The value of information and optimal organization.

Next consider a single-agent mechanism. Let [g.sup.i] = ([g.sup.i.sub.LL], [g.sup.i.sub.LH], [g.sup.i.sub.HL], [g.sup.i.sub.HH]) denote the vector of quantities of good i assigned in this mechanism, and [T.sub.KJ] denote the transfer to the agent who announces costs ([c.sub.K], [c.sub.J]), where K, J [member of] {H, L}. The single-agent mechanism has to satisfy the following incentive and individual rationality constraints for all K, J, U, V [member of] {L, H}:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The structures of incentive constraints in the two-agent and single-agent mechanisms are depicted in Figure 2, ignoring upward incentive constraints, which, as I show, never bind in an optimal mechenism. The downward incentive constraint IC(LL - HH), as well as the horizontal incentive constraints IC(LH - HL) and IC(HL - LH) in the single-agent mechanism have no counterparts in the two-agent mechanism, because agents choose their reports independently in the latter one. When any one of these constraints is binding, it reduces the profitability of the single-agent mechanism, that is, the extra deviation factor is effective. On the other hand, the constraints IC(LL - HL), IC(LL - LH), and IC(LL - HH) in the single-agent mechanism are mutually exclusive, and so the principal can ensure that all these three constraints hold by paying the agent a single informational rent in state LL. This is a manifestation of the internalization factor. The main results of this article, which will be established in the following sections, show that the optimal organizational form depends on the degree of complementarity and substitutability between the inputs measured by the ratios [absolutely 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])]. These ratios provide an appropriate measure of complementarity and substitutability because they determine the relative rate at which the quantities of the two inputs change in response to a change in the cost of one of the inputs. Specifically, if [absolute value of [v.sub.12]([g.sub.1], [g.sub.2])]/[v.sub.12]([g.sub.1], [g.sub.2])] is large, then a change in the cost of jth input, j [not equal to] i, causes the optimal quantity of input i to change by a relatively large amount compared to the change in the quantity of input j.

[] Optimal two-agent mechanism. As a first step in the analysis, I characterize, the optimal two-agent mechanism. The optimal single-agent mechanism turns out to be quite sensitive to the degree of complementarity and substitutability. Therefore, this mechanism is characterized separately under complementarity (see the proof of Proposition 2) and under substitutability (see the proof of Proposition 3).

In the rest of this subsection, I consider the optimal two-agent mechanism. Essentially, it consists of two submechanisms, one for each agent. In each of them, the individual rationality constraint of the high-cost type and the incentive constraint of the low-cost type are binding. Substitutability and complementarity cause the quantity assigned to one of the agents to depend on the cost type of the other agent, but does not affect the set of binding constraints.

Lemma 1. The optimal two-agent mechanism is unique. The optimal quantities are determined by the following first-order conditions:

[v.sub.1]([q.sup.1.sub.LL], [q.sup.2.sub.LL]) = [v.sub.2]([q.sup.1.sub.LL], [q.sup.2.sub.LL]) = [v.sub.1]([q.sup.1.sub.LH], [q.sup.2.sub.HL]) = [v.sub.2]([q.sup.1.sub.HL], [q.sup.2.sub.LH]) - [c.sub.L] (1)

[v.sub.1]([q.sup.1.sub.HL], [q.sup.2.sub.LH]) = [c.sub.H] + [DELTA] [p.sub.1]/1 - [p.sub.1] (2)

[v.sub.2]([q.sup.1.sub.LH], [q.sup.2.sub.LH]) = [c.sub.H] + [DELTA] [p.sub.2]/1 - [p.sub.2] (3)

[v.sub.1]([q.sup.1.sub.HH], [q.sup.2.sub.HH]) = [c.sub.H] + [DELTA] [p.sub.1]/1 - [p.sub.1] (4)

[v.sub.2]([q.sup.1.sub.HH], [q.sup.2.sub.HH]) = [c.sub.H] + [DELTA] [p.sub.2]/1 - [p.sub.2] (5)

The optimal quantity of an input is:

(i) decreasing in its cost, that is, [q.sup.i.sub.LL] > [q.sup.i.sub.HL], [q.sup.i.sub.LH] > [q.sup.i.sub.HH];

(ii) decreasing in the cost of the other input, that is, [q.sup.i.sub.LL] > [q.sup.i.sub.LH] and [q.sup.i.sub.HL] > [q.sup.i.sub.HH], under complementarity;

(iii) increasing in the cost of the other input, that is, [q.sup.i.sub.LL] < [q.sup.i.sub.LH] and [q.sup.i.sub.HL] < [q.sup.i.sub.HH], under substitutability;

The transfers are given by [t.sup.i.sub.HK] = [c.sub.H][q.sup.i.sub.HK], = [c.sub.L][q.sup.i.sub.LK] + [DELTA][q.sup.i.sub.HK] for K [member of] {L, H}.

Thus, to reduce the agents' informational rents, the principal sets quantity allocations [q.sup.i.sub.HL] and [q.sup.i.sub.HH] in the two-agent mechanism below the first-best. The quantities [q.sup.i.sub.LL] are set at the first- best level (no distortion "at the top"), whereas [q.sup.i.sub.LH] is set above (below) the first-best level when the inputs are substitutes (complements).

4. Complementarity

* In this section, I compare the profitability of the single-agent and the two-agent mechanisms under complementarity. The outcome of this comparison depends on the degree of complementarity. The following proposition shows that the single-agent mechanism dominates when the degree of complementarity is not too large.

Proposition 1. Suppose that the inputs are complementary, that is, [v.sub.12](., .) [greater than or equal to] 0. Then the single-agent mechanism is more profitable for the principal than the two-agent mechanism if the degree of complementarity between the inputs is not too large, that is, [absolute value of [v.sub.12]([g.sub.1], [g.sub.2])/[v.sub.11]([g.sub.1], [g.sub.2])] [less than or equal to] 1 for all i [member of] {1, 2}, ([q.sub.1], [q.sub.2]) [member of] [R.sup.2.sub.+].

When the degree of complementarity is less than 1, the value of information is subadditive and the principal can implement the quantity profile from the optimal two-agent mechanism via a single-agent mechanism with lower expected payments. Specifically, in states HH, LH, and HL, the payments in the single-agent mechanism can be set equal to the sum of payments in the two-agent mechanism, whereas in state LL, a lower payment can be made in the single-agent mechanism due to the internalization factor.

To see the latter, note that in state LL, the total informational rent paid by the principal in the two-agent mechanism is equal to [DELTA]([q.sup.1.sub.HL] + [q.sup.2.sub.HL]), because each agent can independently misrepresent her cost as high. If the same allocation profile is assigned in the single-agent mechanism, then the agent can deviate by misrepresenting only one input cost, or the costs of both inputs. The latter deviation is least attractive, because under complementarity the optimal quantity of one input decreases in the cost of the other input and, in particular, [q.sup.i.sub.HL] > [q.sup.i.sub.HH] by Lemma 1. If the agent misrepresents only the cost of the ith input, for i [member of] {1, 2}, then she earns a rent equal to [DELTA][q.sup.i.sub.HL] on her information regarding the cost of this input, but her rent on the information regarding the cost of the input j, j [not equal to] i, will be at most [DELTA][q.sup.i.sub.HH]. So, in state LL in the single-agent mechanism, the principal needs to pay informational rent equal to [DELTA]max {[q.sup.1.sub.HL] + [q.sup.2.sub.HH], [q.sup.i.sub.HH] + [q.sup.2.sub.HL]}, which is less than the informational rent [DELTA]([q.sup.1.sub.HL] + [q.sup.2.sub.HL]) paid in the two-agent mechanism. Thus, the value of information is subadditive. Finally, the restriction that the degree of complementarity should not exceed 1 ensures that the horizontal incentive constraints IC (LH - HL) and IC(HL - LH) remain nonbinding in the single-agent mechanism.

When the degree of complementarity is sufficiently large and there is a strong asymmetry between the inputs, then an increase in the quantity of one input affects the marginal product of this input to a lesser degree than the marginal product of the other input. Consequently, one of the horizontal incentive constraints becomes binding, and the result of Proposition 1 may be reversed due to this extra deviation factor. A sufficient condition for this is given in the following proposition. It is stated in terms of the inverses of the degrees of complementarity.

Proposition 2. The two-agent mechanism is more profitable than the single-agent mechanism under complementarity if for some i and j [member of] {1, 2}, i [not equal to] j, the following condition holds for all ([q.sub.1], [q.sub.2]) [member of] [R.sup.2.sub.+]:

- [v.sub.jj]([q.sub.1], [q.sub.2])/[v.sub.12]([q.sub.1], [q.sub.2] [(1 - [p.sub.j]).sup.2][p.sub.i] - [v.sub.11]([q.sub.1], [q.sub.2])/[v.sub.12]([q.sub.1], [q.sub.2] ([p.sub.i](1 - [p.sub.j]) + [p.sub.j]) < ((1 - [p.sub.j])(2[p.sub.i] + [p.sub.j](1 - [p.sub.i])). (6)