Contracts under wage compression: a case of beneficial collusion.

• Author: Shin, Dongsoo

1. Introduction

   In 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]

   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 -...