The value of information and optimal organization.

Similarly, IC(HL - LH) holds if [q.sup.1.sub.LH] - [q.sup.2.sub.HL] [greater than or equal to] [q.sup.1.sub.HH] - [q.sup.2.sub.HH]. To see that (A1) holds, note that by (1)-(5), [v.sub.2]([q.sup.1.sub.HL], [q.sup.2.sub.LH]) < [v.sub.2]([q.sup.1.sub.HH], [q.sup.2.sub.HH]) and [v.sub.1]([q.sup.1.sub.HL], [q.sup.2.sub.LH]) = [v.sub.1]([q.sup.1.sub.HH], [q.sup.2.sub.HH]). Because [absolute value of [v.sub.2]([q.sub.1], [q.sub.2]) [greater than or equal to] [v.sub.12]([q.sub.1], [q.sub.2]), Property 2 implies that (A1) holds. Similarly, the first-order conditions in Lemma 1, the assumption that [absolute value of [v.sub.22]([q.sub.1], [q.sub.2]) [greater than or equal to] [v.sub.12]([q.sub.1], [q.sub.2]) and, Property 2 imply that IC(HL - LH) holds.

Proof of Proposition 2. Let Condition (6) hold for i = 2 and j = 1. The proof for i = 1 and j = 2 is symmetric. To prove the proposition, we compare the profitability of the optimal two-agent mechanism and the relaxed single-agent mechanism RM(1) derived by omitting the incentive constraints IC(HL - HH), IC(HH - LH), and IC(LL - HH) from the principal's profit maximization problem with a single agent. Obviously, RM(1) is more profitable for the principal than the optimal single-agent mechanism, as all constraints have to be imposed in the latter. Hence, if the two-agent mechanism is more profitable than RM(1), then it is also more profitable than the single-agent mechanism.

The proof consists of seven steps. Step 0 is preliminary. In Steps 1-4, I characterize the relaxed single-agent mechanism RM(1). In Step 5, I develop a method for comparing the profitability of RM(1) and the optimal two-agent mechanisms. In Step 6, this method is used to show that the two-agent mechanism is more profitable than RM(1) under Condition (6) of the proposition.

Step 0. Let us show that 2[v.sub.12] + [v.sub.22[/1 - [p.sub.1] + [v.sub.11](1 - [p.sub.1]) < 0. Indeed, consider [v.sub.22]/1 - [p.sub.1] + [v.sub.11](1 - [p.sub.1]) as a function of [p.sub.1]. If [absolute value of [v.sub.22]] < [absolute value of [v.sub.11]], then it reaches a unique maximum equal to -2[square root of [v.sub.11][v.sub.22]] at [p.sub.1 = 1 - [square root of [v.sub.22]/[v.sub.11]]. If [absolute value of [v.sub.22]] [greater than or equal to] [absolute value of [v.sub.11]], then it reaches a unique maximum equal to [v.sub.11] + [v.sub.22] at [p.sub.1] = 0. But because v(x) is concave, we have - 2[square root of [v.sub.11][v.sub.22]] + 2[v.sub.12] < 0. The last inequality implies that [v.sub.11] + [v.sub.22] + 2[v.sub.12] < 0.

Further, simple rearrangement shows that with i = 2 and j = 1, Condition (6) of the proposition can be rewritten as follows:

([v.sub.12](1 - [p.sub.1]) + [v.sub.22])[p.sub.1](1 - [p.sub.2]) + (2[v.sub.12] + [v.sub.22]/1 - [p.sub.1] + [v.sub.11](1 - p.sub.1))[p.sub.2](1 - [p.sub.1]) > 0.

Because 2[v.sub.12] + [v.sub.22]/1 - p.sub.1]) + [v.sub.11](1 - [p.sub.1]) < 0, it follows that [v.sub.12] + [v.sub.22]/1 - [p.sub.1] > 0.

Step 1. To characterize the relaxed mechanism RM(1), we first solve RM(1)', the principal's profit maximization problem in a single-agent mechanism subject to IR(HH), the individual rationality constraint of HH type, and the downward and horizontal incentive constraints IC(LL - LH), IC(LL - HL), IC(LH - HH), IC(HL - LH), and IC(LH - HL). In Step 4 we show that the solution to RM(1)' satisfies the remaining incentive constraints of RM(1), IC(HL - LL), IC(LH - LL), IC(HH - HL), and IC(HH - LL).

The Lagrangian associated with the problem RM(1)' is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A2)

The Lagrange multipliers [eta], [[lambda].sub.LH], [[lambda].sub.HL], [mu], [kappa] and [[delta].sup.1.sub.HH] are nonnegative and satisfy the complementary slackness conditions, [eta]([T.sub.HH] - [C.sub.H]([g.sup.1.sub.HH] + [g.sup.2.sub.HH])) = 0 and similarly for the other constraints. The first-order conditions with respect to transfers are

[T.sub.LL] : [p.sub.1][p.sub.s2] = [[lambda].sub.LH] + [[lambda].sub.HL] (A3)

[T.sub.LH] : [p.sub.1](1 - [p.sub.2]) = [[delta].sub.1.sup.HH] - [[lambda].sub.LH] - [mu] + [kappa] (A4)

[T.sub.HL] : (1 - [p.sub.1])[p.sub.2] = -[[lambda].sub.HL] + [mu] - [kappa] (A5)

[T.sub.HH] : (1 - [p.sub.1])(1 - [p.sub.2]) = [eta] - [[delta].sup.1.sub.HH] (A6)

The equations (A3)-(A6) imply that [eta] = 1 and [[delta].sup.1.sub.HH] = [p.sub.1](1 - [p.sub.2]) + [p.sub.2], which can be used to simplify the other first-order conditions as follows:

[v.sub.1]([g.sup.1.sub.LL], [g.sup.2.sub.LL]) = [v.sub.2]([g.sub.1.sup.LL], [g.sup.2.sub.LL]) = [c.sub.L] (A7)

[v.sub.1]([g.sup.1.sub.LH], [g.sup.2.sub.HL]) = [c.sub.L] - [mu]/[p.sub.1](1 - [p.sub.2]) [DELTA] (A8)

[v.sub.2]([g.sup.1.sub.HL], [g.sup.2.sub.LH]) = [c.sub.L] - [kappa]/[p.sub.2](1 - [p.sub.1]) [DELTA] (A9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A11)

[v.sub.1]([g.sup.1.sub.HH], (g.sup.2.sub.HH]) = [c.sub.H] + [p.sub.1][DELTA]/1 - [p.sub.1] + [[lambda].sub.LH] + [mu] - [kappa]/(1 - [p.sub.1])(1 - [p.sub.2]) [DELTA] (A12)

[v.sub.2]([g.sup.1.sub.HH], (g.sup.2.sub.HL]) = [c.sub.H] + [p.sub.1][DELTA]/1 - [p.sub.2] + [[lambda].sub.LH] + [mu] - [kappa]/(1 - [p.sub.1])(1 - [p.sub.2]) [DELTA] (A13)

Step 2. In this step, we determine which constraints are binding in the relaxed problem (A2) and compute the values of the multipliers. Obviously, the constraint IR(HH) must be binding, because otherwise [T.sub.HH] could be lowered without violating any other constraint. So, [T.sub.HH] = [c.sub.H]([g.sup.1.sub.HH] + [g.sup.2.sub.HH]). Also, IC(HL - LH) must be binding because otherwise we could lower [T.sub.HL] by a positive amount without violating any incentive constraints.

Next, suppose that constraints IC(LH - HL) and IC(LL - LH) are nonbinding in the solution to problem (A2) (we show this in Step 3). This supposition has several implications: (i) the multipliers [kappa] and [[lambda].sub.LH] corresponding to these

(1) Particularly, while developing a new defense system, the Department of Defense has to decide whether to procure all its components from the same manufacturer or from different ones. The government may allow the existence of a multiproduct monopoly, or break it up into several firms, as in the AT&T case. In more recent examples of deregulation in the electric power industry, the regulators were called to determine whether a public utility producing the bulk of power could also maintain control over the transmission grid or the latter should be controlled by a separate entity.

(2) In the economic literature one can find examples of situations where more information either hurts or benefits the informed party. For example, in the Stackelberg oligopoly game, information about a competitor's action, that is, the competitor's quantity choice, hurts a firm.

(3) These notions are defined below based on the sign of the cross-partial derivative of the principal's benefit function.

(4) Armstrong and Sappington (2004) provide a comprehensive survey of the literature.

(5) Iossa (1999) studies the optimal regulatory regime in a two-good economy with one-dimensional uncertainty with one-dimensional uncertainty about the demand for one of the goods that is privately known by the monopolist or one of the duopolists. She reaches a different conclusion that the regulator prefers monopoly (duopoly) when the goods are substitutes (complements). Given the differences in informational assumptions, the model in this article is not directly comparable to hers.

(6) The principal would then offer them an allocation profile that is implemented in the optimal single-agent mechanism, rather than in a less-profitable two-agent mechanism.

(7) There is a number of reasons why the principal may want or have to procure all supply of a particular input from one source. The most common of them is the presence of fixed costs. If large fixed costs in the form of R&D, investment in equipment, infrastructure, and training, and so forth, have to be sunk by each producer of the good before she learns her production costs, then having more than one supplier could be prohibitively expensive. Alternatively, the principal's commitment to purchase all supply of an input from a particular agent may be required to alleviate a potential hold-up problem and induce this agent to make necessary investments, or to perform R&D.

Consider, for example, the development of a new defense system. In the initial stage of procurement, the government normally considers bids from a number of suppliers. However, only one supplier of each major part is ultimately chosen. Moreover, the final price is usually determined after the contracts have already been awarded. According to Rogerson (1989), "economies of scale together with very small production runs render it economically infeasible to have two or more firms build fully functioning production lines.... The prices for all production runs may be left to be determined by future negotiations. Transaction costs together with constantly evolving technological requirements are thought to render long-term contracts infeasible."

(8) In the online supplement available at http://www.severinov.com/organization_AppendixB.pdf, I show that the relaxed mechanism satisfies the omitted incentive constraints and hence is equivalent to the optimal single-agent mechanism if, in addition to condition (6), we also require that [v.sub.12][p.sub.i](1 - [p.sub.j]) + [v.sub.ii]([p.sub.i](1 - [p.sub.j]) + [p.sub.j]) [less than or equal to] 0.