# Contracts under wage compression: a case of beneficial collusion.

1. IntroductionIn principal-agent models, collusion among agents generally lowers the principal's welfare in the presence of asymmetric information. Under collusion, the agents can have more opportunities to take advantage of information possessed only by them and not by the principal. Standard treatment for collusion in the literature is to "deter it" if affordable and to "allow it" otherwise. Although a few studies show that the prospect of collusion can be beneficial for the principal ex ante, a common result in hidden information models is that when side contracting among agents takes place, the agents' information is not revealed to the principal. In this paper, we examine a situation in which the principal may take advantage of collusion among agents when inducing revelation of hidden information.

We consider an organization in which the top management has limited power to discriminate transfers to different subunits, while it observes each subunit's performance perfectly. An organization facing such a restriction is said to be under "external wage compression." (1) Our results suggest that, under wage compression, the top management may increase efficiency in the output schedule by inducing collusion among subunits. For simplicity, we model the situation with one principal and two agents. Each agent's "type," such as his efficiency or cost parameter, is not known to the principal, and prior to production, each agent reports his type to the principal. An agent can quit if he anticipates a payoff strictly less than his reservation payoff. In our model, externalities due to wage compression provide a potential free ride when the types of the agents are different--the less efficient agent can free ride on the more efficient agent's production.

This free-riding opportunity due to limited wage discrimination leads to an incentive problem associated with hidden information. An efficient agent facing another efficient agent has an incentive to misreport his type to free ride on the other agent. Thus, when both agents are the efficient type, they have information rent for the potential situation in which one of them misreports. When the agents cannot collude, the principal distorts the allocation of production when the types reported by the agents are different--in the optimal contract, the principal increases (decreases) the proportion of the output assigned to the inefficient (efficient) agent. By doing so, the principal removes an inefficient agent's free riding on an efficient agent, which in turn discourages an efficient agent from misreporting his type when paired with another efficient agent.

Under collusion, however, the principal can improve her payoff for two reasons. First, she can make agents of different types internalize the externality by inducing side transfers (2) between them. When the agents are of different types, the inefficient agent needs the efficient agent to be truthful in order to free ride on him. Therefore, the inefficient agent has an incentive to offer the efficient agent a side transfer for a truthful report. As a result, the principal's burden of rent provision to the efficient agent is partly transferred to the inefficient agent. In the optimal contract, side contracting takes place and the agents exchange side transfers when their types are different.

Second, inducing side transfers between agents of different types mitigates misreporting incentives when both agents are efficient. Misreporting by one of the efficient agents, in an attempt to free ride on the other, is no longer an issue. An efficient agent would never pay the induced amount of the side transfer to the other agent in order to free ride because his payoff after paying the side transfer would be less than the payoff without free riding at all. The other agent, however, would not let him free ride without receiving the side transfer (the other agent would also misreport his type without being paid the side transfer). Thus, the optimal contract under collusion removes the potential situation in which one agent misreports his type when both agents are efficient. As a result, the distortion in the output when reported types are different is recovered. We show that the principal is better off when the agents are able to collude.

To be sure, this is not the first study to show that collusion among subunits can be beneficial to an organization. (3) Holmstrom and Milgrom (1990) and Itoh (1993) argue that collusion between risk-averse agents results in an efficient risk allocation and thus allows the principal to save on risk compensation. Their studies, unlike ours, employ moral hazard frameworks in which risk sharing is the source of the incentive provision to the agents. We employ an adverse selection model in which information rent is the source of incentive provision. In our model, collusion between the agents results in an efficient rent allocation, which enables the principal to improve output efficiency. In this regard, the current paper is the adverse selection counterpart of their studies.

In adverse selection models, Tirole (1992), Kofman and Lawarree (1996), and Lambert-Mogiliansky (1998) consider situations in which collusion is allowed in the optimal contract. In these studies, however, the principal allows collusion between the agents because it is too costly to deter it--the principal's payoff would be higher if the agents could not collude in the first place. In their models, therefore, collusion is "detrimental." The studies on "beneficial" collusion in adverse selection models are those by Che (1995) and Olsen and Torsvik (1998), who show that the principal can increase her welfare when the agents can side contract. In Che (1995), for example, although collusion results in ex post inefficiency, it is optimal ex ante because under collusion the supervisor has more incentive to monitor the agent's type in order to receive a bribe. In their models, if side contracting occurs then hidden information remains hidden, whereas in our model, side contracting is induced with the revelation of information in the optimal contract. (4)

This paper is technically related to the models in Martimort (1997), Laffont and Martimort (1997, 1998), and Baron and Besanko (1992, 1999). The first three papers also study an optimal contract under wage compression. A key difference between their models and the model in our paper is that in their models, the outputs from the agents are assumed to be complements. In such a setting, collusion is detrimental because wage compression generates negative externality between the different agents. By contrast, in our model, outputs from the agents are substitutes and the wage compression generates positive externality. Using similar models, Baron and Besanko (1992, 1999) compare various organizational structures. They show that the principal prefers to have two agents acting like one (consolidation). Unlike ours, however, their result relies on the assumption that two agents play cooperatively even before participation--this relaxes the ex post participation constraints for each agent. In our model, the agents can play cooperatively only after participation (collusion). Collusion is beneficial in our model not only because wage payments to the agents become more flexible, but also because the principal can mitigate the misreporting incentives associated with free-riding opportunities.

Finally, McManus (2001) studies the optimal two-part pricing strategy to show that a monopolist's profit can increase if the consumers can share the product through postsale arbitrage among them. In his paper, however, there is no hidden information, and thus the monopolist faces no incentive problem. In our paper, the key issue is an agent's misreporting incentive to the principal in order to free ride on the other agent--collusion between the agents not only internalizes the externality, but also mitigates incentive problems.

The rest of the paper is organized as follows. The model is presented in the next section. In section 3, we discuss the optimal contract with and without collusion between the agents. We conclude with some remarks in section 4. All proofs are relegated to the appendices.

2. Model

A risk-neutral principal hires two risk-neutral agents for a project of a total outcome Q = [q.sup.A] + [q.sup.B], where [q.sup.A] and [q.sup.B] are the output levels assigned to agents A and B, respectively. (5) Each agent's type (cost parameter) [beta] can be low ([[beta].sub.L]) or high ([[beta].sub.H]), with [[beta].sub.L] < [[beta].sub.H] ([DELTA][beta] [equivalent to] [[beta].sub.H] - [[beta].sub.L]), Prob([[beta].sub.L]) = [[phi].sub.L], Prob([[beta].sub.H]) = [[phi].sub.H], and [[phi].sub.L] + [[phi].sub.H] = 1. The agents' types are stochastically independent and not known to the principal. The probability distribution of [beta] for each agent is public information.

The agents do not know each other's type. However, as in Laffont and Martimort (1997, 1998), we can treat the problem as if they learn each other's type after participation even though an agent reports only his own type. That an agent only reports his own type while he learns the other agent's type is justified in that [beta] is "soft information," i.e., no verifiable evidence on an agent's type can be obtained, and hence a court cannot assess it. (6) Before the agents engage in production, each agent reports his type [beta] to the principal. The agents can collude in reporting their types if side contracting is possible. The side contract is assumed to be enforceable. (7)

After each agent reports his cost parameter [beta], production takes place. Each agent produces his individual output (q) that corresponds to his report and sends it to the principal. The output levels are monitored perfectly, i.e., the principal receives [q.sup.A] and [q.sup.B] separately. The principal values the total output level Q = [q.sup.A] + [q.sup.B] according to a strictly concave value function V(Q), which satisfies the Inada condition. The principal's ex post payoff is V(Q) - ([t.sup.A] + [t.sup.B]), where [t.sup.A] and [t.sup.B] are the transfers paid to agents A and B, respectively. The cost of producing q to an agent is given by [beta]q, and hence each agent's ex post payoff is t - [beta]q. The agents can quit if they anticipate that their ex post payoffs are below the reservation level, which is normalized to zero.

To model wage compression in the simplest way, we assume that the agents receive equal transfers. (8) Thus, transfer to each agent is tL when both agents report [[beta].sub.L], [t.sub.M] when one agent reports [[beta].sub.L] and the other reports [[beta].sub.H], and [t.sub.H] when both report [[beta].sub.H].

We denote by [q.sub.ij] the individual output level assigned to an agent reporting [[beta].sub.i] (i = L, H) paired with an agent reporting [[beta].sub.j] (j = L, H). Therefore, the agents' total output is denoted by [Q.sub.ij] = [q.sub.ij] + [q.sub.ji] (by symmetry, [Q.sub.ij] = [Q.sub.ji]). To simplify the notation, we let [Q.sub.H] [equivalent to] [Q.sub.HH], [Q.sub.M] [equivalent to] [Q.sub.LH], and [Q.sub.L] [equivalent to] [Q.sub.LL]. We now can express each agent's individual output in terms of the total output. When the reported types are the same, both agents produce the same amount: [q.sub.LL] = [Q.sub.L]/2 and [q.sub.HH] = [Q.sub.H]/2. (9) When the reported types are different, [q.sub.LH] = r[Q.sub.M] and [q.sub.HL] = (1 - r)[Q.sub.M] with 1 [greater than or equal to] r [greater than or equal to] 0. That is, when the agents' types are different, the [[beta].sub.L] agent produces the proportion r of [Q.sub.M] and the [[beta].sub.H] agent produces 1 - r of [Q.sub.M]. The contract is contingent on the agents' reports on their types, and hence it specifies {[Q.sub.L], [Q.sub.M], r, [Q.sub.H], [t.sub.L], [t.sub.M], [t.sub.H]}. We summarize the timing of the game as follows.

* Each agent learns his type, followed by the contract offer from the principal.

* The agents accept or refuse the contract.

* Once the contract is accepted, the agents learn each other's type.

* If side contracting is possible, the agents can collude to coordinate their reports.

* Reports are made to the principal and the contract is executed (side transfers are exchanged between the agents if the side contract took place).

Benchmark. The First-Best Outcome

Before we move on, we look at the first-best outcome. This is the outcome under perfect wage discrimination and full information. The first-best outcome is characterized by the following expressions: V'([Q.sup.*.sub.L]) = [[beta].sub.L], V'([Q.sup.*.sub.M]) = [[beta].sub.L] with [r.sup.*] = 1, and V'([Q.sup.*.sub.H])= [[beta].sub.H]. An agent obtains no rent. Notice that when agents of different types are paired with each other, the [[beta].sub.L] agent produces the whole [Q.sup.*.sub.M] ([r.sup.*] = 1), and the [[beta].sub.H] agent gets paid zero transfer. This is an extreme result due to the constant marginal costs of production, which allows us to amplify the intuition without altering the main points of the paper. (10) In the following section, we will discuss the optimal contracts under hidden information when the principal must pay an equal amount of transfers to the agents.

3. Results

In this section, we first present the optimal contract when the agents cannot collude and then derive the optimal contract when collusion between the agents is possible. We will discuss incentive issues that cause distortion in the outcome in each case and compare the optimal outcomes to show that the principal's payoff is higher when the agents can collude with each other.

Contract under No Collusion ([C.sup.n])

As mentioned before, for expositional purposes, we assume that the agents learn each other's type after participation, but each agent reports his own type only ([beta] is soft information, and no verifiable evidence on an agent's type can be obtained by the other agent). Since the principal does not know the agents' types, the optimal contract under no collusion must satisfy the following individual incentive constraints (hereafter IC):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The ICs above assure an agent that his payoff will be higher when he reports his type truthfully. For example, ([IC.sup.n.sub.LH]) prevents a [[beta].sub.L] (low cost) agent from exaggerating his cost parameter when he is paired with a [[beta].sub.H] (high cost) agent. Since the agents can quit, the optimal contract must satisfy nonnegative payoff of the agents ex post for production to occur. The participation constraints (hereafter PC) below guarantee the reservation payoff of the agents in every state:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The optimal contract under no collusion, [C.sup.n], maximizes the principal's expected payoff,

[[phi].sup.2.sub.L][V([Q.sub.L])- 2[t.sub.L]]+ 2[[phi].sub.L][[phi].sub.H][V([Q.sub.M]) - 2[t.sub.M]] + [[phi].sup.2.sub.H][V([Q.sub.H]) - 2[t.sub.H]], (P)

subject to ([IC.sup.n.sub.LH]) ~ ([PC.sup.n.sub.HH]). The outcome in [C.sup.n] is characterized in the following lemma.

LEMMA 1. In [C.sup.n] x [Q.sup.n.sub.L] = [Q.sup.*.sub.L], [Q.sup.n.sub.M] < [Q.sup.*.sub.M], [Q.sup.n.sub.H] < [Q.sup.*.sub.H]. Moreover, 1/2 < [r.sup.n] < [r.sup.*] (= 1). A [[beta].sub.L] agent obtains a rent [DELTA][beta](1 - [r.sup.n]) [Q.sup.n.sub.M] if the other agent is [[beta].sub.L], and [DELTA][beta] [Q.sup.n.sub.H]/2 if the other agent is [[beta.sub.H], while a [[beta].sub.H] agent obtains no rent. See Appendix A for proof of Lemma 1.

The traditional result known as efficiency at the top (when both agents are [[beta].sub.L]) holds, and [Q.sub.L] is at its efficient level. There are, however, downward distortions in [Q.sub.M] and [Q.sub.H]. To explain the downward distortion in [Q.sub.H], we use binding ([IC.sup.n.sub.LH]) and ([PC.sup.n.sub.HH]) to express a [[beta].sub.L] agent's rent when he is paired with a [[beta].sub.H] agent as

[t.sub.M] - [[beta].sub.L] r[Q.sub.M] = [DELTA][beta][Q.sub.H]/2.

From the equation above, the principal distorts [Q.sub.H] downward to reduce the rent of a [[beta.sub.L] agent paired with a [[beta].sub.H] agent. Intuitively, a [[beta].sub.L] agent has an incentive to exaggerate his cost parameter as [[beta].sub.H] for a higher compensation. Therefore, by decreasing [Q.sub.H], the principal can reduce [t.sub.H], which in turn discourages misrepresentation by the [[beta].sub.L] agent when he is paired with a [[beta].sub.H] agent.

Our main interest in Lemma 1 is the distortion in [Q.sub.M] and r, the outcome when the different agents receive the same amount of transfer. To see the distortions, we express the rent of a [[beta].sub.L] agent (paired with another [[beta].sub.L] agent) as follows using binding constraints ([IC.sup.n.sub.LL]) and ([PC.sup.n.sub.HL]):

[t.sub.L] - [[beta].sub.L][Q.sub.L]/2 = [DELTA][beta](1 - r)[Q.sub.M]. (1)

It is clear from Equation 1 that the rent of a [[beta].sub.L] agent paired with another [[beta].sub.L] agent depends on [Q.sub.M] and r. From the right-hand side (RHS) of Equation 1, it seems that the principal can extract the rent by setting r = 1 (without distorting [Q.sub.M]). This, in fact, would be the case if the principal could discriminate perfectly the transfers to the agents--when the agents' types are different, the principal allocates the entire production to the [[beta].sub.L] agent (r = 1) and pays zero transfer to the agent reporting [[beta].sub.H]. Under wage compression, however, if the principal sets r = 1 then she must let the [[beta].sub.H] agent free ride on the [[beta].sub.L] agent because they both receive [t.sub.M]. In other words, with r = 1, the [[beta].sub.H]. agent collects [t.sub.M] for free. Therefore, when both agents are [[beta].sub.L], one of them may misreport his type in attempt to free ride on the other agent. To prevent this, the principal removes the potential free-riding opportunity by setting r < 1. As a result, the RHS of Equation 1 becomes strictly positive, and to reduce the rent to the agents, the principal distorts [Q.sub.M] downward in the optimal contract.

We summarize the intuition behind the main result without collusion between the agents. The positive externality between the different agents resulting from wage compression generates an incentive problem when both agents are efficient types--an efficient agent has an incentive to misreport his type to free ride on the other efficient agent. To discourage an efficient agent from such misrepresentation, the principal increases (decreases) an inefficient (efficient) agent's proportion of total production from the efficient level when the types of the agents are different.

Contract under Collusion ([C.sup.c])

In this subsection, we analyze the structure of the optimal contract when side contracting between the agents is possible. Although the possibility of collusion provides the agents with more opportunities to manipulate their information, it can be advantageous to the principal because by inducing side transfers, she may be able to make the agents internalize the positive externality between them.

To explain, we return to the contract under no collusion, [C.sup.n], as the starting point. Recall that, under no collusion, the principal distorts r downward (r < 1) to prevent the incentive problem associated with free-riding opportunities. Suppose the principal chooses [r.sup.*] and [Q.sup.*.sub.M], the efficient allocation and the efficient output level when the agents' types are different. Based on ([IC.sup.n.sub.LH]) in the previous subsection, to induce truth telling from the [[beta].sub.L] agent, transfer [t.sub.M] must be at least

[t.sub.M] = [[beta].sub.L][r.sup.*][Q.sup*.sub.M] + [t.sub.H] - [[beta].sub.L][Q.sub.H]/2, (2)

where the first term, [[beta].sub.L][r.sup.*][Q.sup.*.sub.M], is the [[beta].sub.L] agent's production cost and the last term, [t.sub.H] -- [[beta].sub.L][Q.sub.H]/2, is his information rent. Since [r.sup.*] = 1, the [[beta].sub.H] agent produces zero while collecting [t.sub.M]. Now, if the principal reduces [t.sub.M] from the level in Equation 2, while keeping [r.sup.*] and [Q.sup.*.sub.M], then the [[beta].sub.L] agent will lose the truth-telling incentive. If the [[beta].sub.L] agent misreports his type as [[beta].sub.H], however, the [[beta].sub.H] agent receives no rent from binding (P[C.sup.n.sub.HH]). Therefore, when side contracting is possible, the [[beta].sub.H] agent will pay a side transfer to the [[beta].sub.L] agent up to the following amount for being truthful:

[s.sub.M] [equivalent to] [t.sub.M] - [[beta].sub.H](1 - r)[Q.sub.M] - [[t.sub.H] - [[beta].sub.H][Q.sub.H]/2]. (3)

Thus, when the agents are of different types, a burden ([s.sub.M]) of rent provision to the [[beta].sub.L] agent can be transferred from the principal to the [[beta].sub.H] agent. With the possibility of inducing side transfers between the agents, the PCs are written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the definition of [s.sub.M] in Equation 3, if [s.sub.M] > 0 in (P[C.sup.c.sub.LH]) and ([PC.sup.n.sub.HL]), then there is a positive side transfer from the [[beta].sub.H] agent to the [[beta].sub.L] agent and vice versa. If [s.sub.M] = 0 in the optimal contract, then no side transfer is exchanged between the agents of different types. Notice that when the agents' types are the same, no side transfer is induced in the optimal contract. For example, if a side transfer is induced when both agents are [[beta].sub.H], the PC for one agent would be [t.sub.H] - [[beta].sub.H][Q.sub.H]/2 + [s.sub.H] [greater than or equal to] 0 and for the other agent [t.sub.H] - [[beta].sub.H][Q.sub.H]/2 - [s.sub.H] [greater than or equal to] 0. Clearly, [s.sub.H] = 0 at the optimum. Similarly, [s.sub.L] = 0 when both agents are [[beta].sub.L].

When the side transfer [s.sub.M] between a [[beta].sub.L] and a [[beta].sub.H] agent is induced, it affects not only the PCs, but the ICs as well. The individual incentive constraints that prevent an agent from misreporting his type are written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since the side transfer [s.sub.M] may take place in equilibrium, the ICs become more complex. The side transfer [s.sub.M] does not appear in the RHS of the ICs because the expressions in the RHSs are an agent's payoffs from misreporting without side contracting. For example, in ([IC.sup.c.sub.LL]), a [[beta].sub.L] agent's payoff from misreporting his type as [[beta].sub.H] is [t.sub.M] - [[beta].sub.L](1 - r)[Q.sub.M] if the other agent reports [[beta].sub.L] without being paid [s.sub.M], and [t.sub.H] - [[beta].sub.L][Q.sub.H]/2 if the other agent reports [[beta].sub.H]. Notice that with zero side transfer ([s.sub.M] = 0), the ICs here become identical to those in [C.sup.n].

With the possibility of side contracting, the agents would cooperatively misreport their types if their joint payoff would improve by doing so. Therefore, the agents have extra room for manipulating information under collusion, and ICs may fail to induce truth telling from the agents. To assure truthful reports from the agents, the principal must impose the following coalition incentive constraints (hereafter CIC) in her maximization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The RHS of a CIC is the joint payoff of the agents when one or both of them misreport his/ their type(s) with side contracting, and the CICs prevent all combinations of misreporting. The positive and negative flows of the side transfer [s.sub.M] between the agents of different types cancel each other out in the left-hand side (LHS) of ([CIC.sub.LH,HH]) and ([CIC.sub.LH,LL]), and the RHS of ([CIC.sub.LL,LH]) and ([CIC.sub.HH,LH]).

The optimal contract under collusion, [C.sup.c], maximizes the principal's expected payoff in (P), subject to ([PC.sup.c.sub.LH]) ~ ([CIC.sub.HH,LL]). The outcome in C is characterized below.

LEMMA 2. In [C.sup.c], [Q.sup.c.sub.L] = [Q.sup.*.sub.L], [Q.sup.c.sub.M] = [Q.sup.*.sub.M], [Q.sup.c.sub.H] < [Q.sup.*.sub.H]- Moreover, [r.sup.c] = [r.sup.*](= 1). A side transfer [s.sup.c.sub.M] > 0 takes place in the optimal contract. A [[beta].sub.L] agent obtains a rent [DELTA][beta][Q.sub.c.sub.H]/2, while a [[beta].sub.H] agent obtains no rent. Compared with the output schedule in [C.sub.n], [Q.sup.c.sub.L] = [Q.sup.n.sub.L], [Q.sup.c.sub.M] > [Q.sup.n.sub.M], [Q.sup.c.sub.H] [??] [Q.sup.n.sub.H]. For proof of Lemma 2, see Appendix B.

The main points in Lemma 2 are that a strictly positive side transfer occurs with truthful reports and that the efficient allocation [r.sup.*] and the output level [Q.sup.*.sub.M] are restored. This result comes from the fact that, under collusion, it is not costly to the principal to prevent the potential situation in which one of the agents misreports his type when both agents are [[beta].sub.L]. To see the intuition behind this result, first consider ([IC.sup.c.sub.LH]), which is biding in the optimal contract:

[t.sub.M] - [[beta].sub.L][Q.sub.M] + [s.sub.M] = [t.sub.H] - [[beta].sub.L] [Q.sub.H]/2. (4)

From the above equation, a [[beta].sub.L] agent would truthfully report his type to produce [Q.sub.M] ([??] r = 1) only if the amount of the side transfer [s.sub.M] (= [t.sub.M] from binding ([PC.sup.c.sub.HL]) with r = 1) is side contracted with the [[beta].sub.H] agent. This means that the principal transfers the [[beta].sub.H] agent half the rent provision to the [[beta].sub.L] agent. Suppose now that both agents are [[beta].sub.L]. If one of the [[beta].sub.L] agents tries to free ride on the other agent by misreporting his type as [[beta].sub.H], then he must side contract with the other [[beta].sub.L] agent to pay the side transfer [s.sub.M] to him. Otherwise, Equation 4 implies that the other agent will also misreport his type as [[beta].sub.H]. However, a [[beta].sub.L] agent has no incentive to pay [s.sub.M] to the other [[beta].sub.L] agent for free riding (by misreporting his type alone). This is because, as the following inequality shows, a [[beta].sub.L] agent's payoff after paying the side transfer [s.sub.M] to the other [[beta].sub.L] agent becomes lower than his payoff without free riding.

[t.sub.L] - [[beta].sub.L][Q.sub.L]/2 > [t.sub.M] - [s.sub.M](= 0).

Therefore, when side contracting between the agents is possible, the principal does not need to worry about the potential situation in which one of the agents misreports his type when both agents are [[beta].sub.L]. By inducing side contracting between the agents of different types, the principal can costlessly deter a [[beta].sub.L] agent from misreporting his type in an attempt to free ride on the other [[beta].sub.L] agent.

More technically, from binding ([IC.sup.C.sub.LH]) as in Equation 4, the following inequality holds in the optimal contract: [t.sub.M] - [[beta].sub.L] r[Q.sub.M] < [t.sub.H] - [[beta].sub.L][Q.sub.H]/2. This inequality implies that binding ([IC.sup.C.sub.LL]) in the optimal contract is

[t.sub.L] - [[beta].sub.L] [Q.sub.L]/2 = [t.sub.H] - [[beta].sub.L][Q.sub.H]/2. (5)

After substituting for [t.sub.H] by its value (11) in the RHS of Equations 4 and 5, a [[beta].sub.L] agent's rent is expressed as [DELTA][beta][Q.sup.c.sub.H]/2 regardless of the other agent's type. Also, among the CICs, the ones that prevent exaggeration of the cost parameter(s) are ([CIC.sub.LH,HH]), ([CIC.sub.LL,LH]), and ([CIC.sub.LL,HH]). However, ([CIC.sub.LL,LH]) has no bite in the optimal contract, which implies that when both agents are [[beta].sub.L], the principal does not need to worry about a potential situation in which one of the [[beta].sub.L] agents misreports his type as [[beta].sub.H]. From the RHS of binding ([CIC.sub.LH,HH]) and ([CIC.sub.LL,HH]), the rent of a [[beta].sub.L] agent is again expressed as [DELTA][beta][Q.sup.c.sub.H]/2. Since [Q.sub.H] is the only source of the information rent in every case, although the principal distorts [Q.sub.H] downward to reduce the rent, she does not need to distort r and [Q.sub.M] in the optimal contract.

It is also noteworthy that, although [Q.sub.H] is distorted downward, it is possible that the distortion is smaller than in the case of no collusion ([Q.sup.c.sub.H] > [Q.sup.n.sub.H]), depending on parameters. Recall that in [C.sup.n] (under no collusion), [Q.sub.H] is the source of rent for a [[beta].sub.L] agent only when the other agent is [[beta].sub.H]. In [C.sup.c] (under collusion), [Q.sub.H] is the source of rent for a [[beta].sub.L] agent regardless of the type of the other agent. Therefore, the principal may distort [Q.sub.H] further downward. However, in [C.sup.c], the principal and a [[beta].sub.H] agent share rent provision to a [[beta].sub.L] agent, which gives the principal some room to recover the distortion in [Q.sub.H]. If this effect is significant enough, then the distortion in [Q.sub.H] under collusion becomes smaller than the distortion under no collusion.

In summary, when the agents can collude, the principal designs a contract such that an efficient agent misreports his type unless an inefficient agent pays a side transfer when they face each other. This way, the principal can effectively remove free riding by an inefficient agent, which in turn mitigates an efficient agent's misreporting incentive to free ride on the other agent.

By comparing the optimal contract in [C.sup.n] and [C.sup.c], we present our main result in the following proposition.

PROPOSITION 1. The principal's payoff is higher in [C.sup.c] (under collusion) than her payoff in [C.sup.n] (under no collusion). For proof of Proposition 1, see Appendix C.

The result above suggests that under wage compression, collusion between the agents is beneficial to the principal. There are two sources from which the principal can benefit by inducing side contracting between agents of different types. First, the principal can make a [[beta].sub.L] agent and a [[beta].sub.H] agent internalize the externality between them. Therefore, some burden of rent provision to a [[beta].sub.L] agent is transferred from the principal to a [[beta].sub.H] agent. Second, inducing a side transfer between a [[beta].sub.H] and a [[beta].sub.L] agent automatically prevents a misreporting incentive when both agents are [[beta].sub.L]. As mentioned, a [[beta].sub.L] agent would not let the other [[beta].sub.L] agent misreport his type alone (so that he can free ride) without receiving the induced amount of the side transfer. However, there will be no side transfer between the two [[beta].sub.L] agents because one agent's payoff after paying the side transfer to the other is lower than his payoff without free riding at all. Therefore, under collusion, it is no longer the principal's concern that one of the [[beta].sub.L] agents misreports his type as [[beta].sub.H] to free ride on the other agent's production. As a result, the distortion in r and [Q.sub.M] under no collusion can be recovered when collusion is possible.

4. Conclusion with Remarks

In this paper, we have presented a model in which side contracting between agents improves the principal's welfare. In our model, hidden information and the free-riding problem interact with each other because wage discrimination against the agents of different types is limited. We have shown that, under collusion, the principal can reduce the amount of the rent provision by inducing an inefficient agent to bribe an efficient agent, who in turn removes a misreporting incentive for free riding when both agents are efficient. As a result, the output distortions associated with hidden information are partly recovered, and the principal's payoff becomes higher under collusion between the agents.

We close this paper with several remarks. First, if the wage transfers to the different agents can be perfectly discriminated, then, as usual, the principal's payoff is higher under no collusion. In our model, limited wage discrimination does not affect the optimal outcome if the principal only needs to satisfy the ex ante participation constraints for the agents (i.e., the agents cannot quit). In such a setting, the principal can let an inefficient agent free ride when the other agent is efficient, but makes his ex post rent negative when the other agent is also inefficient (thus the expected rent of an inefficient agent is zero). Similarly, the principal lets an efficient agent enjoy a relatively large rent when the other agent is also efficient, but pays a relatively small transfer when the other agent is inefficient. Since all parties are risk neutral, wage compression with such averaging out has the same effect ex ante as perfect wage discrimination, and the principal prefers the outcome under no collusion. Second, for expositional purpose, we assumed that there is no transaction cost related to side transfers between the agents. In our model, the principal always will induce side contracting unless the transaction cost is less than the amount of the side transfer. (12) Third, as mentioned before (footnote 4), hiring multiple agents in our model is justified because the principal can hire an efficient agent with a higher probability--it is possible that the principal wants to hire only one agent depending on the parameters. (13) Finally, although the possibility of collusion is beneficial to the principal, it may not be for the agents. (14) In such cases, if the agents have commitment power before the principal's contract offer, they may commit to avoid collusion after participation.

Appendix A

PROOFOF LEMMA 1. Since the objective function is concave and the constraints are linear thus convex, the solution is unique. Thus, we show that ([PC.sup.n.sub.HH]), ([PC.sup.n.sub.HL]), ([IC.sup.n.sub.LH]), and ([IC.sup.n.sub.LL]) are binding in deriving the solution. It is straightforward to show that the other constraints are satisfied with the solution without them. With ([PC.sup.n.sub.HH]), ([PC.sup.n.sub.HL]), ([IC.sup.n.sub.LH]), and ([IC.sup.n.sub.LL]), the Lagrangian is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[partial derivative]L/[partial derivative][Q.sub.L] = [[phi].sup.2.sub.L]V'([Q.sub.L]) - [phi][[beta].sub.L]/2 = 0, (A1)

[partial derivative]L/[partial derivative][Q.sub.M] = 2[[phi].sub.L][[phi].sub.H]V'([Q.sub.M]) - [mu][[beta].sub.H] (1 - r) - [delta][[beta].sub.L]r + [sigma][[beta].sub.L](1 - r) = 0,(A2)

[partial derivative]L/[partial derivative][Q.sub.H] = [[phi].sup.2.sub.H]V'([Q.sub.H]) - [lambda][[beta].sub.H]/2 + [delta][[beta].sub.L]/2, (A3)

[partial derivative]L/[partial derivative][t.sub.L] = 2[[phi].sup.2.sub.L] + [sigma] = 0, (A4)

[partial derivative]L/[partial derivative][t.sub.M] = -4[[phi].sub.L][[phi].sub.L] + [mu] + [delta] - [phi] = 0,(A5)

[partial derivative]L/[partial derivative][t.sub.H] = 2[[phi].sup.2.sub.H] + [lambda] - delta] = 0, (A6)

[partial derivative]L/[partial derivative][lambda] = [t.sub.H] - [[beta].sub.H][Q.sub.H]/2 [greater than or equal to] 0, [lambda] [greater than or equal to] 0, [lambda] [partial derivative]L/[partial derivative][lambda] = 0, (A7)

[partial derivative]L/[partial derivative][mu] = [t.sub.m] - [[beta].sub.H](1 - r)[Q.sub.M] [greater than or equal to] 0, [mu] [greater than or equal to] 0, [mu] [partial derivative]L/[partial derivative][mu] = 0, (A8)

[partial derivative]L/[partial derivative][delta] = [t.sub.M] - [[beta].sub.L]r[Q.sub.M] - [[t.sub.H] - [[beta].sub.L][Q.sub.H]/2] [greater than or equal to] 0, [delta] [greater than or equal to] 0, [delta] [partial derivative]L/[partial derivative][delta] = 0, (A9)

[partial derivative]L/[partial derivative][phi] = [t.sub.L] - [[beta].sub.L][Q.sub.L]/2 - [[t.sub.M] - [[beta].sub.L] (1 - r)[Q.sub.M]] [greater than or equal to] 0, [phi] [greater than or equal to] 0, [phi][partial derivative]L/[partial derivative][phi] = 0. (A10)

From Equation A4, [phi] = 2[[phi].sup.2.sub.L] (>0) and thus Equation A10 implies that ([IC.sup.n.sub.LL]) is binding. Similarly, by Equations A6 and A7, ([PC.sup.n.sub.HH]) is binding. Also, from Equation A2, [mu] = [2[[phi].sub.L][[phi].sub.H] V'([Q.sub.M]) - [[delta][[beta].sub.L] r + 2[[phi].sup.2.sub.L] (1 -- r)]/[[beta].sub.H] (1 - r)], and hence r [not equal to] 1. Thus, by differentiating the Lagrangian with respect to r, we have [mu][[beta].sub.H] - [delta][[beta].sub.L] - [phi][[beta].sub.L] = 0 or [mu] = ([delta] + [phi])[[beta].sub.L] /[[beta].sub.H] (>0). This, together with Equation A8, implies that ([PC.sup.n.sub.HL]) is binding. Next, Equations A5 and A6 give

[mu]+ [lambda] = 2. (A11)

Since [mu] = ([delta] + [phi])[[beta].sub.L]/[[beta].sub.H] > 0, [[phi] = 2[[phi].sup.2.sub.L], and k = 2[[phi].sup.2.sub.H] + [delta], Equation A11 gives [delta] = 2[[phi].sup.2.sub.L] ([[beta].sub.H - [[beta].sub.L])/([[beta].sub.H] + [[beta].sub.L]) (>0). Thus, ([IC.sup.n.sub.LH]) is binding by Equation A9. From biding ([PC.sup.n.sub.HH]), ([PC.sup.n.sub.HL]), and ([IC.sup.n.sub.LH]), we have

r = [[beta].sub.H]/[[beta].sub.H] + [[beta].sub.L] - ([[beta].sub.H] - [[beta].sub.L)[Q.sub.H]/2([[beta].sub.H] + [[beta].sub.L])[Q.sub.M]. (A12)

Likewise, the transfers are obtained by the binding constraints ([PC.sup.n.sub.HH]), ([IC.sup.n.sub.LH]), and ([IC.sup.n.sub.LL]). Substituting for r and the transfers by their values in the objective function and differentiating with respect to [Q.sub.L], [Q.sub.M], and OH yields

V'([Q.sup.n.sub.L]) = [[beta].sub.L], V'([Q.sup.n.sub.M]) = [[beta].sub.H] + [[beta].sub.H] - [[beta].sub.L]/[[beta].sub.H] + [[beta].sub.L] [[beta].sub.L] ([[phi].sub.L]/[[phi].sub.H]) [[beta].sub.H] + [[beta].sub.H] - [[beta].sub.L]/[[beta].sub.H] + [[beta].sub.L] [[beta].sub.L],

and

V'([Q.sup.n.sub.H]) = [[beta].sub.H] + (2[[phi].sub.L]/[[phi].sub.H])[[beta].sub.H] - [[beta].sub.L]/[[beta].sub.H] + [[beta].sub.L] [[beta].sub.H] + ([[phi].sub.L]/[[phi].sub.H])[sup.2][([[beta].sub.H] - [[beta].sub.L]).sup.2]/[[beta].sub.H] = [[beta].sub.L].

From the above expressions, [Q.sup.n.sub.L] = [Q.sup.*.sub.L], [Q.sup.n.sub.M] < [Q.sup.*.sub.M], [Q.sup.n.sub.H] < [Q.sup.*.sub.H]. Since [Q.sup.n.sub.L] > [Q.sup.n.sub.M] > [Q.sup.n.sub.H], together with the expression in Equation A12, we can verify that 1/2 < [r.sup.n] < 1 (= [r.sup.*]). The rent of each agent is obtained from binding constraints ([PC.sup.n.sub.HH]), ([IC.sub.n.sub.LH]), and ([IC.sup.n.sub.LL]). QED.

Appendix B

PROOF OF LEMMA 2. AS usual in the model of this type, the CICs encompass the ICs since the CICs prevent all combinations of misreporting. Thus, we construct the Lagrangian with ([CIC.sub.LH,HH]), ([CIC.sub.LL,HH]), and ([PC.sub.HH]) and show that these constraints are binding in deriving the optimal outcome. It is straightforward to verify that other constraints are satisfied by our solution. The Lagrangian is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[partial derivative]L/[partial derivative][Q.sub.L] [[phi].sup.2.sub.L]V'([Q.sub.L]) - [theta][[beta].sub.L] = 0, (B1)

[partial derivative]L/[partial derivative][Q.sub.M] = 2[[phi].sub.L][[phi].sub.H]V'([Q.sub.M]) - [sigma][r[[beta].sub.L] + (1 - r)[[beta].sub.H] = 0, (B2)

[partial derivative]L/[partial derivative][Q.sub.H] = [[phi].sup.2.sub.H]V'([Q.sub.H]) + [sigma]([[beta].sub.H] + [[beta].sub.L])/2 + [theta][[beta].sub.L] - [omega][[beta].sub.H]/2 = 0, (B3)

[partial derivative]L/[partial derivative][t.sub.L] = -2[[phi].sup.2.sub.L] + 2[theta] = 0, (B4)

[partial derivative]L/[partial derivative][t.sub.M] = -4[[phi].sub.L][[phi].sub.H] + 2[sigma] = 0, (B5)

[partial derivative]L/[partial derivative][t.sub.H] = -2[[phi].sup.2.sub.H] - 2[sigma] - 2[theta] - [omega] = 0, (B6)

[partial derivative]L/[partial derivative][sigma] = 2[t.sub.M] - [B.sub.L]r[Q.sub.M] - [[beta].sub.H](1 - r)[Q.sub.M] - [2[t.sub.h] - [[beta].sub.L][[Q.sub.H]/2] [greater than or equal to] 0, [theta] [partial derivative]L/[partial derivative][sigma] = 0, (B7)

[partial derivative]L/[partial derivative][theta] = 2[t.sub.L] - 2[[beta].sub.L][Q.sub.L]/2 - [2[t.sub.H] - 2[[beta].sub.L][Q.sub.H]/2] [greater than or equal to] 0, [theta] [greater than or equal to] 0, [theta] [partial derivative]L/[partial derivative][theta] = 0, (B8)

[partial derivative]L/[partial derivative][omega] = [t.sub.H] - [[beta].sub.H][Q.sub.H]/2 [greater than or equal to] 0, [omega] [greater than or equal to] 0, [omega][partial derivative]L/[partial derivative][omega] = 0. (B9)

From Equation B4, [theta] = [[phi].sup.2.sub.L] (>0), and hence Equation B8 implies that ([CIC.sub.LL,HH]) is binding. From Equation B5, we have [sigma] = 2[[phi].sub.L][[phi].sub.H], and binding ([CIC.sub.LH,HH]) is implied by Equation B7. Also from Equation B6, [omega] = 2[[phi].sup.2.sub.H] + 2[sigma] + 2[theta] (>0). Therefore, Equation B9 implies that ([PC.sup.c.sub.HH]) is binding. Differentiating the Lagrangian with respect to r gives [sigma]([[beta].sub.H] [[beta].sub.L])[Q.sub.M] 0, which implies that r = 1 at the optimum. The transfers and each agent's rent are obtained from the binding constraints ([CIC.sub.LH,HH]), ([CIC.sub.LL,HH]), and ([PC.sub.HH]), and with r = 1 in Equation 3, we have [S.sub.M] = [t.sub.M] > 0. Replacing the transfers with their values in the objective function and differentiating with respect to [Q.sub.L], [Q.sub.M], and [Q.sub.H] yields

V'([Q.sup.c.sub.L]) = [[beta].sub.L]

V'([Q.sup.c.sub.M]) = [[beta].sub.L],

and

V'([Q.sup.c.sub.H]) = [[beta].sub.H] + ([[PHI].sub.L]/[[PHI].sub.H]) ([[beta].sub.H] - [[beta].sub.L] + [([[PHI].sub.L]/[[PHI].sub.H]).sup.2] ([[beta].sub.H] - [[beta].sub.L]).

From the above equations, we have [Q.sup.c.sub.L] = [Q.sup.*.sub.L] (= [Q.sup.n.sub.L]), [Q.sup.c.sub.M] = [Q.sup.*.sub.M] (> [Q.sup.n.sub.M]), [Q.sup.c.sub.H] < [Q.sup.*.sub.H] (and [Q.sup.n.sub.H] [greater than or less than to] [Q.sup.n.sub.H] if [[beta].sub.H] + [[beta].sub.L] / [[beta].sub.H] - [[beta].sub.L] [greater than or less than to] [[phi].sub.L]/ [[phi].sub.H] 2[[beta].sub.L]/ [[beta].sub.H] + [[beta].sub.L]). QED.

Appendix C

PROOF OF PROPOSITION 1. We denote by [[PHI].sup.n] and [[PHI].sup.c] the principal's optimal payoff in [C.sup.n] and [C.sup.c] respectively. Each payoff can be expressed as

[[PHI].sup.n] = [[phi]].sup.2.sub.L][[V.sup.*.sub.L] - 2[[beta].sub.L](2[r.sup.n] - 1)[Q.sub.M]-([[beta].sub.H] - [beta].sub.L])[Q.sup.n.sub.H]] + 2[[phi].sub.H] [[phi].sub.L][[V.sup.n.sub.M] - [[beta].sub.H] - [[beta].sub.L]) [Q.sup.n.sub.H]/2] + [[phi].sup.2.sub.H][[V.sup.n.sub.H]]

[[PHI].sup.c] = [[phi]].sup.2.sub.L][[V.sup.*.sub.L] - [[beta].sub.H] - [[beta].sub.L]) [Q.sup.c.sub.H]] + 2[[phi].sub.H] [[phi].sub.L] [[V.sup.*.sub.M] - ([[beta].sub.H] - [[beta].sub.L]) [Q.sup.c.sub.H]/2] + [[phi].sup.2.sub.H][[V.sup.c.sub.H]],

where [V.sup.*.sub.L] [equivalent to] V([Q.sup.*.sub.L]) - [[beta].sub.L] [Q.sup.*.sub.L] [V.sup.n.sub.M] [equivalent to] V([Q.sup.n.sub.M] - [[beta].sub.L] [r.sup.n] + [[beta].sub.H] (1 - [r.sup.n])] [Q.sup.n.sub.M], [V.sup.n.sub.H] [equivalent to] V([Q.sup.n.sub.H] - [[beta].sub.H] [Q.sup.n.sub.H], [V.sup.*.sub.M] [equivalent to] V([Q.sup.*.sub.M] - [[beta].sub.L] [Q.sup.*.sub.M], and [V.sup.c.sub.H] [equivalent to] V([Q.sup.c.sub.H]) - [[beta].sub.H] [Q.sup.c]. In [[PHI].sup.c], we can adjust the level of [Q.sub.H] from [Q.sup.c.sub.H] to [Q.sup.n.sub.H] without violating the constraints. We define this adjusted payoff as

[[PHI].sup.c.sub.A] = [[phi].sup.2.sub.L][[V.sup.*.sub.L] - ([[beta].sub.H] - [[beta].sub.L]) [Q.sup.n.sub.H]] + 2[[phi].sub.H] [[phi].sub.L] [[V.sup.*.sub.M] - ([[beta].sub.H] - [[beta].sub.L]) [Q.sup.n.sub.H]/2] + [[phi].sup.2.sub.H] [[V.sup.n.sub.H]].

By comparing [[PHI].sup.c.sub.A] and [[PHI].sup.n], it is clear that [[PHI].sup.c.sub.A] > [[PHI].sup.n], which is followed by [[PHI].sup.c] > [[PHI].sup.n]. QED.

I would like to thank Ingela Alger, Fahad Khalil, Jacques Lawarree, two anonymous referees, and the editor for detailed comments and suggestions. I also thank Ching-To Albert Ma, Bill Sundstrom, Gerald Roland, and the seminar participants at the 2006 spring Midwest Economic Theory Conference for helpful comments. Received March 2005; accepted October 2006.

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(1) See Baron and Kreps (1999). Eccles (1985), in his empirical findings, shows that organizations often face a "fairness" restriction because subunits in similar positions within an organizational hierarchy see as unfair the fact that they may receive different transfers resulting from exogenous parameters.

(2) As Mintzberg (1983) notes, not only direct monetary exchanges, but also sharing resources with other subunits, can be interpreted as side transfers.

(3) As mentioned above, collusion among agents typically limits the welfare of the principal. See Tirole (1986), Kofman and Lawarree (1993), and Kessler (2000) for examples.

(4) See also Lawarree and Shin (2005) for information revelation when side contracting takes place. In their study, however, side contracting is not actively induced by the principal as in the current paper.

(5) Since outputs are perfect substitutes, the principal may want to hire one agent. Hiring multiple agents is justified by increasing the likelihood of having an efficient agent.

(6) Therefore, we are implicitly assuming that if an agent's type is also reported by the other agent, an agent has a right to protest the other agent's report in the court of law, and resolution is prohibitively costly, for there is no hard evidence.

(7) This is a standard assumption. See Tirole (1992) for a discussion of enforceability of side contracts.

(8) Our main result holds as long as the principal cannot perfectly discriminate the transfers.

(9) In the principal's problem, constraints are identical for agents of the same type. which implies that [q.sub.LL] = [Q.sub.L]/2 and [q.sub.HH] = [Q.sub.H]/2.

(10) For example, our main result holds with quadratic cost functions. As mentioned before, hiring multiple agents is justified by increasing the likelihood of having an efficient agent.

(11) From binding ([PC.sup.c.sub.HH]), [t.sub.H] = [[beta].sub.H][Q.sub.H] /2.

(12) If the transaction cost is prohibitively large, the principal's payoff in [C.sup.c] equals her payoff in [C.sup.n].

(13) In this paper, we focused on ranges of the parameters where the following inequality holds: [[phi].sup.2.sub.L] [V([[phi].sup.n.sub.L]) - [[beta].sub.L][Q.sup.n.sub.L] - 2(2[r.sup.n] - 1)[[beta].sub.L][Q.sup.n.sub.M] - [DELTA][beta][Q.sup.n.sub.H]] + 2[[phi].sub.H][[phi].sub.L][V([Q.sup.n.sub.M]) - [[r.sup.n][[beta].sub.L] + (1 - [r.sup.n])[[beta].sub.H]] [Q.sup.n.sub.M] - [DELTA][beta][Q.sup.n.sub.H]/2] + [[phi].sup.2.sub.H] [V([Q.sup.n.sub.H]) - [[beta].sub.H] [Q.sup.n.sub.H]] > [[phi].sub.L][V([Q.sup.b.sub.L]) - [[beta].sub.L][Q.sup.b.sub.L] - [DELTA][beta][Q.sup.b.sub.H]] + [[phi].sub.H][[V([Q.sup.b.sub.H]) - [[beta].sub.H][Q.sup.b.sub.H]], where [Q.sup.b.sub.L] and [Q.sup.b.sub.H] are the well-known Baron and Myerson (1982) outcome in the standard one-agent framework (V'([Q.sup.b.sub.L]) = [[beta].sub.L] and V'([Q.sup.n.sub.H]) = [[beta].sub.H] + [DELTA][beta]([[phi].sub.L]/[[phi].sub.H])). It can be easily verified that [Q.sup.n.sub.M] > [Q.sup.b.sub.H], which implies that the principal may want to hire two agents.

(14) Since a [[beta].sub.H] agent's rent is zero in both [C.sub.n] and [C.sub.n], the agents are weakly better off in [C.sup.c] when the following inequality holds for a [[beta].sub.L] agent's rent: [[phi].sub.L](1 - [r.sup.n]) [Q.sup.n.sub.M] + [[phi].sub.H][Q.sup.n.sub.H]/2 < [Q.sup.c.sub.H]/2.

Dongsoo Shin, Department of Economics, Leavey School of Business, Santa Clara University, Santa Clara, CA 95953, USA; E-mail dshin@scu.edu.

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Comment: | Contracts under wage compression: a case of beneficial collusion. |
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Author: | Shin, Dongsoo |

Publication: | Southern Economic Journal |

Article Type: | Report |

Date: | Jul 1, 2007 |

Words: | 8986 |

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