The value of information and optimal organization.

Corollary. For all [p.sub.1], [p.sub.2] < 1, there exists k < 1 s.t. the two-agent mechanism is optimal under substitutability if [absolute value of [v.sub.12]([g.sub.1],[g.sub.2])/[v.sub.11]([g.sub.1],[g.sub.2])] [absolute value of [v.sub.12]([g.sub.1],[g.sub.2])/[v.sub.22]([g.sub.1],[g.sub.2])] [greater than or equal to] for all ([g.sub.1], [g.sub.2]) [member of] [[[g.bar].sub.1], [[bar.g].sub.1]]] x [[[g.bar].sub.2], [[bar.g].sub.2]]].

Corollary 2 is immediately applicable to the case of perfect substitutes, that is, when v([q.sub.1], [q.sub.2]) = u([q.sub.1] + [q.sub.2]). In this case, [v.sub.12]([q.sub.1],[q.sub.2])/[v.sub.11]([q.sub.1],[q.sub.2]) [equivalent to] 1 for i [member of] {1, 2}, so the two-agent mechanism is optimal.

Next, suppose that the degree of substitutability is low. Then, in the single-agent mechanism, the principal exploits the internalization factor by making the quantity of one input decrease in the cost of the other input--much like under complementarity. However, in the two-agent mechanism we have the opposite ordering: the optimal quantity of one input is increasing in the cost of the other input. Because the optimal quantities are ordered differently in the single-agent and the two-agent mechanisms, a simple method of proof based on the comparison of informational rents is not applicable. Instead, to compare the principal's expected profits in the optimal single-agent and two-agent mechanisms, we use the homotopy technique developed in the Appendix. The following proposition describes the result of this comparison.

Proposition 4. Suppose that the inputs are substitutes and that there exist [bar.K] and [K.bar], 0 < [bar.K] [less than or equal to] [bar.K] < [infinity], s.t. [K.bar] < [v.sub.11]([q.sub.1], [q.sub.2])/[v.sub.22]([q.sub.1], [q.sub.2]) < [bar.k] 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]]. Then for any ([p.sub.2], [p.sub.2]) [member of] [(0, 1).sup.2] there exist [w.sub.1] and [w.sub.2], with [w.sub.i] increasing in [p.sub.j] [not equal to] j, such that the single-agent mechanism is optimal if [absolute value of [v.sub.12]([q.sub.1],[q.sub.2])/[v.sub.11]([q.sub.1],[q.sub.2])] < [w.sub.i] for all i [member of] {1, 2} and ([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].

Proposition 4 holds because the effect of the internalization factor outweighs the efficiency losses from distorting the quantity profile in the single-agent mechanism when the degree of substitutability is low, and when both [p.sub.1] and [p.sub.2] are sufficiently high so that state LL occurs with a high likelihood. Low substitutability implies that the efficiency losses from exploiting the internalization factor and making the quantity profile decrease in the cost of the other input in the single-agent mechanism (in particular, setting [g.sup.i.sub.HL] > [g.sup.i.sub.HH] for i [member of] {1, 2}) is not too large, whereas the high likelihood of state LL makes the effect of the internalization factor sufficiently large in expected terms.

In the symmetric case, that is, when v([q.sub.1], [q.sub.2]) = v([q.sub.2], [q.sub.1]) for all ([q.sub.1], [q.sub.2]) [member of] [R.sup.2.sub.+] and [p.sub.1] = [p.sub.2] = p, we can obtain somewhat tighter bounds on the regions of optimality of the single-agent and two-agent mechanisms under substitutability. Revisiting the proofs of Propositions 3 and 4 under symmetry, (10) we obtain that the single-agent mechanism remains optimal if [absolute value of [v.sub.12]([q.sub.1],[q.sub.2]/[v.sub.11]([q.sub.1],[q.sub.2]] [less than or equal to] min{p/4, [p.sup.2]/4(1 - p) + [p.sup.2]}, whereas the two-agent mechanism is optimal if [absolute value of [v.sub.12]([q.sub.1],[q.sub.2]/[v.sub.11]([q.sub.1],[q.sub.2] [greater than or equal to] p/2(1 - p) + [p.sup.2] for all ([q.sub.1], [q.sub.2]), [member of] [R.sup.2.sub.+] and i [member of] {1, 2}.

[FIGURE 5 OMITTED]

Finally, in the special case of the quadratic benefit function, we can use the proofs of Propositions 3 and 4 to derive a necessary and sufficient condition for the optimality of each organizational structure under substitutability. It illustrates the results of these propositions by showing exactly how large the degree of substitutability has to be for the two-agent mechanism to dominate.

Corollary 3. Suppose that v([q.sub.1], [q.sub.2]) = A + a([q.sub.1] + [q.sub.2]) - [b.sub.1]/2 [q.sup.2.sub.1] - [b.sub.2]/2 [q.sup.2.sub.2] + d[q.sub.1][q.sub.2], where A, a, [b.sub.1], [b.sub.2] are positive constants, d is negative (substitutability), and [b.sub.1][b.sub.2] [greater than or equal to] [d.sup.2] (concavity). Then the two-agent mechanism is more profitable than the single-agent mechanism if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Figure 4 provides a graphical illustration of Corollary 3 for [p.sub.1] = [p.sub.2] = 1/2. The lower-left quadrant in this figure corresponds to the substitutability region, where d < 0. The lower boundary of the admissible parameter space in this region is given by [b.sub.1][b.sub.2] = [d.sup.2] (concavity restriction), while the upper boundary of the region of optimality of the two-agent mechanism is given by (8) holding as equality.

Finally, Figure 5 illustrates the regions of optimality of the two-agent and the single-agent mechanisms with symmetry and quadratic benefit function, that is, when [b.sub.1] = [b.sub.2] and [p.sub.1] = [p.sub.2]. Note that in this case condition (8) simplifies to [absolute value of d/b] [greater than or equal to] p/4(1 - p) + [p.sup.2].

6. Delegation

* In this section, I consider another form of organization---delegation, with agents organized in a hierarchy. The two agents contribute different inputs, but the principal directly contracts only with the primary contractor and delegates to her the task of contracting with the other agent, the subcontractor (see Figure 1). Delegation is common in the allocation of tasks within an organization, in procurement and in the construction industry. In large corporations, senior managers typically delegate some supervisory functions and authority to middle managers.

Hierarchial delegation with asymmetric information was studied by a number of authors, in particular, Melumad, Mookherjee, and Riechelstein (1995), Baron and Besanko (1992), Gilbert and Riordan (1995), Laffont and Martimort (1998), and Mookherjee and Riechelstein (2001). This literature points out that the advantages of delegation include an economy of communication costs achieved by shifting some of the contracting tasks from me principal to one of me subordinates. (11) On the other hand, delegation leads to a loss of control by the principal which may negatively affect the incentives within hierarchies. The last point is made by McAfee and McMillan (1995) in the context of a model where intermediate layers of supervision separate the principal from the agent engaged in production. Riordan and Sappington (1987) show that the principal's decision whether to delegate both stages of the production process to the agent or only one stage depends on whether the costs at the two stages are positively or negatively correlated.

This section has two goals. The first goal is to compare the profitability of the delegation mechanism vis-a-vis the two-agent and the single-agent mechanisms in several contractual environments. The second goal is to examine the issue of the optimal choice of the primary contractor. In our model, agents 1 and 2 can have different cost distributions and different productivities, as reflected in the asymmetry of the production function. It is natural to ask whether these asymmetries imply differential performance by agents 1 and 2 in the role of primary contractor.

To make legitimate comparisons across organizational forms, I make the same assumptions regarding input observability as in the single-agent and the two-agent organizations studied in the previous sections. Specifically, I assume that under delegation, the principal can monitor the quantity of an input supplied by each agent. I will consider four different contractual setups referred to as delegation hierarchies [H.sub.1], [H.sub.D], [H.sup.ep.sub.D], and [H.sup.ep.sub.D]. The following sequence of moves characterizes hierarchy [H.sub.1] (named so by Melumad, Mookherjee, and Riechelstein 1995):

(i) The principal offers the contract to the primary contractor.

(ii) The primary contractor decides whether to accept or reject the contract. If she rejects, the game ends and all players obtain their reservation payoffs. If the primary contractor accepts the contract, then the game proceeds through the following stages.

(iii) The primary contractor reports her cost type to the principal.

(iv) The primary contractor offers a contract to the subcontractor. If the subcontractor rejects it, then the game ends and all players obtain their reservation payoffs.

(v) If the subcontractor accepts, she reports her cost to the primary contractor, who then reports it to the principal.

(vi) Both contractors produce their inputs, the final output is delivered to the principal, and the transfers take place according to the two contracts.