The value of information and optimal organization.

To understand the significance of these two effects, we will consider three alternative contractual arrangements. First, consider hierarchy [H.sub.D] in which the primary contractor does not make a cost report to the principal before communicating with the subcontractor. Formally, the sequence of steps in [H.sub.D] is the same as in [H.sub.1], except that stage 3 is eliminated, and in stage 5 the primary contractor reports both costs to the principal. Because the primary contractor accepts the contract with the principal before interacting with the subcontractor, only the interim participation constraints of the primary contractor have to hold. Hierarchy [H.sub.D] appears to be a good representation of contracting in the defense industry, where marginal costs of production are not learned until significant fixed costs have been incurred, production lines have been built, and supplier relationships have been established.

In [H.sub.D], the primary contractor has a larger set of possible deviations than in [H.sub.1], as she may decide to misrepresent her cost for one realization of the subcontractor's cost but not for the other realization. Consequently, the primary contractor can try to appropriate some of the informational rents intended by the principal for the subcontractor. In particular, under complementarity, the primary contractor will have an incentive to misrepresent the state LH as HH in order to reduce the informational rent that she pays to the subcontractor in state LL. As a result, [H.sub.D] attains the performance of the two-agent mechanism under more restrictive conditions than [H.sub.1]. Precisely, we have the following.

Proposition 6. If agent i [member of] {1, 2} serves as the primary contractor, then [H.sub.D] attains the same performance as the two-agent mechanism if [absolute value of [v.sub.12]([q.sub.1], [q.sub.2])/[v.sub.ii]([q.sub.1, [q.sub.2])] [less than or equal to] 1 - [p.sub.i]/1 - [p.sub.j], j [not equal to] i, for all ([q.sub.1], [q.sub.2]) [member of] [[[q.bar].sub.1, [[bar.q].sub.1]] x [[[q.bar].sub.2], [[bar.q].sub.2]]. Conversely, the hierarchy [H.sub.D] with agent i [member of] {1, 2} as the primary contractor is strictly less profitable for the principal if [absolute value of [v.sub.12]([q.sub.1], [q.sub.2])/[v.sub.ii]([q.sub.1, [q.sub.2])] [less than or equal to] 1 - [p.sub.i]/1 - [p.sub.j] > 1 - [p.sub.i]/1 - [p.sub.j], j [not equal to] i, for all ([q.sub.1], [q.sub.2]) [member of] [[[q.bar].sub.1, [[bar.q].sub.1]] x [[[q.bar].sub.2], [[bar.q].sub.2]].

If either agent can serve as the primary contractor, then [H.sub.D] attains the same performance as the two-agent mechanism if [absolute value of [v.sub.12]([q.sub.1], [q.sub.2])/[v.sub.ii]([q.sub.1, [q.sub.2])] [less than or equal to] 1 - [p.sub.i]/1 - [p.sub.j] [less than or equal to] 1/1 - [p.sub.j] for each i [member of] {1, 2} and j [not equal to] i, and all ([q.sub.1], [q.sub.2]) [member of] [[[q.bar].sub.1], [[bar.q].sub.1]] x [[[q.bar].sub.2], [[bar.q].sub.2]].

Comparison of Propositions 5 and 6 shows that the additional deviations available to the primary contractor in hierarchy [H.sub.D] have real consequences, and in some cases [H.sub.1] is strictly more profitable for the principal than [H.sub.D]. Specifically, under intermediate degrees of complementarity, the principal in [H.sub.D] has to leave a higher informational rent to the primary contractor to prevent the latter from exaggerating her cost in state LH (without a misreport in state LL). This deviation--unavailable in [H.sub.1]--allows the primary contractor to reduce the informational rent which she pays to the subcontractor in state LL. Similarly, under intermediate degrees of substitutability, [H.sub.D] becomes more costly for the principal because she has to prevent the primary contractor from exaggerating her cost only in state LL, with state LH announced truthfully. This deviation is also unavailable in [H.sub.1].

Finally, suppose that the primary contractor could opt out of the contract after receiving the subcontractor's report. Then the individual rationality constraints of the primary contractor have to hold ex post. Accordingly, let [H.sup.ep.sub.D]/([H.sup.ep.sub.D]) be a modification of hierarchy [H.sub.1] ([H.sub.D]) obtained by giving the primary contractor an option to withdraw after receiving the subcontractor's cost report in Stage 5. We then have the following.

Proposition 7. Under substitutability, both [H.sup.ep.sub.D] and [H.sup.ep.sub.D] are strictly less profitable for the principal than the two-agent mechanism.

Under complementarity, we have:

(i) [H.sup.ep.sub.D] attains the same performance as the two-agent mechanism if [H.sub.1] attains such performance.

(ii) If agent i [member of] {1, 2} serves as the primary contractor, then [H.sup.ep.sub.D] attains the same performance as the two-agent mechanism if [H.sub.D] attains the same performance and, additionally, [absolute value of [v.sub.12]([q.sub.1],[q.sub.2])/[v.sub.jj]([q.sub.1], [q.sub.2])] [less than or equal to] 1 - [p.sub.j]/[p.sub.j] for all ([q.sub.1], [q.sub.2]) [member of] [[[q.bar].sub.2], [[bar.q].sub.2]], [p.sub.j] is sufficiently small and [p.sub.i] is sufficiently large. (14)

In [H.sup.ep.sub.D] and [H.sup.ep.sub.D], the principal no longer has the freedom to distribute expected payments to the primary contractor across the states of the world in an arbitrary way. This restricts her ability to mitigate the primary contractor's incentives to manipulate the subcontractor's information and/or to capture some of the informational rents intended for the subcontractor. Specifically, because in [H.sup.ep.sub.D] and [H.sup.ep.sub.D] the primary contractor has to earn a nonnegative payoff in state HH, under substitutability the primary contractor has a stronger incentive to report HH in states LH and LL. For this reason, implementation in [H.sup.ep.sub.D] and [H.sup.ep.sub.D] is strictly more costly under substitutability.

Under complementarity, [H.sup.ep.sub.D] performs as well as [H.sub.1]. But in [H.sup.ep.sub.D] the primary contractor has an even stronger incentive to misrepresent her cost in state LH in order to capture a part of the informational rent intended for the primary contractor in state LL. So, [H.sup.ep.sub.D] attains the same performance as the two-agent mechanism under more restrictive conditions than either [H.sub.1] or [H.sub.D].

Finally, a few words about the choice of the primary contractor are in order. Propositions 5-7 demonstrate that asymmetries in the cross-effects between the two inputs and differences of cost distributions affect the agents' relative performance as primary contractors. Propositions 5 and 6 show that the principal is better off when the primary contractor is the agent who produces an input that has a smaller effect on the marginal product of the other input and who is more likely to be a high-cost producer. Moreover, the principal benefits when she can choose either agent to serve as the primary contractor. Propositions 5-7 demonstrate that in some cases, the ability to choose the primary contractor ensures that the principal gets the same payoff as in the two-agent mechanism. These results have policy implications for optimal assignment of tasks within hierarchies.

7. Collusion

* The results of the previous sections can be used to address the issue of collusion in organizations. Laffont and Martimort (1987, 1998)--LM in the sequel--analyze this issue in a similar framework. They consider the same two-agent model as in this article, restricting consideration to the perfect complementarity case that is, when v([q.sub.1], [q.sub.2]) = S(min {[q.sub.1], [q.sub.2]}). So, it is natural to compare the results of this article to theirs and consider how our analysis helps to better understand the effect of collusion.

An opportunity for collusion exists if the agents can communicate with each other and adopt a joint reporting strategy in the mechanism offered by the principal. Formally, the outcome of collusion can be represented by a pair of functions r(x) : [{L, H}.sup.2] [??] [{L, H}.sup.2] and [t.sup.c](x) : [{L, H}.sup.2] [??] [{L, H}.sup.2]. For every state of the world (i.e., LL, HL, LH, or HH), r(*) specifies the state of the world which the agents report in the mechanism and [t.sup.c](x) specifies a side transfer from agent 1 to agent 2.

Because each agent has private information about her cost, the collusion game will typically involve some frictions in communication and bargaining between the agents. So, the outcome of the collusion game may have to satisfy certain incentive constraints, and therefore some outcome pairs (r(*), [t.sup.c](x)) may not be feasible. Which incentive constraints have to hold depends on the specification of the collusion game and the enforceability of collusion.

To avoid model-specific details, in this section I will focus on an important benchmark case of perfect collusion in which there is no friction in bargaining between the agents, and any joint reporting strategy r(*) and side transfer function [t.sup.c](*) are feasible. In this case, for any mechanism offered by the principal, the agents will choose a joint reporting strategy [r.sup.*](*) maximizing the sum of agents' payoffs in each state of the world. We will say that a stake of perfect collusion exists in the two-agent mechanism [{[t.sup.1.sub.KJ], [q.sup.1.sub.KJ], [t.sup.2.sub.KJ], [q.sup.2.sub.KJ]}.sub.K, J [member of] {L, H}], if by using such joint reporting strategy [r.sup.*](*) the agents can attain a strictly higher sum of payoffs in some state of the world than in this mechanism without collusion.