The value of information and optimal organization.

Clearly, if there is no stake of perfect collusion in a mechanism, then the agents cannot benefit from any collusion game which is less than perfect, that is, when the set of feasible joint reporting strategies is restricted due to informational asymmetry and/or bargaining frictions between the agents. The proof of this assertion is immediate: because perfectly colluding parties can adopt any joint reporting strategy, they can adopt the one that arises as an outcome of any collusion game.

Note that collusion reduces the principal's profits only if a stake of perfect collusion exists in the optimal two-agent mechanism characterized in Section 3. So, it is important to understand the conditions under which such a stake exists. We can answer this question by applying the results of previous sections. Specifically, suppose that the allocation profile and transfers from the optimal two-agent mechanism are assigned in the single-agent mechanism. Then a stake of perfect collusion exists if such mechanism is not incentive compatible, that is, if in some state of the world the single agent earns the highest profit by misrepresenting the costs of both inputs. To understand this, note that perfectly colluding agents maximize the same objective function as the single agent, so they would also be strictly better off by misrepresenting both costs in this state of the world. In the two-agent mechanism, such deviation is not feasible without collusion, but it is feasible under perfect collusion. Using this logic, we can establish that a stake of perfect collusion exists whenever a two-agent mechanism is more profitable for the principal than the single-agent mechanism, and also under substitutability. The latter follows from the fact that under substitutability [q.sup.i.sub.HL] [greater than or equal to] [q.sup.i.sub.HH] for i [member of] {1, 2} in the optimal two-agent mechanism (see Lemma 1). Hence, when this mechanism is assigned to a single agent, the latter would deviate in state LL by reporting two high costs.

On the contrary, there is no stake of collusion if the allocation profile from the two-agent mechanism remains incentive compatible in the single-agent mechanism. The proof of Proposition 1 shows that this is the case under the conditions of that proposition. The same argument also works under perfect complementarity. The following proposition summarizes these conclusions.

Proposition 8. A stake of perfect collusion exists in the following two cases: (i) under substitutability; (ii) under complementarity, when the two-agent mechanism is more profitable for the principal than the single-agent mechanism.

A stake of perfect collusion does not exist when the inputs are complementary and max {[absolute value of [v.sub.12]([q.sub.1],[q.sub.2])/[v.sub.11]([q.sub.1], [q.sub.2])], [absolute value of [v.sub.12]([q.sub.1],[q.sub.2])/[v.sub.22]([q.sub.1], [q.sub.2])]} [less than or equal to] for all [q.sub.1], [q.sub.2] [member of] [R.sup.2.sub.+], and also under perfect complementarity that is, when v([q.sub.1], [q.sub.2]) = S(min {[q.sub.1], [q.sub.2]}), with S'(*) > 0 and S"(*) < 0.

Proposition 8 allows us to understand why LM, who focus on the perfect complementarity case, had to impose additional restrictions on the set of feasible mechanisms in order to generate a stake of collusion. Specifically, they require the principal to offer an anonymous mechanism so that both agents get the same transfer in each state of the world. The anonymity generates a stake of collusion equal to [DELTA]([q.sub.HL] - [q.sub.HH]) where [q.sub.HL] and [q.sub.HH] are the solutions to the first-order conditions (2)-(5) characterizing the optimal two-agent mechanism in the symmetric case with [p.sub.1] = [p.sub.2] (see Laffont and Martimort, 1998). Obviously, this stake of collusion disappears under substitutability and separability, because in those cases we have [q.sub.hHH] [less than or equal to] [q.sub.HL] (see Lemma 1).

LM (1998) demonstrate that the principal can avoid the cost of preventing collusion in an anonymous mechanism through delegation. Their delegation mechanism (equivalent to our [H.sub.D] hierarchy) is more profitable for the principal than a two-agent mechanism with collusion. Yet, without an anonymity restriction, this result is not always true. In particular, suppose that inputs are complementary and agent 1 is the primary contractor. Proposition 6 shows that [H.sub.D] is strictly less profitable than 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])] < 1 - [p.sub.1]/1 - [p.sub.2], whereas, by Proposition 8, there is no stake of perfect collusion if max {[absolute value of [v.sub.12]([q.sub.1], [q.sub.2])/[v.sub.11]([q.sub.1], [q.sub.2])], [absolute value of [v.sub.12]([q.sub.1], [q.sub.2])/[v.sub.22]([q.sub.1], [q.sub.2])]. Both of these conditions hold, and so a nonanonymous two-agent mechanism dominates delegation via [H.sub.D] hierarchy, for a fairly large class of benefit functions. For example, take a quadratic benefit function with appropriate restrictions on the parameters.

8. Conclusions

* In this article, I have studied optimal organization of production in an environment where agents have private information about the cost of producing their inputs. The optimality of centralizing production in the hands of a single agent or decentralizing it between two agents depends on whether the value of cost information is sub- or superadditive for the agent(s) which in turn depends on whether the inputs are complementary or substitutable in their final use. Under complementarity or low degrees of substitutability, centralization is optimal, unless the production function is highly asymmetric so that the quantity of one of the inputs affects the marginal benefit of the other input in a significant way. In such case, decentralization is optimal even under complementarity. Decentralization is also optimal when the degree of substitutability is large.

The degree of substitutability/complementarity between the inputs also affects the performance of delegation mechanisms. When it is large, the interdependence between quantities of different inputs is also large in the optimal mechanism. This allows the primary contractor to benefit from her position of an informational intermediary either by increasing the informational rent that she obtains on her cost information or by appropriating part of the informational rent intended for the subcontractor. I have also considered which of the two agents should be chosen as the primary contractor to maximize the performance of a hierarchy. As I have shown, it is optimal to choose 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. The latter result has policy implications for optimal allocation of supervisory functions and assignment of tasks within organizations.

Appendix

* The following properties of concave functions, established by differentiation, will be useful below.

Property 1. Let v(x, x) be a twice-continuously differentiable, increasing, concave function, and suppose that [v.sub.1](([q.sub.1], [q.sub.2]) = [c.sub.1] and [v.sub.2]([q.sub.1], [q.sub.2]) = [c.sub.2] for some [c.sub.1], [c.sub.2] [member of] (0, [infinity]). Then, d[q.sub.1]/d[c.sub.1] = [v.sub.22] < 0, d[q.sub.2]/d[c.sub.2] = [v.sub.11] < 0, d[q.sub.1]/d[c.sub.2] = - [v.sub.12].

Property 2. Suppose that [v.sub.1]([q.sub.1], [q.sub.2]) = [c.sub.1] and [v.sub.2]([q.sub.1], [q.sub.2]) = [c.sub.2] for some [c.sub.1], [c.sub.2] > 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Proposition 1. Consider the following single-agent mechanism. In any state of the world KJ, assign the same quantity allocation ([q.sup.1.sub.KJ], q2K) as in the optimal two-agent mechanism, and the corresponding transfer from the following list: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This mechanism is more profitable for the principal than the optimal two-agent mechanism, because her total payment is the same in all states of the world except LL, where her total payment is lower than in the two-agent mechanism by [DELTA]min {[q.sup.1.sub.HL] - [q.sup.1.sub.HH], [q.sup.2.sub.HL] - [q.sup.2.sub.HH]} > 0. This mechanism satisfies all individual rationality constraints. Let us show that it is incentive compatible. Clearly, it satisfies the downward incentive constraints IC(LL - HL), IC(LL - LH), IC(LH - HH), IC(HL - HH), IC(LL - HH). In particular, the latter holds because [q.sup.i.sub.HH] < [q.sup.i.sub.HL]. The upward constraints IC(HL - LL), IC(LH - LL), and IC(HH - LL) hold because [q.sup.i.sub.KL] > [q.sup.i.sub.KH] < by Lemma 1.

Finally, consider the horizontal incentive constraints IC(LH - HL) and IC(HL - LH). Because IC(LH - HH) and IC(HL - HH) are binding, IC(LH - HL) holds if

[q.sup.2.sub.LH] - [q.sup.1.sub.HL] [greater than or equal to] [q.sup.2.sub.HH] - [q.sup.1.sub.HH]. (A1)