The value of information and optimal organization.

Condition (6) requires the degree of complementarity to be sufficiently large. In particular, combining (6) with the fact that v(*) is concave, we obtain that [absolute value of [v.sub.12](x)/[v.sub.11](*)] > 1/1 - [p.sub.j], that is, the degree of complementarity exceeds 1/1 - [p.sub.j]. To understand the implications of this condition for the single. agent mechanism, suppose that i = 2. Then, as we move from state HH to state LH, that is, as the marginal cost of the first input goes down while the cost of the second input remains high, the optimal quantity of the first input in the single-agent mechanism increases by a smaller increment than the optimal quantity of the second input. As a result, the incentive constraint IC(HL - LH) becomes binding, as it gets more attractive for the agent in state HL to misrepresent both costs and report LH. By doing so, the agent sustains a small loss on the first input, because her true cost of producing this input is high, but she earns a large informational rent on her low cost of the second input. This extra deviation factor makes the two-agent mechanism more profitable, because the nonlocal incentive constraint IC(HL - LH) does not have to hold in the two-agent mechanism.

Proposition 2 is proven by showing that a relaxed single-agent mechanism, in which constraints IC(HL - HH), IC(HH - LH), and IC(LL - HH) are omitted and which therefore is more profitable for the principal than the optimal single-agent mechanism, is dominated by a two-agent mechanism. When condition (6) holds, the set of binding incentive constraints in this relaxed single-agent mechanism consists of IC(LL - HL ), IC(HL - LH), and IC(LH - HH) (see Figure 3). (8)

[FIGURE 3 OMITTED]

In the special case of the quadratic benefit function, we can use the proofs of Propositions 1 and 2 to obtain a necessary and sufficient condition for optimality of a particular organizational structure under complementarity. As in Proposition 2, this condition is also stated in terms of the inverses of the degrees of complementarity. It is slightly different from (6) because in the quadratic case, we can directly compute the expressions for the difference in expected payoffs between the single-agent and the two-agent mechanisms rather than just bound them, as in the general case.

Corollary 1. 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], d are positive constants (so that [v.sub.12] = d > 0) satisfying [b.sub.1][b.sub.2] [greater than or equal to] [d.sup.2]. Then the two-agent mechanism is optimal if and only if for some i and j [member of] {1, 2), i [not equal to] j, we have

[b.sub.j]/d [(1 - [p.sub.j]).sup.2] [p.sub.i]/2 + [b.sub.i]/d ([p.sub.j](1 - [p.sub.i]) + [p.sub.i]/2) < (7)

A graphical illustration of Corollary 1 is provided in Figure 4 for the case [p.sub.1] = [p.sub.2] = 1/2. In particular, its northeastern quadrant corresponds to the complementarity region, d > 0. The upper 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 lower boundary of the region of optimality of the two-agent mechanism is given by (7) holding as equality. Furthermore, under the normalization d = 1, with i = 2 and j = 1, the inequality (7) holds and [b.sub.1][b.sub.2] > 1 when [b.sub.1] = 1 + [[member of].sub.1]/1 - [p.sub.1] > 0 and [b.sub.2] = (1 - [p.sub.1])(1 - [[member of].sub.2]) > 0, where both [[member of].sub.1] and [[member of].sub.2] are positive numbers satisfying [[member of].sub.1] + 1 < [[member of].sub.1]/[[member of].sub.2] < 2[p.sub.1](1 - [p.sub.2])/[p.sub.2] + 1.

Da Rocha and de Frutos (1999) establish a result related to our Proposition 2. They show that the two-agent mechanism can outperform the single-agent mechanism under perfect complementarity. These authors emphasize the asymmetry of the supports of the cost distributions (i.e., is [c.sup.1.sub.H] - [c.sup.1.sub.L]/[c.sup.2.sub.H] - [c.sup.2.sub.L] sufficiently larger than 1) as an explanation. Yet, as our analysis indicates, a strong complementarity in their production function must also play a role in their result. Indeed, Proposition 1 implies that, for any value of [c.sup.1.sub.H] - [c.sup.1.sub.L]/[c.sup.2.sub.H] - [c.sup.2.sub.L], the single-agent mechanism remains optimal when the degree of complementarity is sufficiently small. (9) Conversely, performing renormalization, it is easy to show that the result of Da Rocha and de Frutos (1999) also holds when the cost distributions have a common support, and the production/benefit function is given by min {[q.sub.1]/[r.sub.1], [q.sub.2]/[r.sub.2]} when [r.sub.1]/[r.sub.2] a is large enough. This condition is similar to condition (6) in Proposition 2. (It is not identical because of the nondifferentiability of the Leontieff production function at the corner points.)

[FIGURE 4 OMITTED]

5. Substitutability

* Compared to the complementarity, there are several differences in the nature and strength of the internalization and extra deviation factors under substitutability. The main reason for this is that, under substitutability, the efficiency requires the quantity of one input to increase in the cost of the other input. In particular, it is efficient to set [g.sup.i.sub.HH] > [g.sup.i.sub.HL] for i [member of] {1, 2} in the single-agent mechanism. If the quantity profile in the single-agent mechanism satisfies this ordering, then the extra deviation factor manifests itself in the form of binding incentive constraint IC(LL - HH), that is, the most profitable deviation for the agent with two low costs is to report that both are high.

In the two-agent mechanism, the principal does not need to be concerned about this deviation because the agents could not coordinate their strategies. So, if IC(LL - HH) is binding in the single-agent mechanism, then the value of information is superadditive, and the two-agent mechanism dominates. This is shown in the proof of Proposition 3 in the Appendix.

Still, the potential to exploit the internalization factor can make it optimal for the principal to violate the efficient ordering in the single-agent mechanism and implement a profile of quantities decreasing in the cost of the other input, in particular, by setting [g.sup.i.sub.HL] > [g.sup.i.sub.HH] for i [member of] {1, 2}. Then the value of information will be subadditive in the single-agent mechanism, as the principal will pay a lower informational rent than in the two-agent mechanism with the same quantity assignment. This, however, will be achieved at the cost of productive distortions. In contrast, by Lemma 1, the quantity profile in the two-agent mechanism under substitutability is always increasing in the cost of the other input, which is more efficient. In this case, the optimal organizational form is determined by the tradeoff between a lower informational rent in the single-agent mechanism and a higher efficiency of the two-agent mechanism.

We use the homotopy technique to determine which of these two factors dominates. Similarly to the complementarity case, the results depend on the degree of substitutability measured by [absolute value of [v.sub.12]([g.sub.1],[g.sub.2])/[v.sub.11]([g.sub.1],[g.sub.2])] and [absolute value of [v.sub.12]([g.sub.1],[g.sub.2])/[v.sub.22]([g.sub.1],[g.sub.2])] Also, the threshold degree of substitutability above which the two-agent mechanism dominates turns out to depend on the relative likelihood of low- and high-cost states of the world.

At first, we will focus on the conditions under which the two-agent mechanism is optimal. Let [[g.bar].sub.1], [[bar.g].sub.2] solve [v.sub.1]([[g.bar].sub.1], [[bar.g].sub.2]) = [C.sub.H] + [DELTA] [p.sub.1]/(1 - [p.sub.1])([1 - [p.sub.2]) and [v.sub.2]([[g.bar].sub.1], [[bar.g].sub.2]) = [C.sub.L]. Also, let [[g.bar].sub.1], [[bar.g].sub.2] solve [v.sub.1]([[bar.g].sub.1], [[g.bar].sub.2]) = [C.sub.L] and [v.sub.2]([[g.bar].sub.1], [[bar.g].sub.2]) = [C.sub.H] + [DELTA] [p.sub.2]/(1 - [p.sub.1])(1 - [p.sub.2]). Then we have the following.

Proposition 3. Suppose that the inputs are substitutes and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the two-agent mechanism is optimal if either r [greater than or equal to] [p.sub.2]/1 - [p.sub.2] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

According to Proposition 3, the two-agent mechanism is optimal whenever the degree of substitutability between the inputs is sufficiently high. In this case, the quantity distortions (compared to the efficient levels) needed to neutralize the extra deviation factor in the single-agent mechanism become too large. The principal then sets [g.sup.i.sub.HH] [greater than or equal to] [g.sup.i.sub.HL] for all i [member of] {1, 2} in the single-agent mechanism, making the constraint IC(LL - HH) binding, and so the two-agent mechanism becomes more profitable. Furthermore, the threshold degrees of substitutability [p.sub.2]/1 - [p.sub.2] and [[p.sub.1]/[p.sub.2]] [[p.sub.2]-r(1 - [p.sub.2])/r[p.sub.1](2-[p.sub.1]-[p.sub.2]+[p.sub.1] [p.sub.2]) + (1 - [p.sub.1])] are increasing in [p.sub.2] and in both [p.sub.1] and [p.sub.2], respectively. This is so because the state LL is more likely when both [p.sub.1] and [p.sub.2] are high, and the internalization factor makes the single-agent mechanism more profitable for the principal precisely in state LL.

The following corollary shows that the degree of substitutability required for the two-agent mechanism to be optimal is less than 1.