Informational costs and benefits of creating separately identifiable operating segments

Journal of Accounting and Economics 33 (2002) 69–90

Informational costs and benefits of creating separately identifiable operating segments$ Frank Giglera,*, Thomas Hemmerb a

J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60608-2002, USA b Graduate School of Business, University of Chicago, Chicago, IL 60637, USA Received 6 January 2000; received in revised form 30 May 2001

Abstract We provide an informational theory for how the ownership claims to a firm might be structured. When the market price of equity provides valuable contracting information there is a benefit to creating separate ownership claims to each of a firm’s divisions. However, creating this information also generally has adverse incentive effects because it enriches the agent’s strategy space. We show in a complete contracting setting that under a large class of agencies the firm is strictly better off bundling the ownership claims to divisions that are sufficiently similar and creating separate ownership claims only to divisions that are sufficiently different. r 2002 Elsevier Science B.V. All rights reserved. JEL classification: G32; L22; M41 Keywords: Contracting; Tracking stock; Core competency

1. Introduction It is commonly accepted that providing more detailed information about the performance of stand alone business segments within a firm allows for improvements in monitoring and/or contracting between owners and managers. Such expected improvements in stewardship appear to be at least part of the motivation behind the $ We wish to thank Ronald Dye, Sunil Dutta, Arijit Mukherji, Michael Smith, Sri Sridhar, and seminar participants at Duke University and the University of Texas-Austin 2000 BMAS Conference for their helpful comments and suggestions. *Corresponding author. Tel.: +1-847-491-3427; fax: +1-847-467-1202. E-mail address: [email protected] (F. Gigler).

0165-4101/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 4 1 0 1 ( 0 1 ) 0 0 0 4 3 - X

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new rules for segment disclosure in SFAS 131 that require publication of segment information along internal operating lines rather than along industry or geographic lines. Another less obvious manifestation of the drive toward the provision of more detailed segment information is the increasing popularity of equity carveouts and the creation of tracking stock. One of the often cited benefits of creating independent equity claims to different divisions under the same control is that the new information contained in the separate market prices improves contracting between owners and managers relative to the single market price when the divisions are traded as one.1 While the economics of this argument are intuitively appealing, as a theory, it suffers from its logical conclusion that all assets should then be traded and accounted for separately. This corner solution is of course absurd in its empirical prediction. Consequently, more recent theories have suggested a friction to, for example, endless asset carveouts in the form of economies of scale. In these models the optimal organizational form is determined by weighing the productive benefit to grouping assets together against the informational benefit of separating ownership (see the complementary articles of Aron (1988, 1991)). The problem with this type of explanation, however, is that it violates an elementary result of modern finance theory; the seperation of investing from financing. Without the introduction of transaction costs or contracting incompleteness, theoretically there is nothing to prevent establishing equity claims on separate inputs to the same production function, as is arguably the case with most tracking stock. This allows the informational benefits of divisional market prices without the loss of scale economies. The conclusion of the theory would then again be the corner solution that all assets are separately traded. In order to avoid the corner solution within a complete contracting setting it must be the case that the very information created by the action is the source of the cost. It is therefore our opinion that theories that identify benefits to creating such information must also recognize informational costs to successfully avoid the corner solution. In this paper we provide a model in which both informational benefits and frictions to creating more detailed segment information arise endogenously. While the creation of this information can arise through a variety of actions, such as equity carveouts, the formation of tracking stocks or providing more detailed segment accounting, we focus our analysis on the specific practice of creating tracking stock to make our points as precisely as possible. We show that even though creating tracking stock does improve the information available for contracting between principal and agent, it can also make it more costly to overcome the agent incentive problem. This is because the agent can use the information contained in the additional market price in choosing his subsequent actions and guarding against this richer agent strategy space can be costly to the agency as a whole. In fact, when divisions are sufficiently similar, an agency is strictly better off not creating division specific equity. On the other hand, it is always beneficial to create division specific equity when the agency problems of the two divisions are sufficiently different. 1

For example, see discussions in Holmstrom and Tirole (1989), Schipper and Smith (1984, 1986).

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Interestingly, these results are suggestive of popular notions of ‘‘core competency’’ and the prescription that firms concentrate ownership of assets in similar business lines. The issue we address is related to that of Indjejikian and Nanda (1999). That paper shows how aggregate performance measures can be less susceptible than disaggregate measures to the ratchet effect common in dynamic principal agent models with incomplete contracting. Incomplete contracting models assume some form of inability to commit on the part of the principal as well as the agent. In these settings, it may be preferable to use an aggregate measure even though a disaggregate one contains more information because the principal is unable to commit from using the more detailed information. While our results are qualitatively similar to theirs, the procedure and philosophy of our analysis is substantially different. We take a more traditional agency theory focus, that is on the agent’s inability to commit rather than the principal’s. Rather than expanding the traditional agency model by also limiting the principal’s ability to commit, we show how the agent’s inability to commit alone creates a value to aggregation. Specifically, we show how a commitment not to produce information can be valuable in an agency framework because the agent cannot commit to a set of actions conditional on the (public) realization of the information if it is produced. The philosophy of our analysis is more similar to Dana (1993). He uses an agency framework with complete contracting to determine when it is optimal to produce two goods in the same firm and when it is optimal to produce the two goods in separate firms. Apart from several other technical differences, the main difference between the two papers is that Dana studies when two agents are better than one, while we fix the number of agents at one and study when it is better to produce more or less detailed information about the agent’s performance. In this respect, his results apply more directly to spinoffs and divestitures which create a change in control. And ours are more germane to carveouts and the formation of tracking stock, where the original firm maintains control of the new entity. Qualitatively the two models provide opposite predictions. Dana (1993) shows the value of decentralization increases in the similarity of the two products. Our results support carving apart divisions that are sufficiently different. There are technical similarities between our model and two other models in the literature addressing fundamentally different issues than the issues we address here. Specifically, the source of the informational cost in this model is very similar to that identified in Gigler and Hemmer (1998). Gigler and Hemmer (1998) showed in a repeated agency problem with a binary outcome structure that there is a weak benefit to less frequent financial reporting and that this benefit is strict if the reporting delay is sufficiently long. The benefit of delay comes from keeping the agent ignorant about his interim performance and thus from conditioning his subsequent choices on the results of the past. In this model we take the frequency of reporting as fixed and study the effects of aggregating two divisions’ outputs within each of two periods. Aggregation generally makes it more difficult to infer the agent’s separate actions in each of the two divisions. Most importantly, aggregated first period output provides less information

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to the agent about his interim performance than disaggregated first period output. This in turn provides the agent with less information on which to condition his second period actions and thereby reduces the agency cost. So, economically, the benefit of cross-divisional aggregation stems from keeping the agent ignorant of his interim performance, as does the benefit of delay identified in Gigler and Hemmer (1998). Arya et al. (1999) introduce a potential cost of delayed reporting to a two-period version of the Gigler and Hemmer (1998) model by assuming that delaying reporting until the end of the second period results in an inter-temporal aggregation of the two periods’ outputs. If the agency wishes to take advantage of the strategic benefits of delay identified in Gigler and Hemmer (1998), then it must live with less detailed information at the end of the second period.2 In their model, as well as ours, the potential cost of (second period) aggregation is in not being able to determine output for each individual division. The key difference is that in our setting there is also a benefit of aggregation. Since reporting takes place every period in our model, it is the cross-division aggregation of output in the first period that keeps the agent from inferring his interim performance in some states of the world. By contrast, in Arya et al. (1999) the benefit arises from delay, and delay keeps the agent from inferring his interim performance in all states of the world because there is no first period reporting. Aggregation, in their setting, is the assumed cost of delay and does not have the potential for providing any benefits. Arya et al. (1999) find that, when restricting attention to agents exhibiting constant absolute risk aversion, delay with aggregation is strictly preferred to no delay. This is jointly a result of the facts that aggregation is costless in a binary setting where the agency is identical across periods and that the specific utility function employed by Arya et al. (1999) admits strict benefits to delay regardless of the information environment.3 By contrast, in this model the loss of information due to aggregation introduces a real cost. This is because we introduce an asymmetry across divisions which creates a cost of contracting on the aggregate variable, not present in a binary setting with symmetric agencies. A final important difference between this paper and the Gigler and Hemmer (1998) and Arya et al. (1999) papers is the nature of the results. Specifically, Gigler and Hemmer (1998) relies on a limiting result, requiring a large number of periods, to show a strict benefit to delay for any set of agent preferences (that are unbounded below). And Arya et al. (1999) relies on a specific form of agent preferences to generate the benefit of delay for any information structure. Here instead we derive necessary and sufficient conditions on the information structure and the agent’s preferences for which aggregation of only two divisions generates a strict benefit over only two periods. These results also give additional insight into the source of benefit from keeping the agent ignorant in both of the other models.

2 This setting is representative of, say, a comparison between quarterly and annual reporting, where the annual report contains only the sum of the quarters’ results 3 Following Gigler and Hemmer (1998) this is not generally the case.

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2. Model Consider a two period principal agent model in which a single risk and effort averse agent, with preferences additively separable in utility defined over terminal wealth, UðÞ; and cost of effort, V ðÞ; controls output by use of two separate and independent assets. Hereafter we refer to these two assets as divisions and index the two divisions by i ¼ 1; 2: The two periods are indexed by t ¼ 1; 2: Let real valued (binary) output in period t of division i be represented by xit Af0; 1g; representing the preferred output (a ‘‘success’’) with a 1 and the other (a ‘‘failure’’) with a zero. The agent controls output by exerting period and division specific effort, eit Afl; hg; which generates a personal cost to him in utility terms each time effort is exerted of Vti ðhÞ > Vti ðlÞ: Effort h is more productive than effort l because probðxit ¼ 1jeit ¼ hÞ > probðxit ¼ 1jeit ¼ lÞ for all t; i pairs. Effort h is said to be high and effort l low. For simplicity we restrict attention to production functions of the form probðxit ¼ 1jeit ¼ hÞ  probð1jhÞ and probðxit ¼ 1jeit ¼ lÞ  probð1jlÞ 8i; t: Also we assume that the agent’s capacity for effort is limited such that he can exert effort in at most one division per period. While the agent’s level of effort is unobservable to the principal, the division the agent is working in each period is observable. Without loss of generality call the division in which the agent works in the first period division 1 and the division where the agent works in the second period division 2. Before the agent has exerted any effort in a division i; the output of that division is high with probability 1/2. Once effort eit Afl; hg has been exerted in division i and period t; output xit Af0; 1g is determined randomly with probabilities probðxit jeit Þ and output in all subsequent periods for that division is the same (i.e., x12  x11 ). These restrictions on the production function are useful in that they effectively reduce a four dimensional problem to two dimensions. They are not in themselves responsible for any of the qualities of the results.4 The critical feature of the hypothesis we examine is that the price set by the capital market in trading equity claims contains information that is otherwise unavailable for contracting between the principal and agent. This requires that neither the principal nor the agent otherwise have access to that information. To model this in the simplest possible way, we assume that the capital market is privately aware of each period’s output and that period output is measured in terms of economic value, or price.5 Once price is set it is publicly observable. The tradeoff we examine is between whether it is better to make each division’s output separately observable by creating separate equity in the two divisions, or whether it is better to make only the sum of the division’s outputs observable by trading both divisions under one claim. Specifically, if tracking stock is created the market will set two prices in each period,

4

Literally this production function resembles that of a ‘‘design’’. The design puts the process in control, is either good or bad, and is irrevocable over some horizon. 5 We have simplified the information setting by ‘‘normalizing’’ output to be equivalent to the private information contained in price.

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Pit  xit ; i ¼ 1; 2 and t ¼ 1; 2: Otherwise the market will only set one price each period, Pt  x1t þ x2t ; t ¼ 1; 2: The informational role of price in our setting is a simplification of that modeled in Dye and Sridhar (1999). Price, through its function as an aggregator of all market information, can be informative to an agent who does not know the value that markets will attach to his actions. In Dye and Sridhar (1999) the agent makes use of this by communicating his intentions to the market and deciding whether or not to follow through with these intentions based upon the market’s (informative) reaction. In our setting, the agent will condition his subsequent period actions upon the market’s reaction to his current period action. The principal’s problem is to maximize the expected two-period sum of the divisions’ outputs net of the expected wage paid to the agent, CðoÞ; conditional on the information available at the end of the second period, oAfðP11 ; P21 ; P12 ; P22 Þ; ðP1 ; P2 Þg: To avoid confusion, about which information set is in use, we will adopt the notation of calling the contract CD ðoÞ when o ¼ ðP11 ; P21 ; P12 ; P22 Þ to denote disaggregate and CA ðoÞ when o ¼ ðP1 ; P2 Þ to denote aggregate information. We assume that it is always optimal to have the agent exert the high effort whenever possible. So given the organizational form (equivalently the available information, o), the principal chooses the expected wage minimizing contract which makes it in the agent’s best interest to exert high effort in division 1 in the first period and high effort in division 2 in the second period. Finally, the principal chooses the organizational form, o; that results in the lower of the two expected wages.

3. Analysis 3.1. Overview Our analysis proceeds as follows. First we derive the optimal contract when the two divisions are separately traded. We call this contract the disaggregate solution and it represents the solution to the problem given that tracking stock has been n created, CD ðÞ: Next we show that the disaggregate solution is feasible when the two divisions are traded together, provided the agency problems across the two divisions are identical. This means that when the divisions are identical aggregation of the two divisions weakly dominates disaggregation. We then derive conditions under which the aggregate contract, CAn ðÞ; strictly dominates the disaggregate contract when the divisions are identical. Finally we establish that disaggregation always strictly dominates aggregation for divisions that are sufficiently different in terms of their agency costs. 3.2. Disaggregate problem Recall that in the disaggregate setting the information available for contracting is described by the four-tuple, ðP11 ; P21 ; P12 ; P22 Þ: Because each of these variables is binary, generally there are sixteen possible contingencies to consider in solving for the

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disaggregate contract. However, the program has been simplified by the form of the production function described in Section 2.0. First note that the agent will be forced to work in division 1 in the first period and in division 2 in the second period. Then P21 is simply noise, not affected by any of the agent’s chosen actions. There is no reason to write the contract contingent on that price.6 Next notice that because output stays the same in each division in all periods after the agent has worked in that division, P12  P11 : So there is no need to contract on P11 : Consequently, only the second period prices, ðP12 ; P22 Þ; are relevant for contracting, meaning that only four contingencies need be considered in the disaggregate contract. Furthermore, since ðP12 ; P22 Þ  ðx11 ; x22 Þ; we have the following program: Program I min Eðx1 ;x2 Þ ½CD ðx11 ; x22 Þje11 ¼ h; e22 ¼ h CD ðÞ

1

2

subject to: individual rationality Eðx1 ;x2 Þ ½UðCD ðx11 ; x22 ÞÞje11 ¼ h; e22 ¼ h V11 ðhÞ V22 ðhÞZU 1 2 % first period incentive compatability ðIRIÞ

ðICh1 Þ

Eðx1 ;x2 Þ ½UðCD ðx11 ; x22 ÞÞje11 ¼ h; e22 ¼ h V11 ðhÞ V22 ðhÞ 1

2

ZEðx1 ;x2 Þ ½UðCD ðx11 ; x22 ÞÞje11 ¼ l; e22 ¼ h V11 ðlÞ V22 ðhÞ 1

2

second period incentive compatibility ðIC12 Þ

Eðx2 Þ ½UðCD ð1; x22 ÞÞje22 ¼ h V22 ðhÞ 2

ZEðx2 Þ ½UðCD ð1; x22 ÞÞje22 ¼ l V22 ðlÞ; 2

ðIC02 Þ

Eðx2 Þ ½UðCD ð0; x22 ÞÞje22 ¼ h V22 ðhÞ 2

ZEðx2 Þ ½UðCD ð0; x22 ÞÞje22 ¼ l V22 ðlÞ: 2

Assuming that it is always optimal to induce the high action, the objective of the principal is to minimize the agent’s expected wage while observing the standard constraints. The IR I constraint gives the agent his required expected utility for the two period contracting horizon. The first period incentive constraint, ICh1 ; provides incentives for the agent to take the high action in the first period conditional on taking the high action in the second period.7 The second period incentive constraints, IC12 and IC02 ; provide incentives for the agent to take the high action in the second period conditional on having observed a first period price consistent with division 1 output of x11 ¼ 1 and x11 ¼ 0; respectively. Note that in the second period there is no 6 There is no role for P21 in inducing first period effort (i.e., ex post randomization) by Holmstrom (1979). From proposition 5 of Arnott and Stiglitz (1988) there is no role for P21 in inducing second period effort (i.e., ex ante randomization) because the optimal deterministic contract we derive results in expected profit to the principal which is strictly concave decreasing in the agent’s expected utility. 7 Only one first period incentive constraint is necessary due to backwards induction.

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need to guard against the agent taking the low effort in both periods. Once the agent knows first period output, first period effort is irrelevant. This is because the contract depends only on first and second period output, the agent knows first period output, and second period output is independent of first period effort. Lemma 1 provides us with a result useful in solving program I. Lemma 1. All of the constraints of program I hold with equality. Proof. All proofs are contained in Appendix A. Solving this system of linear equations gives us the disaggregate solution in our first proposition. n Proposition 1. The solution to program I, CD ðÞ; is characterized by n UðCD ð0; 0ÞÞ ¼ U þ V11 ðhÞ þ V22 ðhÞ probð1jhÞðK1 þ K2 Þ; % n UðCD ð0; 1ÞÞ ¼ U þ V11 ðhÞ þ V22 ðhÞ þ K2 probð1jhÞðK1 þ K2 Þ; % n UðCD ð1; 0ÞÞ ¼ U þ V11 ðhÞ þ V22 ðhÞ þ K1 probð1jhÞðK1 þ K2 Þ; % n UðCD ð1; 1ÞÞ ¼ U þ V11 ðhÞ þ V22 ðhÞ þ K1 þ K2 probð1jhÞðK1 þ K2 Þ; % where

K1 ¼

V11 ðhÞ V11 ðlÞ probð1jhÞ probð1jlÞ

and

K2 ¼

V22 ðhÞ V22 ðlÞ : probð1jhÞ probð1jlÞ

The values of K1 and K2 capture the agent’s effectiveness at working in division 1 and division 2, respectively. If K1 is less than K2 (K2 is less than K1 ) the agent is more productive in division 1 than division 2 (division 2 than division 1). For a given level of productivity to effort the marginal cost of exerting high effort in each division (which appears as the numerators of K1 and K2 ) measures the agent’s productivity in that division. Likewise, for a given marginal cost of exerting high effort, the level of productivity to effort (which appears as the denominators of K1 and K2 ) measures the agent’s productivity in that division. Notice that the agent is paid (in utility terms) Ki at the margin for generating a high output in division i: This of course is what gives the agent the incentive to exert the high effort in each division i: The important feature of this contract for our study is that, unless K1 ¼ K2 ; implementation of the contract requires knowledge of which division generated a success. In general then, there is a cost to aggregating the performance of the two divisions. With an aggregate performance measure only the number of successes are known, and not their locations. The informational benefit of carving out a division is the improvement in contracting available with the division specific performance measures.

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3.3. Aggregate problem Here we present the principal’s problem when the two divisions are traded together. The information available for contracting is now the pair ðP1 ; P2 Þ where Pt  x1t þ x2t ; t ¼ 1; 2: While aggregation is costly because the information structure, and therefore the contract, is unable do distinguish between first and second period outcomes, there is also a benefit to this. The agent too has less information about the first period outcome going into the second period and this can make it less costly to induce the second period effort. This results in the weaker set of second period incentive constraints below. Formally we have the following program. Program II min EðP1 ;P2 Þ ½CA ðP1 ; P2 Þje11 ¼ h; e22 ¼ h CA ðÞ

subject to: individual rationality EðP1 ;P2 Þ ½UðCA ðP1 ; P2 ÞÞje11 ¼ h; e22 ¼ h V11 ðhÞ V22 ðhÞZU % first period incentive compatibility ðIRIÞ

ðICh1 Þ

EðP1 ;P2 Þ ½UðCA ðP1 ; P2 ÞÞje11 ¼ h; e22 ¼ h V11 ðhÞ V22 ðhÞ ZEðP1 ;P2 Þ ½UðCA ðP1 ; P2 ÞÞje11 ¼ l; e22 ¼ h V11 ðlÞ V22 ðhÞ

second period incentive compatibility for each P1 : 1 2 2 1 ðICh;P 2 Þ EðP2 Þ ½UðCA ðP1 ; P2 ÞÞje1 ¼ h; e2 ¼ h; P1 V2 ðhÞ

ZEðP2 Þ ½UðCA ðP1 ; P2 ÞÞje11 ¼ h; e22 ¼ l; P1 V22 ðlÞ 1 2 2 1 ðICl;P 2 Þ EðP2 Þ ½UðCA ðP1 ; P2 ÞÞje1 ¼ l; e2 ¼ h; P1 V2 ðhÞ

ZEðP2 Þ ½UðCA ðP1 ; P2 ÞÞje11 ¼ l; e22 ¼ l; P1 V22 ðlÞ: Notice that if the first period price is zero or two, the division 1 output is known with certainty to be zero and one, respectively. Because the division 1 output stays the same in the second period, the second period price then reveals the division 2 output. As a result, when the first period price is either zero or two, the information available in the aggregate setting is identical to that available in the disaggregate setting. Uncertainty regarding the separate divisions’ outputs results only when the first period price is one. Then it is impossible to tell whether the first period’s single success was due to the agent’s effort in division 1 or to pure chance in division 2. Furthermore, if the second period price is also one, neither the principal nor the agent will ever be able to tell which of the agent’s actions was successful. This results in two primary differences between programs I and II when the first period aggregate price is equal to one. First, the contract must be written on the sum of the two second period outputs rather than the separate second period outputs. This reduces the number of possible contingent payoffs when P1 ¼ 1 from four in program I to three in this program. Note that this would be equivalent to adding a

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constraint to program I requiring that CD ðx12 ; x22 Þ  CD ðx22 ; x12 Þ: This is the cost of aggregation. The second difference is that in program I the second period incentive constraints are conditional on the first period output realizations, whereas in program II the second period incentive constraints are conditional on the expected first period outcomes given the first period aggregate price. When P1 ¼ 1 these constraints are weaker than the program I constraints in that they are each convex combinations of the individual program II constraints. Intuitively, the second period incentive constraints in program II are weaker than those in program I because they need only be met in expectation, whereas the program I constraints must be met state-by-state. This is the benefit of aggregation from keeping the agent ignorant of division 1 output. The first step in our analysis of the aggregate solution is to compare programs I and II in the case where K1 ¼ K2 : Lemma 2. When K1 ¼ K2 aggregation is costless and therefore aggregation weakly dominates disaggregation. To see this note that any disaggregate contract satisfying CD ðx12 ; x22 Þ  CD ðx22 ; x12 Þ is implementable as an aggregate contract. This is because any such contract is measurable with respect to the sum of second period outputs and this is equivalent to being measurable with respect to second period aggregate price. And as Proposition 1 shows, when K1 ¼ K2 the solution to program I satisfies the condition that CD ðx12 ; x22 Þ  CD ðx22 ; x12 Þ: The only other difference between programs I and II is that program II has a weaker set of second period incentive constraints, requiring satisfaction only in expectation rather than state-by-state. Hence, the optimal contract in program I is feasible in program II, making aggregation weakly better than disaggregation. Next we characterize when strict benefits to aggregation exist. The following lemma establishes a necessary and sufficient condition for aggregation to strictly dominate disaggregation in terms of the second period incentive compatibility constraints of program II. Lemma 3. When K1 ¼ K2 ; a necessary and sufficient condition for aggregation to 1 ¼1 strictly dominate disaggregation is that either constraint ICh;P or constraint IC2l;P1 ¼1 2 is slack. The sufficiency comes from the fact that when one of the second period incentive constraints of program II is slack, the optimal contract is no longer linear (in utility terms) in second period output, as it is with the optimal disaggregate contract. Necessity follows because binding both second period incentive constraints in program II is equivalent to satisfying the state-by-state incentive constraints of program I. Hence, the benefit of aggregation exists only when slack may be put into one of these constraints. When there is slack in one of the second period incentive constraints in program II they are less demanding than the second period incentive constraints of program I.

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The degree to which it is more or less demanding to meet the incentive constraints state-by-state is captured by the predictability of the realization of the division 1 output, x12 ; given the chosen action, e11 : The more predictable the division 1 output, the less beneficial meeting the constraints in expectation rather than state-by-state and therefore the less the potential benefit to aggregation. For instance, if the first period output is perfectly predictable when the agent takes the high action (meaning 1 ¼1 probð1jhÞ ¼ 1), ICh;P is identical to IC12 : Likewise, if first period output is perfectly 2 1 ¼1 predictable when the agent takes the low action (meaning probð1jlÞ ¼ 0), ICl;P is 2 0 identical to IC2 : As a result, when output is perfectly predictable for both actions (i.e., probð1jhÞ ¼ 1 and probð1jlÞ ¼ 0), the set of incentive compatible contracts in program II is equivalent to the set of incentive compatible contracts in program I, with no resulting benefit to aggregation. More generally, the feasible region of incentive compatible contracts in program II is monotonically decreasing in 1 ¼1 probð1jhÞ; with ICh;P approaching IC12 ; and monotonically increasing in 2 l;P1 ¼1 probð1jlÞ; with IC2 approaching IC02 : Clearly the information setting fprobð1jhÞ; probð1jlÞg; is instrumental in characterizing conditions under which aggregation is strictly better than disaggregation. So is the form of the agent’s utility function because it determines which of the second period incentive constraints are binding in the aggregate setting. Our next two propositions summarize necessary and sufficient conditions under which strict improvements from aggregation exist when divisions 1 and 2 are symmetric. Proposition 2. When K1 ¼ K2 and U 00 =U 03 is constant in wealth, disaggregation is equivalent to aggregation. Proposition 3. When K1 ¼ K2 and U 00 =U 03 is strictly increasing in wealth, aggregation strictly dominates disaggregation. When K1 ¼ K2 and U 00 =U 03 is strictly decreasing in wealth, aggregation strictly dominates dissagregation if probð1jlÞ=probð1jhÞ is sufficiently close to 1, but disaggregation is equivalent to aggregation if probð1jlÞ=probð1jhÞr1=2: The conditions on the agent’s utility function given in Propositions 2 and 3 are common in the literature on optimal monitoring (Baiman and Demski, 1980; Dye, 1986; Lambert, 1985). Jewitt (1988) also adopts them as part of his set of sufficient conditions for applying the first-order approach to principal-agent problems. Jewitt provides some intuition to these conditions, stating that

U 00 =U 03 nondecreasing means that the agent is ‘‘more easily motivated by sticks rather than carrots’’ (p. 1187). For instance, in a one period principalagent problem with a likelihood ratio that is linear in output, the optimal contract measured in utility is increasing concave in output if U 00 =U 03 is increasing. But with U 00 =U 03 decreasing, the optimal contract measured in utility is increasing convex in output. Hence the interpretation of using punishments when U 00 =U 03 is increasing and rewards when it is decreasing. For the border case, where U 00 =U 03 is constant, the optimal contract measured in utility is linear in output.

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Taken together, Propositions 2 and 3 imply that when divisions are symmetric there exist information settings such that a strict contracting improvement is available through aggregation if and only if U 00 =U 03 is not constant.8 This result deserves some explanation. By Proposition 1, in the disaggregate setting (with K1 ¼ K2 ) the optimal contract in utility terms is linear in output regardless of the agent’s utility function. This is because the second period incentive constraints require the difference in utility contingent on second period output to be the same regardless of first period output. That is meeting second period incentive constraints state-by-state imposes the linearity.9 By contrast, the relaxed incentive constraints of the aggregate setting in program II allow the differences in utility contingent on second period output to differ with first period output. This means that an incentive compatible contract measured in utility may be either strictly concave or convex in output if desired. However, when U 00 =U 03 is constant in wealth, the optimal contract (measured in utility) is linear in output. But since the benefit of aggregation comes from allowing a contract that is nonlinear in output, the benefit is realizable if and only if U 00 =U 03 is not constant. Proposition 3 shows that when U 00 =U 03 is strictly increasing, the benefit of aggregation is realizable if and only if the output is not too predictable. Given an agent utility function of this form, aggregation is beneficial only when the second period incentive constraints in program II are sufficiently relaxed compared to those in program I. Even though the agency is better off with some non-linearity in the contract whenever U 00 =U 03 is not constant, this may hold only for small departures from linearity. Sufficiently small departures will not be feasible unless the constraints are suitably relaxed. 3.4. Asymmetric divisions In the previous section we established that aggregation weakly dominates disaggregation when the two divisions of the firm are identical. In addition, we provided necessary and sufficient conditions for aggregation to strictly dominate disaggregation with identical divisions. In this section we prove that disaggregation strictly dominates aggregation for any agency if the two divisions are sufficiently different in terms of agent productivity. Recall that we parameterize the agent’s productivity in each of the two divisions by the variables K1 and K2 : Holding the probabilities of success conditional on action constant means that K1 and K2 capture the agent’s marginal cost of exerting a high amount of effort in divisions 1 and 2, respectively. Hence, higher Ki means lower productivity in division i. Proposition 4 gives our main result.

8

It should be noted that this is a very large set of preferences. Within the HARA class this condition is satisfied for all but the square root utility function. 9 Meeting the incentive constraints state-by-state is also responsible for the linearity identified by Holmstrom and Milgrom (1987).

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Proposition 4. Disaggregation strictly dominates aggregation whenever the marginal cost of effort in division 1 is sufficiently low compared to division 2. The cost of aggregation is in not knowing which specific division generated a success or a failure. Section 3.2 showed that when the divisions are identical this cost is zero. Because the optimal marginal payment for a success is the same for both divisions, knowing which specific division generated the success is unnecessary for implementation of the optimal contract. However, if the two divisions are different, the incentives in the optimal contract are likewise different across divisions. To implement the optimal contract the principal needs to know which specific division was successful. Consistent with the results of Proposition 4, it is intuitive that this cost of aggregation is maximal when the difference in divisions is greatest. Less obvious is the fact that the benefit of aggregation goes away as the marginal cost of the high action in division 1 goes to zero. Recall that the benefit of aggregation is in relaxing the second period incentive constraints by keeping the agent ignorant of first period output. But Proposition 4 holds constant second period productivity and the predictability of first period output. So how is it that reducing the first period agency problem reduces the benefit of aggregation? The key is that the benefit of aggregation is not in keeping the agent ignorant of first period output per se, but rather in keeping him ignorant of his wage conditional on first period output. As the first period agency cost goes to zero, so does the difference in wage conditional on first period output. So while output is no more predictable given action, wage conditional on first period output is! And similar to the result in Proposition 3, the value of aggregation is decreasing in the predictability of the wage conditional on first period output.

4. Summary and conclusions In this paper we provide an informational theory for how the ownership claims to a firm might be structured. When the market price of equity provides valuable contracting information there is a benefit to creating separate ownership claims to each of the firm’s assets. Where other theories have counteracted this benefit with a productive cost, we recognize a potential informational cost. We feel that this informational theory is an improvement over existing theories because it is robust to the ability to separate investing (or production) from financing activities. As a result, our theory may provide better predictions as to how the ownership of a firm is constructed. For instance, our theory would predict that ownership claims bundle similar assets, where ‘‘similar’’ means that management is equally productive in operating the assets. This would be the case regardless of the scale economies related to these assets. In contrast, if assets are grouped under the same ownership claim because of productive efficiencies, scale economy factors are of primary importance. It would be interesting to empirically discern whether the movement toward ‘‘core competency’’ is fueled by scale economies or by informational concerns. Our theory would predict

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the latter. We show how the information environment in which the firm operates can be responsible for the way it structures its ownership. Specifically, our theory predicts that the agency cost of separating ownership of assets is greatest when the returns from the assets are either most unpredictable or when there is little other information available about the specific asset performance.

Appendix A Proof of Lemma 1. For notational convenience define UD ðP12 ; P22 Þ  UðCD ðP12 ; P22 ÞÞ: Then the incentive constraints can be rewritten with slack variables as IC12 : UD ð1; 1Þ UD ð1; 0Þ ¼ K2 þ e12 IC02 : UD ð0; 1Þ UD ð0; 0Þ ¼ K2 þ e02 ICh1 : probð1jhÞ½UD ð1; 1Þ UD ð1; 0Þ þ probð0jhÞ½UD ð0; 1Þ UD ð0; 0Þ þ ½UD ð1; 0Þ UD ð0; 1Þ ¼ K1 þ eh1 First we prove that the second period incentive constraints are both binding. Suppose for the purpose of contradiction that at least one of the second period incentive constraints has slack. Without loss of generality suppose that IC12 has slack (the proof when IC02 has slack is symmetric), so e12 > 0: Then introducing the following contractual variation, C# D ðP12 ; P22 Þ; will lower the expected wage while maintaining all of the constraints: UðC# D ð1; 1ÞÞ ¼ UðCD ð1; 1ÞÞ d probð1jhÞ d; UðC# D ð1; 0ÞÞ ¼ UðCD ð1; 0ÞÞ þ probð1jlÞ where drprobð1jlÞe12 : It is easy to check that this variation is neutral to the IC02 ; ICh1 and IR I constraints. Therefore any such variation is feasible when IC12 is slack. To see that the variation leads to an improvement note that, with GðÞ  U 1 ðÞ; the change in expected compensation from the variation is: probð1jhÞ2 ½GðUD ð1; 1Þ dÞ GðUD ð1; 1ÞÞ     probð1jhÞ d GðUD ð1; 0ÞÞ þ probð1jhÞprobð0jhÞ G UD ð1; 0Þ þ probð1jlÞ which has the same sign as: GðUD ð1; 0Þ þ d probð1jhÞ=probð1jlÞÞ GðUD ð1; 0ÞÞ d probð1jhÞ=probð1jlÞ GðUD ð1; 1Þ dÞ GðUD ð1; 1ÞÞ

d

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As d-0 this gives G 0 ðUD ð1; 0ÞÞ G0 ðUD ð1; 1ÞÞ which is less than zero by U 00 ðÞo0; generating the contradiction. With IC12 and IC02 binding it is easy to see that ICh1 must be binding. If not, one can lower UD ð1; 0Þ by drprobð1jlÞeh1 and increase UD ð0; 0Þ by dprobð1jhÞ=probð1jlÞ; generating an improvement and a contradiction. Finally, if IR I is slack the utility of each wage contingency can be lowered by an equal amount d > 0: This maintains all incentive constraints exactly and generates a reduction in expected wage, thereby contradicting that IR I is slack. & Proof of Proposition 1. Program I requires solving for four contingent payments: CD ð1; 1Þ; CD ð1; 0Þ; CD ð0; 1Þ and CD ð0; 0Þ: By Lemma 1, all four constraints hold with equality. Solving these four equations for the four utilities gives the results of the proposition. & Proof of Lemma 2. The aggregate problem can be written as min

CA ðP1 ;P2 Þ

ð1=2Þprobð0jhÞ2 ½CA ð0; 0Þ þ CA ð1; 0Þ

þ ð1=2Þprobð0jhÞprobð1jhÞ½CA ð0; 1Þ þ CA ð1; 1Þ þ ð1=2Þprobð1jhÞprobð0jhÞ½CA ð1; 1Þ þ CA ð2; 1Þ þ ð1=2Þprobð1jhÞ2 ½CA ð1; 2Þ þ CA ð2; 2Þ subject to: ðIRAÞ ð1=2Þprobð0jhÞ2 ½UA ð0; 0Þ þ UA ð1; 0Þ þ ð1=2Þprobð0jhÞprobð1jhÞ½UA ð0; 1Þ þ UA ð1; 1Þ þ ð1=2Þprobð1jhÞprobð0jhÞ½UA ð1; 1Þ þ UA ð2; 1Þ þ ð1=2Þprobð1jhÞ2 ½UA ð1; 2Þ þ UA ð2; 2Þ Z U þ V11 ðhÞ þ V22 ðhÞ % ðICAh1 Þ probð0jhÞf½UA ð1; 1Þ UA ð1; 0Þ þ ½UA ð2; 1Þ UA ð0; 0Þ g þ probð1jhÞf½UA ð2; 2Þ UA ð0; 1Þ þ ½UA ð1; 2Þ UA ð1; 1Þ gZ2K1 1 ¼0 1 ¼0 ðICAh;P Þ ðICAl;P Þ UA ð0; 1Þ UA ð0; 0ÞZK2 2 2 1 ¼1 ðICAh;P Þ probð1jhÞ½UA ð1; 2Þ UA ð1; 1Þ þ probð0jhÞ½UA ð1; 1Þ UA ð1; 0Þ ZK2 2

ðICA2l;P1 ¼1 Þ probð1jlÞ½UA ð1; 2Þ UA ð1; 1Þ þ probð0jlÞ½UA ð1; 1Þ UA ð1; 0Þ ZK2 1 ¼2 1 ¼2 ðICAh;P Þ ðICAl;P Þ UA ð2; 2Þ UA ð2; 1ÞZK2 2 2

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To implement the disaggregate solution as an aggregate contract, choose CA ðP1 ; P2 Þ independent of P1 and satisfying: n CA ð0; 0Þ ¼ CA ð1; 0Þ ¼ CD ð0; 0Þ n n ð0; 1Þ ¼ CD ð0; 1Þ CA ð0; 1Þ ¼ CA ð1; 1Þ ¼ CA ð2; 1Þ ¼ CD n ð1; 1Þ CA ð1; 2Þ ¼ CA ð2; 2Þ ¼ CD

Now it is easy to verify, through the results of Proposition 1, that this choice of CA ðP1 ; P2 Þ satisfies all of the constraints as written above when K1 ¼ K2 : & Proof of Lemma 3. From program II as written in the proof of Lemma 2 it can be shown that satisfaction of the second period incentive constraints implies that the first period incentive constraint is satisfied as long as: ðICAh1 Þ UA ð2; 1Þ UA ð0; 1ÞZ0 Now we show that either IC2h;P1 ¼1 or IC2l;P1 ¼1 must be binding. Both slack requires UA ð1; 2Þ UA ð1; 1Þ > K2 and UA ð1; 1Þ UA ð1; 0Þ > K2 : But then lowering UA ð1; 2Þ by e and increasing UA ð1; 0Þ by e probð1jhÞ2 =probð0jhÞ2 decreases the objective function while satisfying IRA and ICAh1 : 1 ¼1 1 ¼1 Sufficiency follows from the fact that if ICh;P (ICl;P ) is slack, binding IC2l;P1 ¼1 2 2 h;P1 ¼1 (IC2 ) requires UA ð1; 2Þ UA ð1; 1Þ > ðoÞUA ð1; 1Þ UA ð1; 0Þ: But this must be an improvement over the disaggregate contract, which is feasible, because that contract has UD ð1; 1Þ UD ð1; 0Þ ¼ UD ð1; 0Þ UD ð0; 0Þ: Next we show necessity. IC2h;P1 ¼1 and IC2l;P1 ¼1 both binding implies UA ð1; 2Þ UA ð1; 1Þ ¼ K2 and UA ð1; 1Þ UA ð1; 0Þ ¼ K2 : Given this, and K1 ¼ K2 ; the first period incentive constraints can be rewritten as: ðICAh1 Þ probð0jhÞ½UA ð2; 1Þ UA ð0; 0Þ þ probð1jhÞ½UA ð2; 2Þ UA ð0; 1Þ ZK2 Now it is easy to show that both IC2P1 ¼0 and IC2P1 ¼2 are binding. For if IC2P1 ¼0 (IC2P1 ¼2 ) has slack, decrease UA ð0; 1Þ (UA ð2; 2Þ) by e and increase UA ð0; 0Þ (UA ð2; 1Þ) by e probð1jhÞ=probð0jhÞ: This variation exactly maintains IRA, is neutral to all other incentive constraints and decreases the objective function. So far we have: UA ð1; 2Þ UA ð1; 1Þ ¼ K2 ; UA ð0; 1Þ UA ð0; 0Þ ¼ K2

UA ð1; 1Þ UA ð1; 0Þ ¼ K2 UA ð2; 2Þ UA ð2; 1Þ ¼ K2

From this we will show that both first period incentive constraints must bind. First note that ½UA ð2; 2Þ UA ð0; 1Þ ¼ ½UA ð2; 1Þ UA ð0; 0Þ : This follows from: ½UA ð2; 2Þ UA ð0; 1Þ ¼ ½UA ð2; 2Þ UA ð2; 1Þ þ ½UA ð2; 1Þ UA ð0; 1Þ ¼ K2 þ UA ð2; 1Þ ½UA ð0; 0Þ þ K2

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Therefore, either both first period incentive constraints have slack or both are binding. We now show by contradiction that both are binding. Suppose both are slack, then: ½UA ð2; 2Þ UA ð0; 1Þ ¼ ½UA ð2; 1Þ UA ð0; 0Þ > K2 But this allows for an improvement by decreasing UA ð2; 2Þ and UA ð2; 1Þ by e and increasing UA ð0; 1Þ and UA ð0; 0Þ by e probð1jhÞ=probð0jhÞ: So now we also have: UA ð2; 2Þ UA ð0; 1Þ ¼ K2

UA ð2; 1Þ UA ð0; 0Þ ¼ K2

Note that UA ð2; 1Þ ¼ UA ð0; 1Þ from UA ð2; 2Þ UA ð0; 1Þ ¼ K2 and UA ð2; 2Þ UA ð2; 1Þ ¼ K2 : To complete the proof we show that UA ð1; 1Þ ¼ UA ð2; 1Þ ¼ UA ð0; 1Þ: Suppose otherwise, that UA ð1; 1Þ > UA ð2; 1Þ (UA ð1; 1ÞoUA ð2; 1Þ). Then UA ð1; 2Þ > UA ð2; 2Þ > UA ð1; 1Þ > UA ð2; 1Þ ¼ UA ð0; 1Þ > UA ð1; 0Þ > UA ð0; 0Þ (UA ð2; 2Þ > UA ð1; 2Þ > UA ð2; 1Þ ¼ UA ð0; 1Þ > UA ð1; 1Þ > UA ð0; 0Þ > UA ð1; 0Þ). Lower (increase) UA ð1; 0Þ; UA ð1; 1Þ and UA ð1; 2Þ by e and increase (lower) UA ð0; 0Þ; UA ð0; 1Þ; UA ð2; 1Þ and UA ð2; 2Þ by e: This variation is neutral to all constraints and generates a reduction in the expected wage. So we have that CA ð1; 1Þ ¼ CA ð2; 1Þ ¼ CA ð0; 1Þ: Then UA ð1; 1Þ UA ð1; 0Þ ¼ K2 and UA ð0; 1Þ UA ð0; 0Þ ¼ K2 gives CA ð1; 0Þ ¼ CA ð0; 0Þ and UA ð2; 2Þ UA ð2; 1Þ ¼ K2 along with UA ð1; 2Þ UA ð1; 1Þ ¼ K2 gives CA ð2; 2Þ ¼ CA ð1; 2Þ: Finally, the program becomes one of choosing three contingent wages independent of first period price, and dependent upon second period price in such a way that UA ð; 2Þ UA ð; 1Þ ¼ UA ð; 1Þ UA ð; 0Þ ¼ K2 while maintaining IRA. Solving these three equations for the three unknowns gives n CA ð0; 0Þ ¼ CA ð1; 0Þ ¼ CD ð0; 0Þ

n n ð0; 1Þ ¼ CD ð1; 0Þ CA ð0; 1Þ ¼ CA ð1; 1Þ ¼ CA ð2; 1Þ ¼ CD

n ð1; 1Þ: CA ð1; 2Þ ¼ CA ð2; 2Þ ¼ CD

&

Proof of Proposition 2. Recall from the proof of Lemma 3 that the following is sufficient for the first period incentive constraints to be satisfied: ðICAh1 Þ UA ð2; 1Þ UA ð0; 1ÞZ0: 1 ¼1 Furthermore, from the proof of Lemma 3 we know that either IC2h;P1 ¼1 or ICl;P 2 must be binding. This in turn implies that UA ð2; 1Þ UA ð0; 1ÞZ0 is also necessary for ICAh1 if IC2h;P1 ¼1 is binding. Therefore, UA ð2; 1Þ UA ð0; 1ÞZ0 is necessary and sufficient for ICAh1 :

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This allows us to write program II as: min ð1=2Þprobð0jhÞ2 ½GðUA ð0; 0ÞÞ þ GðUA ð1; 0ÞÞ

UA ðP1 ;P2 Þ

þ ð1=2Þprobð0jhÞprobð1jhÞ½GðUA ð0; 1ÞÞ þ GðUA ð1; 1ÞÞ þ ð1=2Þprobð1jhÞprobð0jhÞ½GðUA ð1; 1ÞÞ þ GðUA ð2; 1ÞÞ þ ð1=2Þprobð1=hÞ2 ½GðUA ð1; 2ÞÞ þ GðUA ð2; 2ÞÞ subject to: ðIRAÞ ð1=2Þprobð0jhÞ2 ½UA ð0; 0Þ þ UA ð1; 0Þ þ ð1=2Þprobð0jhÞprobð1jhÞ½UA ð0; 1Þ þ UA ð1; 1Þ þ ð1=2Þprobð1jhÞprobð0jhÞ½UA ð1; 1Þ þ UA ð2; 1Þ þ ð1=2Þprobð1jhÞ2 ½UA ð1; 2Þ þ UA ð2; 2Þ Z U þ V11 ðhÞ þ V22 ðhÞ % ðICAh1 Þ UA ð2; 1Þ UA ð0; 1ÞZ0 1 ¼0 Þ and ðICA2l;P1 ¼0 Þ UA ð0; 1Þ UA ð0; 0ÞZK2 ðICAh;P 2 1 ¼1 ðICAh;P Þ probð1jhÞ½UA ð1; 2Þ UA ð1; 1Þ þ probð0jhÞ½UA ð1; 1Þ UA ð1; 0Þ ZK2 2

ðICA2l;P1 ¼1 Þ probð1jlÞ½UA ð1; 2Þ UA ð1; 1Þ þ probð0jlÞ½UA ð1; 1Þ UA ð1; 0Þ ZK2 1 ¼2 ðICAh;P Þ and ðICA2l;P1 ¼2 Þ UA ð2; 2Þ UA ð2; 1ÞZK2 2

This is a convex programming problem with linear constraints. Therefore, the Kuhn–Tucker conditions are necessary and sufficient for its solution. Let lh2 be the 1 ¼1 1 ¼1 lagrange multiplier for ICAh;P ; ll2 be the lagrange multiplier for ICAl;P and m 2 2 for IRA. Differentiating the lagrangian with respect to UA ð1; 2Þ; UA ð1; 1Þ and UA ð1; 0Þ gives the following three equations: qL ¼ ð1=2Þprobð1jhÞ2 G0 ðUA ð1; 2ÞÞ ð1=2Þprobð1jhÞ2 m qUA ð1; 2Þ

probð1jhÞlh2 probð1jlÞll2 qL ¼ probð1jhÞprobð0jhÞG 0 ðUA ð1; 1ÞÞ probð1jhÞprobð0jhÞm qUA ð1; 1Þ þ ½probð1jhÞ probð0jhÞ lh2 ½probð0jlÞ probð1jlÞ ll2 qL ¼ ð1=2Þprobð0jhÞ2 G0 ðUA ð1; 0ÞÞ ð1=2Þprobð0jhÞ2 m qUA ð1; 0Þ þ probð0jhÞlh2 þ probð0jlÞll2 An interior solution requires that these three derivatives be equal to zero. Setting them equal to zero and solving for the lagrange multipliers means that an interior

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solution must have: probð0jhÞprobð1jhÞ lh2 ¼ 1=2 probð1jhÞ probð1jlÞ fprobð1jhÞ½2probð0jlÞ probð0jhÞ ½G 0 ðUA ð1; 2ÞÞ G 0 ðUA ð1; 1ÞÞ

probð0jhÞ½2probð1jlÞ probð1jhÞ ½G0 ðUA ð1; 1ÞÞ G 0 ðUA ð1; 0ÞÞ g and probð1jhÞ2 probð0jhÞ2 probð1jhÞ probð1jlÞ f½G 0 ðUA ð1; 1ÞÞ G 0 ðUA ð1; 0ÞÞ ½G 0 ðUA ð1; 2ÞÞ G 0 ðUA ð1; 1ÞÞ g

ll2 ¼ 1=2

To simplify notation define DH  UA ð1; 2Þ UA ð1; 1Þ and DL  UA ð1; 1Þ UA ð1; 0Þ: 1 ¼1 Recall from the proof of Lemma 3 that if ICh;P (IC2l;P1 ¼1 ) is slack, binding IC2l;P1 ¼1 2 h;P1 ¼1 (IC2 ) requires UA ð1; 2Þ UA ð1; 1Þ > ðoÞUA ð1; 1Þ UA ð1; 0Þ: Lemma 3 then implies that a necessary and sufficient condition for aggregation to strictly dominate disaggregation is DH aDL : The Kuhn–Tucker conditions require that each of the lagrange multipliers be greater than or equal to zero. The expression for ll2 therefore requires that ½G0 ðUAn ð1; 1ÞÞ G 0 ðUAn ð1; 0ÞÞ Z ½G0 ðUAn ð1; 2ÞÞ G 0 ðUAn ð1; 1ÞÞ : When U 00 =U 03 is constant (i.e., G 00 is constant), this is satisfied if and only if DH rDL : So any benefit of aggregation must come from setting DH oDL : It is clear from the two incentive 1 ¼1 constraints that if DH oDL ; satisfaction of IC2h;P1 ¼1 means that ICl;P will have 2 l;P1 ¼1 l slack. IC2 slack in turn implies that l2 ¼ 0; which is the case if and only if ½G0 ðUAn ð1; 1ÞÞ G 0 ðUAn ð1; 0ÞÞ ¼ ½G 0 ðUAn ð1; 2ÞÞ G 0 ðUAn ð1; 1ÞÞ : Again, for U 00 =U 03 constant, this is satisfied if and only if DH ¼ DL : As a result, there is no benefit to setting DH oDL when U 00 =U 03 is constant, proving the proposition. & Proof of Proposition 3. The Kuhn–Tucker conditions require that ll2 Z0: Therefore, ½G0 ðUAn ð1; 1ÞÞ G 0 ðUAn ð1; 0ÞÞ Z½G 0 ðUAn ð1; 2ÞÞ G 0 ðUAn ð1; 1ÞÞ : When U 00 =U 03 is strictly increasing (i.e., G 00 is strictly increasing), this is satisfied only if DH oDL : Since the disaggregate solution is feasible and has the characteristic that DH ¼ DL ; the aggregate solution strictly dominates the disaggregate solution, proving the first part of the proposition. The proof for when U 00 =U 03 is strictly decreasing follows in two parts. First we show that DH ¼ DL whenever probð1jlÞ=probð1jhÞr1=2: This is because the coefficient on ½G 0 ðUA ð1; 1ÞÞ G 0 ðUA ð1; 0ÞÞ in the expression for lh2 is positive iff this condition on the likelihood ratio is satisfied, while the coefficient on ½G 0 ðUA ð1; 2ÞÞ G 0 ðUA ð1; 1ÞÞ is always strictly greater than zero. Therefore, the likelihood ratio condition implies lh2 > 0 and, hence, that IC2h;P1 ¼1 is binding. 1 ¼1 With ICh;P binding, satisfaction of IC2l;P1 ¼1 requires DH rDL : But since U 00 =U 03 2 1 ¼1 is strictly decreasing, this in turn implies that ll2 > 0; so ICl;P is also binding. 2 Both incentive constraints binding implies DH ¼ DL ; proving the first of the two parts.

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Next we show that DH oDL when probð1jlÞ=probð1jhÞ is close to one. It is sufficient to show that DH ¼ DL and probð1jlÞ=probð1jhÞ close to one together imply lh2 o0: Rewriting the expression for lh2 gives: lh2 Z03

probð1jhÞ 2probð0jlÞ probð0jhÞ G 0 ðUA ð1; 1ÞÞ G0 ðUA ð1; 0ÞÞ Z : probð0jhÞ 2probð1jlÞ probð1jhÞ G 0 ðUA ð1; 2ÞÞ G0 ðUA ð1; 1ÞÞ

The left hand side of this equation approaches 1 as probð1jlÞ=probð1jhÞ approaches 1 while the right hand side evaluated at DH ¼ DL is strictly greater than one whenever U 00 =U 03 is strictly decreasing. Therefore, by continuity lh2 o0 for probð1jlÞ=probð1jhÞ sufficiently close to one, proving the result. & Proof of Proposition 4. Denote the optimal aggregate contract as UAn ð; Þ: If the 1 ¼1 optimal aggregate contract has both IC2l;P1 ¼1 and ICh;P binding there is no cost to 2 disaggregation, so disaggregation strictly dominates aggregation whenever K1 aK2 : Suppose the optimal aggregate contract has IC2l;P1 ¼1 binding and IC2h;P1 ¼1 slack. Then UAn ð1; 2Þ UAn ð1; 1Þ > K2 > UAn ð1; 1Þ UAn ð1; 0Þ: Let UAn ð1; 2Þ UAn ð1; 1Þ ¼ K2 þ D and consider the following (disaggregate) variation on the optimal aggregate contract: UD ð1; 1Þ ¼ UAn ð1; 2Þ þ K1 D UD ð0; 1Þ ¼ UAn ð1; 1Þ þ D

probð0jhÞprobð1jlÞ probð1jhÞprobð0jlÞ

probð1jlÞ probð0jlÞ

UD ð1; 0Þ ¼ UAn ð1; 1Þ UD ð0; 0Þ ¼ UAn ð1; 0Þ

probð1jhÞ2 K1: probð0jhÞ2

This variation is expected utility neutral and so satisfies IRA. To see that it is incentive compatible in both periods note that: First period UD ð1; 1Þ UD ð0; 1Þ ¼ UAn ð1; 2Þ UAn ð1; 1Þ þ K1   probð0jhÞprobð1jlÞ probð1jlÞ þ

D probð1jhÞprobð0jlÞ probð0jlÞ > K2

probð1jlÞ D þ K1 ¼ UAn ð1; 1Þ UAn ð1; 0Þ þ K1 ZK1 probð0jlÞ

Second period UD ð1; 1Þ UD ð1; 0Þ ¼ UAn ð1; 2Þ UAn ð1; 1Þ þ K1 > K2 þ K1 > K2

probð0jhÞprobð1jlÞ D probð1jhÞprobð0jlÞ

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and UD ð0; 1Þ UD ð0; 0Þ ¼ UAn ð1; 1Þ UAn ð1; 0Þ þ ¼ K2 þ

probð1jlÞ probð1jhÞ2 Dþ K1 probð0jlÞ probð0jhÞ2

probð1jhÞ2 K1 > K2 probð0jhÞ2

The change in expected wage from adopting this variation is:     probð0jhÞprobð1jlÞ 2 n n D GðUA ð1; 2ÞÞ probð1jhÞ G UA ð1; 2Þ þ K1 probð1jhÞprobð0jlÞ     probð1jlÞ n n þ probð0jhÞprobð1jhÞ G UA ð1; 1Þ þ D GðUA ð1; 1ÞÞ probð0jlÞ     probð1jhÞ2 n þ probð0jhÞ2 G UAn ð1; 0Þ þ K ð1; 0ÞÞ :

GðU 1 A probð0jhÞ2 By the convexity of GðÞ this expression is less than   probð0jhÞprobð1jlÞ 0 D K1

G ðUD ð1; 1ÞÞ probð1jhÞprobð0jlÞ probð0jhÞprobð1jlÞ

G 0 ðUD ð0; 0ÞÞK1 þ G 0 ðUD ð0; 1ÞÞD probð1jhÞprobð0jlÞ which is less than zero for sufficiently small K1 : This proves the strict benefit of disaggregation when IC2l;P1 ¼1 binds for sufficiently small K1 : 1 ¼1 1 ¼1 The proof for when the optimal aggregate contract has ICl;P slack and ICh;P 2 2 binding is very similar and so will not be repeated. We give the variation in this setting because it is not the same as the previous one. Then UAn ð1; 2Þ UAn ð1; 1ÞoK2 oUAn ð1; 1Þ UAn ð1; 0Þ: So let UAn ð1; 1Þ UAn ð1; 0Þ ¼ K2 þ D and consider the following (disaggregate) variation on the optimal aggregate contract: UD ð1; 1Þ ¼ UAn ð1; 2Þ þ K1 UD ð0; 1Þ ¼ UAn ð1; 1Þ UD ð1; 0Þ ¼ UAn ð1; 1Þ

probð0jhÞ D probð1jhÞ

UD ð0; 0Þ ¼ UAn ð1; 0Þ þ D K1 : This variation puts slack in IRA. Verifying incentive compatibility in both periods and then showing a decrease in expected wage for small K1 completes the proof. &

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