The value of information and optimal organization.

The hierarchy [H.sub.1] is the most profitable for the principal among all delegation hierarchies with the same observability assumptions, because it endows the principal with the broadest possible contracting abilities. In particular, the principal signs a contract with the primary contractor and receives her cost report before the latter communicates with the subcontractor. Therefore, [H.sub.1] serves as a natural benchmark establishing what is attainable in a delegation mechanism. This hierarchy provides a good representation of contractual schemes in the construction industry where the customer, first, hires a primary contractor and obtains a cost estimate from her. The primary contractor is then typically given the authority to subcontract other providers whose costs are ex ante uncertain.

By the Revelation Principle, the two-agent mechanism is at least as profitable for the principal as [H.sub.1]. So the questions are whether the principal can achieve the same expected profits in [H.sub.1] as in the two-agent mechanism, and how [H.sub.1] compares to the single-agent mechanism. An answer to these questions is provided in the following proposition. Before presenting it, let us introduce the following piece of notation. Recall that the quantity schedule in the optimal two-agent mechanism is denoted by [{[q.sup.i.sub.LL], [q.sup.i.sub.LH], [q.sup.i.sub.HL], [q.sup.i.sub.HH]}.sup.i=2.sub.i=1]. Let [[bar.q].sub.i] = max{[q.sup.i.sub.LL], [q.sup.i.sub.LH]} and [[q.bar].sub.i] = min{[q.sup.i.sub.HL], [q.sup.i.sub.HH]}.

Proposition 5. If agent i [member of] {1, 2} serves as the primary contractor, then the principal obtains the same payoff in [H.sub.1] as in the two-agent mechanism if [absolute value of [v.sub.12]([q.sub.1], [q.sub.12])/[v.sub.11]([q.sub.1], [q.sub.2])] [less than or equal to] 1/1 - [p.sub.j], i [not equal to] j, 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.1] with agent i [member of] {1, 2} as the primary contractor is strictly less profitable for the principal than the two-agent mechanism if [absolute value of [v.sub.12]([q.sub.1], [q.sub.12])/[v.sub.11]([q.sub.1], [q.sub.2])] > 1/1 - [p.sub.j], i [not equal to] j, 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 the principal obtains the same payoff in [H.sub.1] as in the two-agent mechanism in the following cases: (i) under complementarity; (ii) under substitutability, if [v.sub.iii](x) [greater than or equal to] 0, [v.sub.iii] [less than or equal to] 0, and [v.sub.ijj] [greater than or equal to] 0 for some i [not equal to] j 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]].

[H.sub.1] is strictly more profitable for the principal than the single-agent mechanism if IC(LL - HH) is binding in the latter.

According to Proposition 5, if only one of the agents can serve as the primary contractor, then [H.sub.1] is equivalent to the two-agent mechanism when the interdependence between the inputs in their final use is not too large, that is, the marginal benefit/product of one input is not too sensitive to the quantity of the other input.

To understand this result, note that [H.sub.1] is equivalent to the two-agent mechanism only if the principal can implement the quantity profile from the optimal two-agent mechanism via [H.sub.1] at the same expected cost. It is easy to see that in [H.sub.1] each agent obtains at least as much surplus from private information regarding her own cost as in the two-agent mechanism with the same quantity profile. So, [H.sub.1] can only attain the same level of profitability as the two-agent mechanism if the primary contractor cannot exploit her role as an informational intermediary to earn additional surplus, and simply passes on the information from the subcontractor to the principal without manipulating it. Manipulating this information could be profitable for the primary contractor for two reasons: (i) she could appropriate part of the informational rent that the principal intends for the subcontractor; (ii) she could extract more surplus from her own information.

In hierarchy [H.sub.1], option (i) is infeasible because the primary contractor has to report her cost type before communicating with the subcontractor. Given the primary contractor's report, the informational rents on the subcontractor's information can be appropriated only by the subcontractor. However, option (ii) becomes significant when the report regarding the subcontractor's cost has a large effect on the quantity assigned to the primary contractor, which is exactly when the degree of complementarity or substitutability between the inputs is sufficiently large.

Specifically, suppose that the inputs are complementary and consider the following deviation: the primary contractor misrepresents her low cost as high in the first stage, and then always reports that the subcontractor's cost is low, that is, in states LH and LL, the primary contractor reports state HL. Then, in states LH and LL, the primary contractor has to pay [C.sub.H][q.sup.2.sub.LH] to the subcontractor, with a net loss of [DELTA]([q.sup.2.sub.LH] - [q.sup.2.sub.HH]). However, the expected surplus obtained by the primary contractor on the information about her own cost increases from [DELTA]([q.sup.1.sub.HL] [p.sub.2] + [q.sup.1.sub.HH] (1 - [p.sub.2])) to [DELTA][q.sup.1.sub.HL]. In the proof of Proposition 5, I show that this increase outweighs the extra payment to the subcontractor when the degree of complementarity is sufficiently large. This is so because a report that the subcontractor's cost is low rather than high causes a larger increase in the quantity supplied by the primary contractor, and hence in her informational rent, than in the quantity supplied by the subcontractor, and hence the extra payment to her. Then, in [H.sub.1], the principal has to pay a larger informational rent to the primary contractor than in the two-agent mechanism.

Under substitutability, the primary contractor with a low cost has a strong incentive to announce that both costs are high, irrespective of the subcontractor's cost. This incentive is similar to the extra deviation factor and binding incentive constraint IC(LL - HH) in the single-agent mechanism. The principal can offset this incentive to a certain extent by imposing a penalty on the primary contractor when the latter reports that both costs are high. Yet, this penalty cannot be too large, because otherwise the primary contractor will misrepresent her own high cost as low. As a result, the primary contractor's incentive to overstate her cost cannot be mitigated when the degree of substitutability is high, and again the principal has to pay a higher informational rent in [H.sub.1]. (12)

If the principal can choose either agent to serve as the primary contractor, then under complementarity she can always do so in such a way that [H.sub.1] attains the same performance as the two-agent mechanism. This is so because only one of the agents, when serving as the primary contractor, can have an incentive to always report her cost as high and the subcontractor's cost as low. (13) Under substitutability, the ability of the principal to choose either agent to serve as the primary contractor guarantees that [H.sub.1] is equivalent to the two-agent mechanism only under additional restrictions on the signs of the third-order derivatives of the benefit function, because both agents could potentially have an incentive to overstate both costs.

Finally, the hierarchy [H.sub.1] performs better than the single-agent mechanism if IC(LL - HH) is binding in the latter (which could only happen under substitutability), because in this case the principal can implement the optimal single-agent quantity profile at a lower cost via [H.sub.1].

The hierarchy [H.sub.1] has two important properties affecting its performance. First, in [H.sub.1] the primary contractor's decision whether to report her true cost cannot be contingent on the subcontractor's cost. This reduces the set of feasible deviations by the primary contractor and benefits the principal. Second, only the interim, rather than ex post, individual rationality constraints of both agents have to be satisfied in [H.sub.1]. There is a significant difference between these two types of constraints, in particular, as far as the primary contractor is concerned. With the interim constraints, the principal structures her contract with the primary contractor in such a way that the primary contractor with a high cost obtains a negative payoff for one realization of the subcontractor's cost, and a positive payoff for a different realization of the subcontractor's cost. However, this would be impossible if the primary contractor's ex post individual rationality constraint had to be satisfied, as would be the case, for example, if the primary contractor could withdraw from the contract after receiving the subcontractor's cost report.