j], [[gamma].sub.j], [alpha]) = 1. Next,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [x.sub.1] = (1 - [[gamma].sub.1] - [[gamma].sub.2]) [alpha] + [[gamma].sub.i]/[(1 - [[gamma].sub.1] - [[gamma].sub.2]).sup.[alpha]] and [x.sub.2] = [(1 - [[gamma].sub.1] - [[gamma]sub.2]) [alpha] + [[gamma].sub.j]/(1 - [[gamma].sub.1] - [[gamma].sub.2]).sup.[alpha]]. Because ln (x) + 1/x is an increasing function in x for x > 1, we find that ln ([x.sub.1]) + [1/[x.sub.1] > ln ([x.sub.2]) + 1/[x.sub.2] for [x.sub.1] > [x.sub.2]. Thus, [partial derivative]f([[gamma].sub.i], [[gamma].sub.j], [alpha]) / [partial derivative][alpha] > 0 for [[gamma].sub.i] > [[gamma.sub.j] and [partial derivative]f([[gamma].sub.i], [[gamma].sub.j], [alpha])/[partial derivative][alpha] < 0 for [[gamma].sub.i] < [[gamma].sub.j]. Because f([[gamma].sub.i], [[gamma].sub.j], 1) = 1, this implies that [for all][alpha] < 1 we have: f ([[gamma].sub.i], [[gamma].sub.j], [alpha]) < 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Proposition 4

From Lemmas 2 and 3 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [delta] [equivalent to] [[gamma].sub.i]/[[gamma].sub.j]. Now define H([[gamma].sub.P], [delta]) [equivalent to] 1/ (1 - [alpha]) ln [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then H([[gamma].sub.P], [delta]) = 0 implicitly defines [[gamma].sub.P] as a function of [delta]: [[??].sub.P] = f([delta]). The proposition is proved by the following claims:

CLAIM 1. The relative gains of centralization with respect to decentralization to agent i increases for a constant relation of agents' bargaining powers, [delta], with the bargaining power of the principal, [[gamma].sub.P]. That is, H is increasing in [[gamma].sub.P].

Proof We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because in absolute value the third term is smaller than the last term. Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [partial derivative]H/[[partial derivative][[gamma]P] > 0 for 0 < [[gamma] < [[gamma].sub.P]] < 1.

CLAIM 2. For 0 < [[gamma].sub.P] < 1 there exists a critical value of [[gamma].sub.P], say [[??].sub.P], such that, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, [lim.sub.[[gamma]P[right arrow]1] H reaches a minimum at [delta] = [[alpha].sub.i]/[[alpha].sub.j]. The value at the minimum is [lim.sub.[gamma]P[right arrow]1] H ([[gamma].sub.P], [[alpha].sub.i/[[alpha].sub.j]) = 0. Therefore, [lim.sub.[gamma]P[right arrow]1] H [greater than or equal to] 0. Claim 2 follows because H is strictly increasing in [[gamma].sub.P] (Claim 1), negative for small [[gamma].sub.P] and nonpositive for large [[gamma].sub.P].

CLAIM 3. f' ([delta]) < 0 for [delta] > [[alpha].sub.i]/[[alpha].sub.j], f'([delta]) > 0 for [delta] < [[alpha].sub.i]/[[alpha].sub.j] and f'([delta]) = 0 for [delta] = [[alpha].sub.i]/[[alpha].sub.j].

Proof From Claim 2 we know that H is a strictly convex function for [[gamma].sub.P] = 1 that reaches a minimum at [delta] = [[alpha].sub.i]/[[alpha].sub.j], i.e. [[partial derivative].sub.[[gamma]P/[partial derivative][delta] = f'([delta]) = 0 for [delta] = [[alpha].sub.i/[[alpha].sub.j]. Furthermore, from Claim 1 we know that H is increasing in [[gamma].sub.P]. Thus, as [[gamma].sub.P] decreases, the [delta] such that H([[gamma].sub.P], [delta])= 0 increases for [delta] > [[alpha].sub.i]/[[alpha].sub.j], and decreases for [delta] < [[alpha].sub.i]/[[alpha].sub.j]. This means [partial derivative][[gamma].sub.P]/[partial derivative][delta] = f'([delta]) < 0 for [delta] > [[alpha].sub.i]/[[alpha].sub.j], [partial derivative][[gamma].sub.P]/[partial derivative[delta] = f'([delta]) 0 for [delta] < [[alpha].sub.i]/[[alpha].sub.j].

CLAIM 4. f ([[alpha].sub.i])/[[alpha].sub.j]) = 1.

Proof In Claim 2, we have seen that [lim.sub.[[gamma]P[right arrow]1] H ([[gamma].sub.P], [[alpha].sub.i]/[[alpha].sub.j]) = 0. Thus, [[??].sub.P] = 1 for [delta] = [[alpha].sub.i]/[[alpha].sub.j] or f([[alpha].sub.i]/[[alpha].sub.j]) = 1.

REFERENCES

Baliga, S., and T. Sjostrom. "Decentralization and Collusion." Journal of Economic Theory, 83, 1998, 196-232.

Binmore, K. G., A. Rubinstein, and A. Wolinsky. "The Nash Bargaining Solution in Economic Modelling." RAND Journal of Economics, 17, 1986, 176-88.

Chae, S., and P. Heidbues. "A Group Bargaining Solution." Mathematical Social Science, 48, 2004, 37-53.

Dimitri, N., G. Piga, and G. Spagnolo. Handbook of Procurement. Cambridge: Cambridge University Press, 2006.

Harsanyi, J. Rational Behavior and Bargaining Equilibrium in Games and Social Situations. Cambridge: Cambridge University Press, 1977.

Jelovac, I., and I. Macho-Stadler. "Comparing Organizational Structures in Health Services." Journal of Economic Behavior & Organization, 49, 2002, 501-22.

Laffont, J.-J., and D. Martimort. "Collusion and Delegation." RAND Journal of Economics, 29, 1998, 280-305.

--. The Theory of Incentives, Princeton: Princeton University Press, 2002.

Lam, E., A. Chan, and D. Chan. "Barriers to Applying the Design-Build Procurement Method in Hong Kong." Architectural Science Review 42(2), 2006, 189-95.

Mclumad, N., D. Mookherjee, and S. Reichelstein. "Hierarchical Decentralization of Incentive Contracts." RAND Journal of Economics, 26, 1995, 654-72.

--. "Contract Complexity, Incentives, and the Value of Delegation." Journal of Economics and Management Strategy, 6(2), 1997, 257-89.

Mookherjee, D. "Decentralization, Hierarchies and Incentives: A Mechanism Design Perspective." Journal of Economic Literature, 64, 2006, 367-90.

Nash, J. "The Bargaining Problem." Econometrica, 18, 1950, 155-62.

Pitchford, R. "Moral Hazard and Limited Liability: The Real Effect of Contract Bargaining." Economics Letters, 61, 1998, 251-59.

Poitevin, M. "Can the Theory of Incentives Explain Decentralization?" Canadian Journal of Economics, 33(4), 2000, 878-906.

Rubinstein, A. "Perfect Equilibrium in a Bargaining Model." Econometrica, 50, 1982, 97-109.

Suh, S.-C., and Q. Wen. "Multi-agent Bargaining and the Nash Bargaining Solution." Journal of Mathematical Economics, 42, 2006, 61 73.

BERND THEILEN *

* An earlier version of this paper has been presented at several seminars and the XXIII edition of the "Jornadas de Economia Industria." I am very grateful to the participants in these meetings and especially to Matthias Dahm and Ricardo Flores-Fillol, a coeditor, and three referees for their comments and suggestions. Financial supports from the Spanish "Ministerio de Ciencia e Innovacion" under projects SEJ2007-67580-C02-01 and ECO2010-17113 and the "Departament d'Universitats, Recerca i Societat de la Informacio de la Generalitat de Catalunya" under project 2005SGR 00949 are gratefully acknowledged.

Theilen: Professor of Economics, Departament d'Economia and CREIP, Universitat Rovira i Virgili, Avinguda de la Universitat 1, E-43204 Reus, Spain. Phone +34-977759-850, Fax +34-977-758-907, E-mail bernd.theilen @urv.net

(1.) See Dimitri, Piga, and Spagnolo (2006).

(2.) This example is also known as Harsanyi's joint-bargaining paradox. See the study of Harsanyi (1977) and also that of Chae and Heidhues (2004).

(3.) For an overview of this literature, see the studies of Poitevin (2000) and Mookherjee (2006).

(4.) The two-outcome assumption is standard in the literature. A similar probability function has been used by Jelovac and Macho-Stadler (2002). See the study of Laffont and Martimort (2002) for an overview of different model specifications.

(5.) This means that the legislator and the principal have the same objective. Therefore, the main implication of the separation of the principal and the legislator is that it allows commitment to a contracting structure. So, the choice of the contracting structure itself is no issue of bargaining between the agents and the principal. As mentioned before, this is a realistic assumption in public procurement where the principal is a public agency, and where the contracting process is determined ex ante by law.

(6.) Of course, upon partial acceptance of an agreement between agent 1 and the principal, it can also be agent 1 that moves to the next session to bargain with agent 2.

(7.) The offer procedure reflects fairly well the bargaining process in practice. For example, contractors (principals) who order a construction facility can adopt either policy: they contract directly with an architect and a constructor, or they contract only one agent who subcontracts the other agent. In the former case, the architect and the principal would only reach a partial agreement contingent on the constructor's acceptance of the project. Under contract delegation such as design-build, the principal and one of the agents, for example the constructor, reach a final agreement. Then, it is the constructor's responsibility to subcontract and reach an agreement with an architect.

(8.) Sub and Wen (2006) have proven this under the assumption that all players have the same discount rate. A generalization of this proof for the case in which agents have different discount rates (the assumption made in this paper) is available from the author upon request.

(9.) Notice that the assumption c > [bar.x] guarantees that [e.sup.*.sub.i] < 1 if [DELTA][[??].sub.i] < [bar.x] which will be the case in both organizational structures. So, in equilibrium, p([e.sup.*.sub.1], [e.sup.*.sub.2]) < 1.

(10.) See also the work of Chae and Heidhues (2004). They show that this corresponds to a group bargaining problem in which agents 1 and 2 form a group which bargains with the principal. So we have bargaining between groups as well as within groups. The solution to such problems constitutes a Nash solution both within and across groups.

(11.) The assumption of standard principal-agent theory is that the party that proposes the contract has all the bargaining power. So we must calculate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(12.) One can argue that otherwise the agent with lower bargaining power will break down the negotiation with the principal because, as we have seen before, he/she is better off when the agent with higher bargaining power negotiates with the principal in the first round. Furthermore, the main objective of the paper is to show that the principal can be better off under decentralization than under centralization. If this is shown for the principal's least preferred decentralized structure (i.e., the one in which he/she bargains with the agent with higher bargaining power) the result is even stronger for the more favorable case.
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