multistage capital budgeting for shared investments

MANAGEMENT SCIENCEArticles in Advance, pp. 1–16ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.1120.1598

© 2012 INFORMS

Multistage Capital Budgeting for Shared Investments

Nicole Bastian JohnsonHaas School of Business, University of California, Berkeley, Berkeley, California, 94720, [email protected]

Thomas PfeifferDepartment of Business Administration, University of Vienna, 1210 Vienna, Austria, [email protected]

Georg SchneiderFaculty of Business Administration, University of Paderborn, 33098 Paderborn, Germany, [email protected]

This paper studies the performance of delegated decision-making schemes in a two-stage, multidivision capi-tal budgeting problem for a shared investment with an inherent abandonment option. Applying both robust

goal congruence and sequential adverse selection frameworks, we show that the optimal capital budgetingmechanism entails a capital charge rate above the firm’s cost of capital in the first stage but below the cost ofcapital in the second stage. Further, the first-stage asset cost-sharing rule depends only on the relative divisionalgrowth profiles, and equal cost sharing can be optimal even when the divisions receive significantly differentbenefits from the shared investment project. In the presence of an adverse selection problem, all agency costsare incorporated into the second-stage budgeting mechanism, leaving the first-stage capital charge rate andasset-sharing rule unaffected even though the agency problem induces capital rationing at both stages.

Key words : capital budgeting; cooperative investments; two-stage investment decisions; abandonment options;cost allocation

History : Received June 9, 2011; accepted February 20, 2012, by Mary Barth, accounting. Published online inArticles in Advance.

1. IntroductionFirms typically coordinate divisional investment deci-sions via the capital budgeting process. One approachthat seems to be prevalent in practice is to makedivisions accountable for their funding using capitalcharge rates or hurdle rates. Empirical and anecdo-tal evidence suggests substantial variation in capitalcharge rates, even within the same firm (e.g., Poterbaand Summers 1995, Mukherjee and Hingorani 1999).Mukherjee and Hingorani (1999) also find evidencethat firms adapt their capital rationing policies overtime, leading to within-firm variation in investmentdecision rules. These empirical findings suggest thatthe capital budgeting process is more complex thandiscussed in the existing theoretical literature, par-ticularly if investments involve multiple rounds offunding and investment decisions can be revised overtime.

This paper studies the design of a capital budgetingmechanism for a rather simple but instructive two-stage investment problem where multiple divisionscan initiate potentially profitable projects if the firmacquires a shared asset at the first stage. Each divi-sion can complete its project and realize the associatedcash flows if the firm invests further at the secondstage. The benefit each division will derive from theshared asset is privately observed by the divisional

manager incrementally over the two stages. Our setupis descriptive for firms that act in environments wheredivisions share substantial resources.

To coordinate the shared investment decisions, thefirm’s capital budgeting mechanism defines a setof rules that disaggregates the firm’s performanceinto divisional performance measures using an assetcost-sharing rule and capital charge rate. Specifi-cally, the asset-sharing rule assigns each division ashare of the joint investment cost, which is allocatedacross time using a firmwide capital charge rate anddepreciation rule. In the first part of the paper, westudy robust goal-congruent divisional performancemeasures (absent an adverse selection problem) thatprovide each divisional manager with incentives tosupport the firm’s optimal long-term investment strat-egy regardless of his planning horizon or time pref-erences. The second part of the paper shows how toadapt the capital budgeting mechanism to a second-best setting with sequential adverse selection.

The management accounting literature has longpointed to potential problems associated with fixedcost allocations.1 In our analysis, fixed cost alloca-tions are necessary to align the ex ante investment

1 For example, Sunder (1997, pp. 55–56) points out that in design-ing a cost-allocation scheme for a shared asset, a decentralized firmwill often face a trade-off between motivating truthful reporting

1

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Published online ahead of print November 5, 2012

Johnson, Pfeiffer, and Schneider: Multistage Capital Budgeting for Shared Investments2 Management Science, Articles in Advance, pp. 1–16, © 2012 INFORMS

incentives of divisional managers and the ownersof the firm, even if the project is ultimately aban-doned. However, the interaction between multipledivisions and multiple investment stages leads to dif-ferent interdivisional asset-sharing rules and capitalcharge rates at each investment stage even though theassets purchased at each investment stage could beessentially identical.2 In particular, we find that thefirm can only induce robust goal congruence with asingle capital charge rate for (i) the static single-stagecase, i.e., in the absence of the abandonment option,or (ii) the single-division case.

Consistent with earlier work on cooperative, single-stage investments, which finds that positive external-ities lower capital charge rates (Baldenius et al. 2007),the second-stage capital charge rate must be set belowthe firm’s cost of capital to make each division prop-erly internalize the positive externalities created forthe other divisions when the investment is carriedout. Similar to results in Baldenius et al. (2007), thesecond-stage asset-sharing rule allocates the second-stage investment cost according to each division’scontribution toward making the continuation projectbreak even.

In contrast, we find that the first-stage capitalcharge rate must be set above the firm’s cost ofcapital. Although the first-stage investment also cre-ates positive expected externalities from any divi-sional manager’s point of view, those externalities arenot incorporated into the asset-sharing rule or cap-ital charge rate until the second investment stage.Accounting for positive externalities at both stageswould “double count” them, inducing inefficientinvestment and abandonment decisions. Instead, thefirst-stage investment cost must be allocated amongthe divisions in proportion to the present value ofeach division’s intertemporal growth profile with theso-called relative growth profile sharing rule (hence-forth, the RGP sharing rule). For a given capitalcharge rate, the RGP sharing rule allocates a smallershare of the initial investment cost to divisions withmore “back-loaded” cash flow profiles (i.e., cash flowsarrive later). The RGP sharing rule can be interpretedas a multiperiod extension of the equal cost-sharingrule, because it corresponds to equal cost sharingwhen all divisions have comparable growth profilesor when all cash inflows arrive in a single period.This implies that equal cost sharing can be optimal

about expected utilization at the purchase stage and motivating effi-cient ex post utilization of the asset. Similarly, Young and O’Byrne(2000) and Zimmerman (1997) provide examples that show how theimproper allocation of shared resource costs can derail investmentincentives.2 For example, if the first-stage investment is a lease of researchfacilities or computing equipment, the second-stage investmentcould be a renewal of the same lease for an additional period.

even when the shared asset provides quite differentbenefits to the individual divisions. In other words,favoritism in terms of “equal treatment of unequals”can be optimal.

Previous results in the capital budgeting litera-ture have emphasized that agency costs play a keyrole in determining capital charge rates. The single-divisional capital budgeting literature, for instance,has shown that private managerial information leadsto capital rationing (e.g., Antle and Eppen 1985). Onecommon approach to implementing capital rationingis to increase the capital charge rate above the firm’scost of capital. To study how divisional agencyconflicts affect the design of capital charge ratesand asset-sharing rules in our two-stage setting, wemodify our goal-congruence model to a second-best(adverse selection) setting by assuming each managercan increase the profitability of his division via hid-den effort.

A sequential agency problem arises because eachmanager can misreport his project information atboth investment stages in order to supply less effort.Consistent with the one-parametric standard adverseselection problem, the firm has to pay each man-ager informational rents for his precontract informa-tion but not for his postcontract information (becauseit can extract any postcontract informational rentsup front).3 The firm can avoid paying informationalrents, however, by either not initiating or abandoningthe shared investment project because each manager’seffort level can be perfectly inferred in the absenceof the cash flows from investment. Accordingly, thesecond-best solution entails capital rationing in bothstages, so that compared to the goal-congruencebenchmark case, the shared investment project is ini-tiated less often and abandoned more often.

Interestingly, however, only the second-stage cap-ital charge rate is directly affected by the agencyconflicts. Consistent with our previous result, thesecond-stage cost charge makes the divisional man-agers internalize the informational rents that lowerthe firmwide net present value (NPV) when theshared investment project is continued in the sec-ond stage. Accordingly, and consistent with previousresearch (e.g., Antle and Eppen 1985, Baldenius et al.2007), the second-stage capital charge rate increasesin the severity of the agency conflicts.

In contrast, the agency conflicts do not alter thefirst-stage capital charge rate or the RGP sharingrule, which retains its simple structure. Even thoughthe agency conflicts induce capital rationing in thefirst stage, increasing the first-stage capital chargerate would indirectly double count the agency costs

3 See Pfeiffer and Schneider (2007) for a similar finding in the single-divisional case.

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Johnson, Pfeiffer, and Schneider: Multistage Capital Budgeting for Shared InvestmentsManagement Science, Articles in Advance, pp. 1–16, © 2012 INFORMS 3

that determine the second-stage capital charge rate.Instead, capital rationing in both stages is inducedby the increase in the second-stage capital chargerate. These findings provide an interesting empiricalhypothesis in that the first-stage capital charge rateand asset-sharing rule should not be altered by thepresence of agency conflicts.

The first part of our analysis complements previ-ous research on the design of robust goal-congruentperformance metrics. As previously outlined, ouranalysis is most closely related to Baldenius et al.(2007), who show how to induce robust goal con-gruence for a multidivision, single-stage investmentproblem with a single capital charge rate that isbelow the firm’s cost of capital. Our paper is alsorelated to Dutta and Reichelstein (2005), who showhow to construct robust goal-congruent income mea-sures for a single-division, two-stage investment set-ting with a single capital charge rate that equalsthe firm’s cost of capital. Our paper shows howthe presence of multiple divisions and investmentstages alters the asset-sharing rules and associatedcapital charge rates substantially relative to the solu-tions developed in these papers. Similarly, our workalso complements previous research that shows thatrobust goal-congruent investment can be inducedfor a single-division, single-stage investment decisionwith a single capital charge rate equal to the firm’scost of capital (e.g., Rogerson 1997, Reichelstein 1997,Wei 2004, Mohnen and Bareket 2007).

In introducing a multistage adverse selection prob-lem for the individual divisions, we find that oursecond-stage results are essentially the same as inthe multidivision, single-stage investment setting ofBaldenius et al. (2007), but our first-stage results dif-fer from theirs significantly. This part of our anal-ysis also complements the single-division, two-stageinvestment setting of Pfeiffer and Schneider (2007).They show that the firm has to pay informationalrents for the manager’s private precontract informa-tion, but not for his private postcontract information,and as in our model, informational rents are onlypaid when the project is not abandoned. In extend-ing their work to a multiagent setting, we addresshow the other divisions’ informational rents and thesynergy stemming from the cooperative investmentsaffect cost allocations across the individual divisionsand across time. In particular, we show how thesefactors lead to a simple first-stage asset-sharing ruleand an associated capital charge rate that is above thefirm’s cost of capital.

Our paper is also related to Baiman et al. (2012),who study two-stage resource allocation with anabandonment option under adverse selection but ina single-division setting under centralized decisionmaking. In their model a manager creates budgetary

slack in the first stage by overreporting his costs. Themanager can only consume the slack if the project isnot abandoned, but abandoning the project does notallow the firm to recover the slack. Consequently, it isoptimal to commit to overinvest in some states ex postin order to reward lower-cost reports and minimizeslack creation ex ante. In contrast, the informationalrents in our adverse selection model (analogous tothe slack in Baiman et al. 2012) can be avoided if theproject is abandoned, leading to underinvestment.

Levitt and Snyder (1997) and Arya and Glover(2003) examine the role of abandonment options inthe presence of a moral hazard problem between aprincipal and a single agent. In contrast to our paper,exercising the abandonment option in these paperscan be costly because doing so can reduce the avail-able information about the agent’s supplied effortand make it more costly to provide effort incentives.In both papers, the principal’s expected payoff andthe agent’s induced effort levels will be higher if theprincipal can commit not to abandon some projectsthat have negative expected payoffs. Arya and Glover(2003) study optimal information system design andshow further that coarse information can serve asa commitment device to abandon the project lessaggressively.

From a broader perspective, our paper is alsorelated to Arya and Glover (2001) and Antle et al.(2001, 2007), who show in a single-agent setting thatthe option to delay decision making when facedwith mutually exclusive single-parameter investmentprojects can mitigate adverse selection problems.In general, our findings also supplement recent stud-ies on the design of residual income-based incen-tive systems in single-stage, single-agent investmentsettings, which examine factors leading to opti-mal capital charge rates that vary from the firm’scost of capital. See, for example, Baldenius (2003),Christensen et al. (2002), Dutta (2003), and Dutta andFan (2009).4

Finally, our paper is related to the classical one-parametric public choice literature (e.g., Groves 1973,Green and Laffont 1979) and to the recent extensionsof this literature to infinite multistage pivot mech-anisms based on the marginal contribution of eachagent (Bergemann and Välimäki 2010). Our capitalbudgeting mechanism is not a multistage version ofthe pivot mechanism, however, because our mecha-nism only charges divisions for investments that have

4 Our paper further adds to the broader literature on single-investment decisions under adverse selection, including Antle andFellingham (1990), Arya et al. (1993), and Dutta and Reichelstein(2002). It also builds on studies that have examined cost-allocationand budgeting mechanisms for multidivisional firms in one-periodsettings such as Arya et al. (1996) and Balakrishnan (1995).

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Figure 1 Timeline of Events

Date –1

Headquarters choosescapital budgeting

mechanism

Date 0

� realized,I0 determined

Date 1

� realized,I1 determined

Dates 2,… , T

Cash flows (ci2,… , ciT)and profits (�i2,… , �iT)

realized

been undertaken, whereas the pivot mechanism cancharge divisions even if no investment is carried out.

The remainder of the paper is organized as follows.The formal model is introduced in §2. The multistagebudgeting process is analyzed in §3. Section 4 extendsthe results to a multistage adverse selection problemwith a manager who makes unobservable, productiveeffort decisions. Section 5 concludes. A list of modelvariables and all proofs are provided in Appendices Aand B, respectively.

2. ModelWe study a firm’s capital budgeting system wheninvestment decisions have to be made incrementallyin two stages, and divisional managers have privateinformation about investment profitability at eachstage. All parties are risk neutral. At the first stage(date t = 0), the firm (headquarters) can acquire anasset that will allow N divisions to initiate poten-tially profitable projects. Purchasing the asset requiresa cash outlay of �b0 at date 0, where � = 1/41 + r5and r is the firm’s cost of capital.5 The indicator vari-able I0 denotes the initial purchase decision, whereI0 = 1 if the asset is purchased and I0 = 0 otherwise.At the second investment stage, date 1, all divisionalprojects can continue if the firm invests further inthe shared asset at cost b1. The indicator variable I1equals 1 if the second-stage investment is undertakenand 0 otherwise.

If the shared investment receives additional fund-ing at the second stage, each division’s project gen-erates verifiable end-of-period cash flows for theremaining periods:

cit4�i1�i1 I11 I05 = xit4�i +�i5I1I0

for t = 21 0 0 0 1 T and i = 11 0 0 0 1N 0 (1)

The intertemporal growth profile of each division,represented by xi = 4xi21 0 0 0 1 xiT 5 > 0, is commonknowledge.6 The productivity parameters �i and �i

5 All cash flows in the paper are discounted to date 1. The initialasset cost is defined as �b0 at date 0 for ease of notation.6 The common-knowledge and separability assumptions are stan-dard (e.g., Reichelstein 1997, Rogerson 1997), despite being some-what demanding. See, for instance, Pfeiffer and Velthuis (2009) foradditional discussion of the implications of these assumptions.

are observed privately by manager i at the begin-nings of dates 0 and 1, respectively. Larger valuesof �i and �i represent a more favorable investmentenvironment in division i. Common beliefs regard-ing each manager’s private information, �i, are rep-resented by the probability distribution Fi4�i5 witha strictly positive density fi4�i5 on the interval6¯�i1 �i7⊂�+. Similarly, common beliefs regarding �i

are represented by the probability distribution Gi4�i5with a strictly positive density gi4�i5 on the inter-val 6

¯�i1 �i7 ⊂ �+. All random variables �i and �i

are independently distributed. For simplicity, we usethe following vector notation: � = 4�11 0 0 0 1 �N 5, �−i =

4�11 0 0 0 1 �i−11 �i+11 0 0 0 1 �N 5, �= 4�11 0 0 0 1�N 5, and �−i =

4�11 0 0 0 1�i−11�i+11 0 0 0 1�N 5.Figure 1 is a timeline showing the sequence of

events in the model.Throughout the analysis all cash inflows and

outflows are discounted to period t = 1. The presentvalue of division i’s cash flows is given by

PVi4�i1�i5·I1I0 =

T∑

t=2

cit�t−1

=

T∑

t=2

xit4�i+�i5�t−1

·I1I00 (2)

Unless stated otherwise, we impose the followingassumption. Assumption 1 rules out the trivial case inwhich the abandonment decision is deterministic andensures that ex post each division is necessary for theoverall investment project to be profitable.7

Assumption 1. For every division i = 11 0 0 0 1N , thefollowing inequalities hold for all �1�−i: PVi4�i1 �i5 +∑N

j 6=i PVj4�j1�j5 > b1 >PVi4�i1¯�i5+

∑Nj 6=i PVj4�j1�j5.

As a point of reference, we first derive the firm’soptimal multistage investment decisions under fullinformation (i.e., when headquarters observes � atdate 0 and � at date 1). The firm’s overall objec-tive is to choose first- and second-stage investmentrules, I04�5 and I14�1�5, that maximize the expectedfirmwide net present value (discounted to date t = 1):

maxI04�51 I14�1�5

E�1�

[(( N∑

i=1

PVi4�i1�i5− b1

)

· I14�1�5− b0

)

I04�5

]

0 (3)

7 This assumption allows us to avoid case distinctions. Our mainresults hold if only one division is necessary and the other divisionsare assigned zero capital costs.

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Solving (3) yields the following solution to the firm’sproblem:

I∗

1 4�1�5=

1 ifN∑

i=1

PVi4�i1�i5≥ b11

0 otherwise1(4)

and (note the value of the initial cash outlay at date 0,�b0, is b0 at date 1)

I∗

0 4�5

=

1 if E�

[( N∑

i=1

PVi4�i1�i5− b1

)

· I∗

1 4�1�5

]

≥ b01

0 otherwise.(5)

That is, the firm will invest further at date 1 when-ever the continuation net present value is positive andwill abandon the project otherwise. Taking the opti-mal abandonment decision into account, the firm ini-tiates the investment if the associated expected netpresent value of the project is positive.

2.1. Capital Budgeting MechanismNow we turn our attention to the design of the cap-ital budgeting mechanism when the divisional man-agers have private information about their projects’profitability levels and each manager seeks to max-imize his own divisional income measure. Initially,we focus on the design of robust performance mea-sures and ignore problems of adverse selection. Sub-sequently, we show in §4 how our results can beadapted to an optimal second-best capital budgetingmechanism.

To design the capital budgeting mechanism, thefirm has to allocate the costs of the shared assetsacross the individual divisions and across the peri-ods in time. We focus on asset cost-sharing rules thatsatisfy the usual comprehensive income measurementcondition so that the total cost shares allocated acrossthe divisions equal the total investment costs. In par-ticular, the firm chooses the following key items ofthe capital budgeting mechanism:

(i) asset cost-sharing rules, �0i and �1i, assign eachdivision a share of the associated joint asset’s invest-ment costs, �0ib0 and �1ib1. Comprehensive incomemeasurement requires that the sum of the sharesassigned to the individual divisions equals 1, i.e.,∑N

i=1 �ki = 1 for k = 011;(ii) firmwide capital charge rates r0 and r1;(iii) intertemporal cost-allocation rules, zit4r05 and

zit4r15, allocate division i’s share of the investmentcosts, �0ib0 and �1ib1, to period t with the asso-ciated capital charge rate. The intertemporal cost-allocation rule represents a convenient shortcut forthe depreciation schedule plus capital costs per allo-cated unit calculated with the capital charge rate

(e.g., Rogerson 1997, Reichelstein 1997). Formallystated, the intertemporal cost-allocation rule zit4r5consists of the depreciation schedule dit plus the costof capital calculated from the capital charge rate r andthe remaining proportion of division i’s asset bookvalue at the end of the preceding period Ait−1; i.e.,zit4r5= dit+ rAit−1. The book values (per unit) are pre-sumed to satisfy the clean surplus relationship; i.e.,Ai1 = 1, Ait =Ait−1 − dit for t > 1, and AiT = 0. Conse-quently, the intertemporal cost-allocation rule is com-plete with respect to the associated capital charge rate;i.e.,

∑Tt=2 zit4r541 + r 5−4t−15 = 1.

Given the capital budgeting process, division i’sperformance measure at date t is given as

�it = cit − 4zit4r15�1ib1Ii1 + zit4r05�0ib05Ii0

for t = 21 0 0 0 1 T 1 (6)

and �i0 =�i1 = 0. The performance measure �it equalsdivision i’s current cash flow, cit , less a divisional costcharge that consists of two pieces: one that is allo-cated to the division if the project is initiated at date0 and another if the project is continued at date 1.The indicator variables Ii0 and Ii1 denote division i’sinvestment decisions, but because we study capitalbudgeting mechanisms that will always provide dom-inant incentives for optimal unanimous decision mak-ing by all managers, we do not distinguish betweenIk and Iik.8 Each cost charge allocates a share of theinvestment costs to period t according to the asso-ciated intertemporal cost-allocation rule. The class ofperformance measures in (6) encompasses typicallyused accounting-based performance measures suchas income, residual income, and operating cash flow.Further, it allows us to distinguish between circum-stances under which the firm can apply a single cap-ital charge rate to the two investment expendituresand circumstances that require two capital chargerates and asset-sharing rules, supporting observationsin practice that some firms use different capital chargerates for different investment decisions.

Following the terminology in earlier literature,we confine attention to performance measures thatinduce sequentially robust goal congruence in dominantstrategies (hereafter, robust goal congruence). That is,each manager’s dominant strategy is to (i) sequen-tially report his private information truthfully and(ii) undertake, from the firm’s perspective, the opti-mal investment decisions on a period-by-period basis,

8 For the same reason, our results do not change if the divisionscommit ex ante to a report-contingent decision-making rule ratherthan allowing each division to make its own investment deci-sion. Similarly, it is not necessary to specify rules about the groupdecision-making process, because procedures such as the order inwhich they report or the group’s decision-making procedures ateach stage will not be relevant.

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regardless of the weights assigned to his performancemeasure. Only nonzero performance measures cansatisfy this criterion; i.e., 4�i21 0 0 0 1�iT 5 6= 0 for all i =

11 0 0 0 1N .One interpretation for this approach is that the firm

does not know the managers’ planning horizons ortime preferences and, thus, has to ensure nontriv-ially goal-congruent decisions on a period-by-periodbasis. Another interpretation is that the accountingrules have to be chosen before the specifics of themanagerial contracts become known (e.g., Dutta andReichelstein 2005). A capital budgeting mechanism iscalled robust if for all divisions it induces robust goal-congruent incentives, as defined above, and satisfiesthe comprehensive income measurement condition.

3. Robust Capital BudgetingMechanism in the Absence ofAdverse Selection

In this section, we characterize the optimal capi-tal budgeting mechanism that induces robust goal-congruent investment decisions absent an adverseselection problem. Each division reports its privateinformation �i at date 0 and �i at date 1, with (�i, �i)denoting division i’s reported profitability levels. Thefirm can base the components of the capital budgetingmechanism, such as the asset-sharing rules, the capi-tal charges rates, and the intertemporal cost-allocationrules, on the reports.

To characterize the capital budgeting mechanism,it is convenient to define each division’s critical prof-itability parameter, �∗

i 4�1 �−i5, as the solution to

PVi4�i1�∗

i 4�1 �−i55+N∑

j 6=i

PVj4�j1 �j5= b10 (7)

That is, �∗i 4�1 �−i5 is the minimum profitability level

division i must report if the shared project is tobe continued. Assumption 1 ensures that a criticalprofitability parameter exists for each division. Thefollowing result characterizes the capital budgetingmechanism.

Proposition 1. Given Assumption 1, the followingmultistage capital budgeting mechanism is the only capitalbudgeting mechanism that induces robust goal-congruentinvestment incentives for N divisions, N > 1, while satis-fying the comprehensive income requirement. In particular,the date 0 asset-sharing rule, which we denote the rela-tive growth profile asset-sharing rule (RGP rule), isgiven by

�0i =

∑Tt=2 xit41 + r05

−4t−15

∑Tt=2 xit41 + r5−4t−15

1 (8)

the date 1 asset-sharing rule is given by

�1i4�1 �5

=

∑Tt=2 xit41 + r14�1 �55

−4t−154�i +�∗i 4�1 �−i55

b11 (9)

the associated capital charge rate rk4 · 5 satisfies ( for k =

011)N∑

i=1

�ki4 · 5= 11 (10)

and the associated intertemporal cost-allocation rule,

zRBit 4rk4 · 55=xit

∑Tt=2 xit41 + rk4 · 55

−4t−151 (11)

is the relative benefit cost-allocation rule (RBCA rule)calculated with the associated capital charge rate rk4 · 5.

(All proofs are provided in Appendix B.)To show how the metric in Proposition 1 works,

we first restate the date 0 and date 1 divisional costcharges, respectively, as follows:

zRBit 4r05�0ib0 = zRBit 4r5b01 (12)

zRBit 4r15�1ib1 = zRBit 4r5

(

b1 −

N∑

j 6=i

PVj4�j1 �j5

)

0 (13)

That is, the asset-sharing rules, capital chargerates, and intertemporal cost-allocation rules aredesigned such that the date 0 initial investmentcost, b0, and the date 1 opportunity costs, b1 −∑N

j 6=i PVj4�j1 �j5, are allocated to division i at date taccording to the RBCA rule calculated with the firm’scost of capital, r .9 Assumption 1 ensures that the date1 opportunity costs, which consist of date 1 invest-ment costs less the reported net present value of theother divisions, are positive. We discuss the capitalcharge rates in more detail in §3.1.

The RBCA rule provides the following relationbetween division i’s current cash flow and thepresent value of the project: cit4�i1�i1 I11 I05 = zRBit 4r5 ·

PVi4�i1�i5I1I0 (Rogerson 1997, Reichelstein 1997).10

9 The associated relative benefit depreciation schedule (Rogerson1997) can be obtained as a one-to-one mapping from the RBCArule and the capital charge rate (Baldenius et al. 2007, Lemma 1).See Pfeiffer and Velthuis (2009) for a comprehensive summary ofprevious findings for properties of the relative benefit depreciationschedule.10 This follows from zit4r5I1I0 = xit4�i + �i5I1I0/4

∑Tt=2 xit41 + r5−4t−15 ·

4�i +�i55= cit/PVi0

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Applying this property, the optimal performance met-ric for division i can be restated as follows:11

�it = zRBit 4r5

((

PVi4�i1�i5

+

N∑

j 6=i

PVj4�j1 �j5− b1

)

I1 − b0

)

I00 (14)

From division i’s perspective, it receives its “true”divisional net present value plus the reported netpresent value of the other divisions less invest-ment costs, allocated over time with the RBCA rule.Because the metric in (14) is not a function of divisioni’s own reports 4�i1 �i5, each division i will report itsinformation truthfully, independent of the other divi-sions’ decisions. Given truthful reporting of the otherdivisions, each period t performance measure �it is amultiple of the firm’s objective function shown in (3).Accordingly, each manager undertakes the optimalinvestment decisions regardless of his time prefer-ences. Hence, the performance metric ensures robustgoal congruence.

Although the performance metric has a Grovesstructure, it is instructive to note that in a strictsense it is not a Groves mechanism because it is notbased on realized net present values. A Groves mech-anism based on realized net present value woulddelay performance measurement until all cash flowswere realized (back-loading), which violates the goal-congruence criterion requirement 4�i21 0 0 0 1�iT 5 6= 0. Inthis vein, our capital budgeting mechanism is nota multistage version of the so-called pivot mecha-nism under which a division receives an additionalpositive transfer payment for a pivotal report thatalters the investment decisions. The pivot mechanismcan charge the divisions even if no investment isultimately carried out, whereas our capital budget-ing mechanism only allocates costs that have beenincurred (see (12) and (13)).12

11 Note that although the metric in (14) is equivalent to the metricin Proposition 1 in its effect on each manager’s divisional incomeand incentives, it does not satisfy comprehensive income account-ing. In fact, robust goal congruence can be achieved with a varietyof equivalent metrics using different combinations of capital chargerates and asset-sharing rules, but the metric in Proposition 1 isunique in inducing robust goal congruence under comprehensiveincome accounting. Similarly, only (14) induces robust goal con-gruence using a capital charge rate that is equal to the firm’s costof capital, but it violates the comprehensive income accountingrequirement.12 See also Baldenius et al. (2007), who use this line of reason-ing to note that their single-stage pay-the-minimum-necessary(PMN) mechanism is not a multiperiod version of the pivotmechanism. They further note that a key property of pivotmechanisms is that they always achieve a “budget surplus,”whereas their PMN mechanism runs a deficit in terms of presentvalues. Similarly, in our budgeting mechanism the present

3.1. Properties of the Optimal RobustCapital Budgeting Mechanism

The capital budgeting mechanism outlined in Propo-sition 1 requires two different asset-sharing rules andcapital charge rates. If the abandonment decision isnot deterministic, i.e., if Assumption 1 is satisfied,dual capital charge rates will always be requiredin the multidivision case to make the divisions cor-rectly internalize the expected costs and benefits theyimpose on each other at each investment stage.13 Fur-ther, the optimal capital budgeting mechanism doesnot obey a strict “no-play, no-pay” condition that stip-ulates that costs only be allocated to a division if itsproject is not abandoned at the second stage. Instead,all divisions will be allocated a portion of any costsincurred at date 0 regardless of the continuation deci-sion at date 1.14

We consider first the date 0 divisional cost charge.The date 0 RGP sharing rule �0i, as defined in (8), doesnot depend on the divisions’ profitability parametersbecause all opportunity costs are pushed forward tothe date 1 cost-sharing rule to induce efficient decisionmaking in the continuation problem. Instead, the RGPrule allocates investment costs in proportion to thepresent value of each division’s intertemporal growthprofile (as indicated by its name). Ceteris paribus,divisions with more back-loaded cash flow profiles(i.e., cash flows arrive later) will receive smaller costallocations. To see this more clearly, consider the geo-metric cash flow case, xit = �t−1

i , where the geometricparameter �i satisfies �i ∈ 4011 + r5. Then the RGPsharing rule will be

�0i =1 + r −�i

1 + r0 −�i

0 (15)

If �i > �j , division i receives a relatively largerproportion of its cash flows in later periods (i.e., divi-sion i’s cash flows are more back-loaded than divi-sion j’s cash flows). As cash flows in any division

value of the divisional cost charges for the date 1 investmentdecision,

∑Ni=1

∑Tt=2 �

t−1zRBit 4r15�1ib1 =∑N

i=1

∑Tt=2 �

t−1zRBit 4r54b1 −∑N

j 6=i PVi4�i1�i55 = Nb1 − 4N − 15∑N

i=1 PVi4�i1�i5 < b1 (noting that∑T

t=2 �t−1zRBit 4r5 = 1), is less than the associated investment outlay

b1 whenever the NPV of the continuation project is positive (i.e.,∑N

i=1 PVi4�i1�i5 > b1).13 Similarly, dual cost-allocation systems observed in practice usedifferent rules to allocate the fixed and variable portions of sharedcosts (e.g., Horngren et al. 2009).14 Our results are consistent with full cost accounting, whichrequires the capitalization of all costs on the balance sheetindependent of the projects’ success. In contrast, successful effortsaccounting allows the firm to only capitalize costs associated withsuccessful investment projects. Our finding shows that Dutta andReichelstein’s (2005) insight advocating full cost accounting ratherthan successful efforts accounting for the single division case car-ries over to the multiple division case.

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become more back-loaded, comprehensive incomeaccounting can be satisfied with a capital charge ratethat deviates less from the firm’s cost of capital, lead-ing to a lower date 0 capital charge rate for every divi-sion. The optimal mechanism recognizes this positiveexternality by assigning division i a smaller share ofthe initial investment cost.

Because cost allocations depend only on cashflow patterns, two divisions with equally profitableprojects (i.e., PVi4�i1�i5 = PVj4�j1�j5) may be allo-cated different shares of the initial investment cost.Similarly, our capital budgeting mechanism showsthat equal cost sharing, which is frequently appliedin practice, can be optimal in a multiperiod settingeven when divisional projects vary significantly inprofitability level. In other words, “equal treatment ofunequals” can be optimal.15 Corollary 1 describes thegeneral circumstances that lead to equal cost sharingunder the RGP rule.

Corollary 1. The RGP sharing rule �0i correspondsto an equal sharing rule, �0i = 1/N , when (i) there is onlyone period of cash inflows, T = 21 or (ii) the divisionalgrowth parameters exhibit the same pattern across divi-sions, i.e., xit = kij · xjt for all i, j , and t, where kij denotesa positive constant.

In contrast to the static case, where synergiesamong divisions lead to a lower capital charge rate(Baldenius et al. 2007), a nondeterministic abandon-ment option plays an important role in determiningthe optimal capital budgeting mechanism and pushesthe first-stage capital charge rate above the costof capital. In particular, relation (12), zRBit 4r05�0ib0 =

zRBit 4r5b0, implies for r0 = r that each division bearsthe entire investment cost b0 (i.e., �0i = 1). Since �0i isdecreasing in r0, the associated date 0 capital chargerate must exceed the firm’s cost of capital to sat-isfy the comprehensive income measurement require-ment (i.e.,

∑Ni=1 �0i = 1).

Corollary 2. In the multiple division case, N > 11 thedate 0 capital charge rate exceeds the firm’s cost of capital,r0 > r .

The investment problem at date 1 can be treatedas a single-stage problem because the initial invest-ment cost b0 is sunk. Consequently, the date 1 capitalcharge rate and asset-sharing rule are straightforwardmodifications of the single-stage solution derived byBaldenius et al. (2007). In particular, the date 1 asset-sharing rule is the present value of the division’scash flows at its critical profitability level divided bythe date 1 investment cost, and the associated capi-

15 Consistent with Prendergast and Topel’s (1996) positive view offavoritism, the RGP rule can require unequally profitable divisionsto be treated equally to ensure robust goal congruence.

tal charge rate must be set below the firm’s cost ofcapital when the NPV of the continuation project ispositive.16 Because Proposition 1 ensures truth telling,we do not highlight the reports in the subsequentcorollary, which is a straightforward modification ofthe Baldenius et al. (2007, Corollaries 1 and 2) findingsfor the single-stage case.

Corollary 3 (Baldenius et al. 2007). (i) The date 1capital charge rate is below the firm’s cost of capital,r1 ≤ r , if the NPV of the continuation project is positive,∑N

i=1 PVi4�i1�i5≥ b1.(ii) The date 1 asset-sharing rule, �1i4�i1�i5, is increas-

ing in division i’s total profitability, 4�i +�i5.

We close this section by exploring the basic proper-ties of the two capital charge rates further. We returnto the geometric cash flow setting, letting xit = �t−1

i

with �i ∈ 4011 + r5, and setting �i = � for all i. Then,the capital charge rates r0 and r1 can be written as afunction of the geometric parameter, �, the number ofdivisions, N , and the total return rate of the continu-ation project, R14�1 �5= 4

∑Ni=1 PVi4�i1 �i5/b15− 1:

r0 = r + 4N − 1541 + r −�5 and

r1 = r − 4N − 1541 + r −�5 ·R14�1 �50(16)

First, the two capital charge rates will only be equalif N = 1. In this case, both charge rates will be equalto the firm’s cost of capital, r . We show how thisinsight extends to the general case in the next section.Second, for R1 > 01 the difference between the capi-tal charge rates, r0 − r1, is ceteris paribus decreasingin �. The intuition is as described previously—whencash flows are more back-loaded, small changes ineither capital charge rate will have a larger impacton the magnitude of the respective asset-sharing ruleand smaller capital-charge-rate adjustments will berequired to satisfy comprehensive income accounting.Finally, r0 − r1 is increasing in the number of divi-sions, N , and the continuation return rate, R1, becauseboth asset-sharing rules require larger adjustments tosatisfy comprehensive income accounting when N islarge, and the date 1 asset-sharing rule requires largeradjustments when R1 is large.

3.2. Single Capital Charge RateThe two capital charge rates required to induce robustgoal-congruent investment decisions at dates 0 and 1

16 If r1 = r , the sum of the shares equals∑N

i=1 �1i4�1 �5 = 4Nb1 −

4N −15∑N

i=1 PVi4�i1 �i55/b1, and the comprehensive income account-ing requirement is satisfied only if the (reported) net presentvalue of the continuation project is zero (i.e.,

∑Ni=1 �1i4�1 �5 = 1

only if∑N

i=1 PVi4�i1 �i5 = b1). Since the allocated shares �1i4�1 �5are decreasing in r1, the date 1 capital charge rate must be set belowthe firm’s cost of capital when the NPV of the continuation projectis positive.

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are quite different, even though the assets purchasedat each stage could be identical. In this section, weshow that two key ingredients are responsible for thisresult: (i) the presence of a nondeterministic aban-donment option, as required in Assumption 1, and(ii) multiple divisions, N > 1.

Suppose that the firm wants to design a capitalbudgeting mechanism that allows two asset-sharingrules, �0i and �1i, but only a single capital chargerate, r . As mentioned, the associated income mea-sure must be a multiple of the firmwide net presentvalue for all abandonment and investment decisions,I1 and I0. That is, the following equation must be sat-isfied for all t = 21 0 0 0 1 T and i = 11 0 0 0 1N :

kit ·

((

PVi4�i1�i5+N∑

j 6=i

PVj4�j1�j5− b1

)

I1 − b0

)

I0

= cit4�i1�i1 I11 I05− zit4r54�1i4�1�5b1I1 +�0i4�5b05I0

for all 4�i1�i1 I01 I151 (17)

where kit denotes a positive constant. Condition (17)is a necessary condition that aligns the firm’s anddivision i’s investment decisions (and abstracts fromproblems that the other divisions misreport theirinformation). Because only the zero function satis-fies (17) for all 4�i1�i1 I01 I15, the coefficients mustbe zero; i.e., kit = xit/

∑Tt=2 xit�

t−1 = zRBit 4r5, kit4b1 −∑N

j 6=i PVj5 = zit4r5�1ib1 and kitb0 = zit4r5�0ib0 for allt = 21 0 0 0 1 T and i = 11 0 0 0 1N .17 If N > 1, a singlecapital charge rate r cannot solve these equationssimultaneously while also satisfying the comprehen-sive income measurement condition,

∑Ni=1 �ki = 1 for

k = 011. Rather, condition (17) can only be satisfiedfor the single-division case. In this case, only the rel-ative benefit cost-allocation rule based on the firm’scost of capital, i.e., zt4r5 = zRBt 4r5, ensures robust goalcongruence.18

Corollary 4. Given Assumption 1, a single capitalcharge rate is only optimal in the single-divisional case;i.e., r0 = r1 = r if and only if N = 1.

In previous work, Dutta and Reichelstein (2005,Proposition 4) showed that a residual income mea-sure based on a single capital charge rate can obtain

17 We can rearrange (17) as 4xit − kt ·∑T

t=2 xit�t−15 · 4�i + �i5I1I0 +

4zit4r5�1i4�1�5b1 − kt4b1 −∑N

j 6=i PVj 4�j1�j 555I1I0 + 4zit4r5�0i4�5b0 −

ktb05I0 = 0 for all 4�i1�i1 I01 I15. We can interpret this as polynomialP4Z5 in variables Z = 4Z11Z21Z35 = 44�i + �i51 I11 I05, i.e., 4xit − kt ·∑T

t=2 xit�t−15 · Z1Z2Z3 + 4zit4r5�1i4�1�5b1 − kt4b1 −

∑Nj 6=i PVj 4�j1�j 555 ·

Z2Z3 + 4zit4r5�0i4�5b0 −ktb05 ·Z3 = 0 for all 4Z11Z21Z35. Then P4Z5= 0can only be the case if P4Z5 is the zero function. That is, the coeffi-cients must be zero.18 Technically, zt4r5= zRBt 4r5 is the only solution that solves the con-ditions for the single-division case, N = 1 (suppressing the indexi = 1): kt = zRBt 4r5, ktb1 = zt4r5b1, and ktb0 = zt4r5b0 for t = 21 0 0 0 1 T 0

robust goal-congruent investment decisions in thesingle-division case with multiperiod investments.Corollary 4 shows that this result does not transfer tothe multidivisional case.

Similarly, if we relax Assumption 1 and assumethat no abandonment option exists, i.e., I1 = 1 for all4�1�5, the overall capital budgeting problem is similarin structure to the continuation problem. Then robustgoal congruence is achieved if investment costs forboth stages are pooled and allocated with a singleasset-sharing rule and capital charge rate that are sim-ilar to those in the solution to the date 1 continuationproblem.

4. Incentive Contracting UnderAdverse Selection

Previous theoretical and empirical research hasemphasized the importance of the role of agencycosts in influencing the capital budgeting processand determining optimal capital charge rates. Usinga simple separable model that allows us to focuson implementation issues, we discuss in this sec-tion how agency conflicts between the firm and theindividual divisions alter our results. Specifically, weassume manager i provides effort ait from the inter-val 601 ait7 in period t. Cash flows in period t areredefined as

cit4ait1 �i1�i1 I11 I05= ait + xit4�i +�i5I1I00 (18)

The firm can observe cash flows in each period butcannot distinguish between cash flows arising frommanagerial effort and those arising from investmentreturns. We make the standard assumption that theinverse hazard rate Hi4�i5= 41−Fi4�i55/fi4�i5 is weaklydecreasing in �i for all i. Manager i is effort averse,with his utility function Ui4 · 5 given by

Ui4sit4 · 51 ait5=

T∑

t=2

�t−14sit4 · 5− vitait51 (19)

where sit4 · 5 is the compensation payment and vit

measures the disutility of effort.19

In the following, we consider a direct sequential rev-elation mechanism that specifies an investment ruleI04�5 ∈ 80119, an abandonment rule I14�1 �5 ∈ 80119, tar-get cash flows ci4�1 �5 = 4ci24�1 �51 0 0 0 1 ciT 4�1 �55 to bedelivered by division i, and associated compensationpayments si4�1 �5= 4si24�1 �51 0 0 0 1 siT 4�1 �55 contingenton the reports 4�1 �5. For any such revelation mecha-nism, Ui4�1 � � �i1�i5 denotes manager i’s utility con-tingent on the submitted reports 4�1 �5 and the true

19 The linear cost function is primarily chosen for didactic reasons.Our insights continue to hold if the managers’ cost functions areincreasing and convex in effort.

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profitability parameters 4�i1�i5. Then manager i’s util-ity is given by

Ui4�1 � � �i1�i5

=

T∑

t=2

�t−14sit4�1 �5− vitait4�1 � � �i1�i551 (20)

where ait4�1 � � �i1�i5 = arg min8ait � ait + xit4�i + �i5 ·

I14�1 �5I04�5 ≥ cit4�1 �59 is the minimum effortmanager i exerts in order to achieve the target cashflow cit4�1 �5. This equation reveals that each managerhas an incentive to understate the project’s profitabili-ties �i and �i so that he can deliver the required targetcash flows with less effort.

According to the revelation principle, we canrestrict our attention to mechanisms that induce themanagers to reveal their private information truth-fully (e.g., Myerson 1979, 1981). We obtain the follow-ing sequential optimization problem:

max4ci4�1�51 si4�1�51I04�51 I14�1�55

Ni=1

E�1�

[ N∑

i=1

T∑

t=2

�t−14cit4�1�5− sit4�1�55

− 4b0 + b1I14�1�55I04�5

]

subject to

4IC152 Ui4�1�i1 �−i � �i1�i5≥Ui4�1 �i1 �−i � �i1�i5

∀ �1 �1 �i1�i1 i1

4IC052 E�6Ui4�i1 �−i1� � �i1�i57

≥ E�6Ui4�i1 �−i1� � �i1�i57 ∀ �1 �i1 i1

4PC52 E�−i1�6Ui4�1� � �i1�i57≥ 0 ∀�i1 i0

The participation constraints 4PC5 require that eachmanager break even in expectation over the othermanagers’ types. The incentive compatibility con-straints 4IC15 and 4IC05 ensure sequential truth-ful reporting in dominant strategies at dates 1and 0, respectively. In our setting, imposing adominant-strategy equilibrium rather than a sequen-tial Bayesian-Nash equilibrium does not change theequilibrium payoffs of any party.20 Rather, we getimplementation in sequential dominant strategieswithout cost. We denote the second-best solution by4c†i 4�1�51 s

†i 4�1�51 I

†0 4�51 I

†1 4�1�55

Ni=1.

Using the standard approach to solving adverseselection problems, we replace the two incentive con-straints by their local counterparts and determine

20 Similar to Mookherjee and Reichelstein (1992), we calculate the(expected) information premium for each manager under bothequilibrium concepts and find they are the same. However, unlikeBaldenius et al. (2007), we cannot infer this result from Mookherjeeand Reichelstein (1992) directly in our sequential-investment-stagesetting.

each manager’s expected information premium. Sincemanager i’s utility is linear in ait , it is optimal toinduce the boundary values. To concentrate on theimplementation issue, we assume throughout ouranalysis that it is always optimal to induce the max-imum effort level ait for all i, t.21 The sequentialinvestment problem can then be written as follows(ignoring the constant

∑Ni=1∑T

t=2 �t−141 − vit5ait):

maxI04�51 I14�1�5

E�1�

[ N∑

i=1

44PVi4�i1�i5−�iHi4�i5− b15

· I14�1�5− b05I04�5

]

1 (21)

where �iHi4�i5 =∑T

t=2 �t−1vitxitHi4�i5 is the present

value of division i’s virtual costs at date 1. Equa-tion (21) is equivalent to (3) in the goal-congruencesection except that the present value of the virtualcosts associated with each division, �iHi4�i5, has to beconsidered when the project is undertaken ex post,i.e., when I1 = I0 = 1. The optimal contract requiresthe firm to pay an information premium to inducetruth telling at date 0, but it will only be paid expost in states in which the project is continued atdate 1. If the project is abandoned at date 1, the firmcan perfectly infer the managers’ effort levels. Further,the firm pays an information premium only for pre-contract private information because any rents asso-ciated with postcontract private information can beextracted by the firm up front.

The optimal abandonment and investment deci-sions are given below. At date 1, the firm investsfurther if the firmwide present value of the cashinflows from investment less the associated virtualcosts exceeds the additional investment costs:

I†1 4�1�5=

1 ifN∑

i=1

4PVi4�i1�i5−�iHi4�i55≥ b11

0 otherwise0(22)

Considering the optimal abandonment decision, thefirm initiates the common project if the expectedfirmwide net present value less the associated virtualcosts exceeds the initial investment cost:

I†0 4�5=

1 if E�

[( N∑

i=1

4PVi4�i1�i5−�iHi4�i55− b1

)

· I†1 4�1�5

]

≥ b01

0 otherwise0

(23)

21 This assumption is consistent with previous literature (e.g.,Baldenius et al. 2007, Baldenius 2003, Dutta 2003). In our set-ting, it is optimal for the firm to induce the maximum effortlevel for each division in each period, i.e., ait4�1�5 = ait for all4�1�1 i1 t5, if the maximum effort level is sufficiently large, i.e., ait ≥vitxitHi4

¯�i5/41 − vit5 for all i1 t.

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Because the virtual costs that arise from the agencyproblem increase the cost of investment, the optimalsolution calls for investment in a smaller set of statesat both stages relative to the goal-congruence case.

Observation 1. In the presence of the agency con-flict, rationing occurs at both stages so that the firminitiates the project less often and abandons it moreoften; i.e., I†0 4�5 ≤ I∗

0 4�5 and I†1 4�1�5 ≤ I∗1 4�1�5 and

there exist some 4�1�5 such that I†0 4�5 < I∗0 4�5 and

I†1 4�1�5 < I∗1 4�1�5.

In the following analysis we demonstrate how therobust goal-congruent capital budgeting mechanismidentified in the previous section provides dominantincentives for each manager to report his privateinformation truthfully and make efficient investmentdecisions regardless of his incentive weights. A capi-tal budgeting mechanism is said to be optimal if thereexist linear compensation schemes based on the classof performance measures identified in (6),

sit = �it4�1 �5+�it4�1 �5�it4 · 5

for all t = 21 0 0 0 1 T and i = 11 0 0 0 1N1 (24)

that achieve the same payoff for the firm asthe second-best mechanism identified above (e.g.,Melumad and Reichelstein 1989). Here, �it denotesthe fixed payment and �it is the bonus coefficient.

We adjust our previous assumption to incorporatethe virtual costs. Assumption 2 ensures that the aban-donment decision is nondeterministic at date 0 andthat ex post each division is necessary for the overallinvestment project to be profitable.

Assumption 2. For every division i = 11 0 0 0 1N ,the following inequalities hold for all �1�−i: PVi4�i1 �i5− �iHi4�i5 +

∑Nj 6=i4PVj4�j1�j5 − �jHj4�j55 > b1 >

PVi4�i1¯�i5−�iHi4�i5+

∑Nj 6=i4PVj4�j1�j5−�jHj4�j55,

The firm can induce the optimal effort level bysetting the bonus coefficient equal to the manager’sdisutility parameter; i.e., �it = vit . To characterize thecapital budgeting mechanism, we define each divi-sion’s critical profitability parameter under adverseselection, �†

i 4�1 �−i5, as the solution to the follow-ing (noting that Assumption 2 implies that �†

i 4�1 �−i5exists):

4PVi4�i1�†i 4�1 �−i55−�iHi4�i55

+

N∑

j 6=i

4PVj4�j1 �j5−�jHj4�j55= b10 (25)

Ceteris paribus, the critical profitability parameter ofdivision i, �†

i 4�1 �−i5, is increasing in the present valueof division i’s own virtual costs, �iHi4�i5, as well as inthe other divisions’ virtual costs,

∑Nj 6=i �jHj4�j5. Simi-

lar to Proposition 1, we obtain the following result.

Proposition 2. Given Assumption 2, the followingsequential capital budgeting mechanism is optimal. Thedate 0 asset-sharing rule is the relative growth profileasset-sharing rule (RGP rule):

�0i =

∑Tt=2 xit41 + r05

−4t−15

∑Tt=2 xit41 + r5−4t−15

1 (26)

the date 1 asset-sharing rule is given by

�1i4�1 �5

=

∑Tt=2 xit41 + r14�1 �55

−4t−154�i +�†i 4�1 �−i55

b11 (27)

the associated capital charge rate rk4 · 5 satisfies ( for k =

011)N∑

i=1

�ti4 · 5= 11 (28)

and the intertemporal cost-allocation rule,

zRBit 4rk4 · 55=xit

∑Tt=2 xit41 + rk4 · 55

−4t−151 (29)

is the relative benefit cost-allocation rule (RBCA rule)calculated with the associated capital charge rate rk4 · 5.

As before, the optimal period t performance met-ric for division i can be restated as follows (recallzRBit 4r5PVi4�i1�i5= xit4�i +�i5):

�it = zRBit 4r5

((

PVi4�i1�i5−�iHi4�i5+N∑

j 6=i

4PVj4�j1 �j5

−�jHj4�j55− b1

)

I1 − b0

)

I0 + ait1 (30)

using

zRBit 4r14 · 55�1i4 · 5b1

= zRBit 4r5

(

�iHi4�i5−N∑

j 6=i

4PVj4�j1 �j5−�jHj4�j55+ b1

)

and zRBit 4r05�0ib0 = zRBit 4r5b0. Given truthful reportingby the divisions, the performance measure �it is amultiple of the firm’s objective function (21), provid-ing dominant incentives for each manager to makeefficient investment decisions regardless of his incen-tives weights.

Corollary 5. The date 0 capital charge rate, r0, andthe asset-sharing rule, �0, are not affected by the agencyconflict; i.e., they are identical to the date 0 capitalcharge rate and asset-sharing rule as characterized inProposition 1.

Even though the agency conflict does induce cap-ital rationing at date 0 (see Observation 1), the date0 capital charge rate and the asset-sharing rule are

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not functions of the agency costs. All agency costsmust be incorporated into the date 1 cost charge toensure efficient decision making in the continuationproblem; naively including them also at date 0 wouldconstitute double counting. The date 1 asset-sharingrule and capital charge rate, which incorporate theagency costs through the critical profitability param-eter �†

i 4�1 �−i5, are similar to those that arise in thesingle-stage investment problem with adverse selec-tion described in Baldenius et al. (2007). Consistentwith previous results (e.g., Antle and Eppen 1985,Baldenius et al. 2007), more capital rationing as aresult of the agency conflict leads to a higher date 1capital charge rate r14 · 5, but as highlighted by Corol-lary 5, the agency conflict does not translate to ahigher date 0 capital charge rate r04 · 5.

5. ConclusionThis paper has studied the optimal design of a capi-tal budgeting mechanism for multistage investmentsin shared (cooperative) assets. Our model shows thatthe interaction between multiple divisions and mul-tiple investment stages requires two quite differentinterdivisional asset cost-sharing rules and associatedcapital charge rates to arrive at fixed cost allocationsthat provide appropriate investment incentives. Thefirst-stage capital charge rate must be set above thefirm’s cost of capital, whereas the second-stage capitalcharge rate must be set below the firm’s cost of capi-tal. This finding supports anecdotal evidence regard-ing within-firm variation in capital charge rates fordifferent types of projects and at different stages ofinvestment (e.g., Mukherjee and Hingorani 1999).22

Empirical studies suggest that firms, on average,apply capital charge rates above the firm’s cost ofcapital. Typically, theorists explain this empirical find-ing with (i) divisional agency conflicts or (ii) competi-tion between divisions.23 Our result provides anotherrationale. In the absence of an agency conflict, inter-divisional synergy in combination with a nonde-terministic abandonment option leads to a capitalcharge rate above the cost of capital. This is some-what surprising because one might conjecture thatboth factors, synergy and an abandonment option,should lower capital charge rates. As discussed previ-ously, our result arises because the positive externali-ties generated by each division’s investment decisionmust be incorporated into the first- and second-stage

22 The analytical finance literature has also identified the use of mul-tiple hurdle rates across divisions within the same firm (see thediscussion in Bernardo et al. 2006).23 Consistent with agency theory, Baldenius et al. (2007) show that asufficiently severe agency problem in a single-investment-stage set-ting can push the capital charge rate for shared investments abovethe firm’s cost of capital.

asset-sharing rules and capital charge rates differently,even if the two assets are identical. In the presenceof agency conflicts, we find that the second-stagecapital charge rate increases in the severity of thedivisional agency conflicts, consistent with standardresults (e.g., Antle and Eppen 1985). In contrast, thefirst-stage capital charge rate is not altered whenagency conflicts are introduced, demonstrating thatagency costs can lead to capital rationing withoutleading to an increased capital charge rate.

Finally, our mechanism dictates that the individualdivisions will always be held responsible for first-stage investment costs, regardless of the abandon-ment decision at the second stage. The relative growthprofile asset-sharing rule (RGP sharing rule), whichmust be applied in the first stage, allocates the invest-ment cost according to each division’s growth profile,regardless of the relative benefits that each divisionderives from the shared asset. A smaller share of theinvestment cost is allocated to divisions with rela-tively more back-loaded cash flow profiles (i.e., cashflows arrive later), and the RGP rule correspondsto equal cost sharing when all cash inflows arrivein a single period. Accordingly, equal cost sharingcan be optimal even when the shared asset providesquite different benefits to the individual divisions,highlighting that equal cost sharing can be optimalin a wider variety of circumstances than previouslyassumed.

AcknowledgmentsThe authors gratefully acknowledge comments and sugges-tions from Anil Arya, Tim Baldenius, Peter Christensen,Gregor Dorfleitner, John Fellingham, Wolfgang Kuersten,Alexis Kunz, Alexander Muermann, Stefan Reichelstein,Richard Saouma, Doug Schroeder, Andy Stark, Rick Young,department editor Mary Barth, an associate editor, twoanonymous referees, and seminar participants at the OhioState University, HEC Lausanne, Jena-Regensburg-WienWorkshop, Swiss Accounting Research Workshop in Bern,and the European Institute for Advanced Studies in Man-agement Workshop on Accounting and Economics in Milan.The authors give special thanks to Sunil Dutta for extremelyhelpful guidance.

Appendix A

Model VariablesAit Book value per unit for division i at the end of

period tb0 Cash outlay required for initial asset purchase at

date 0b1 Cash outlay required for second asset purchase at

date 1cit Division i’s cash flow at date tdit Depreciation rate for division i at date t

Fi4�i5 Probability distribution of �ifi4�i5 Density function for �i

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Gi4�i5 Probability distribution of �i

gi4�i5 Density function for �i

I0 Indicator variable denoting date 0 investment deci-sion

I∗0 Indicator variable denoting optimal date 0 invest-

ment decisionI1 Indicator variable denoting date 1 investment

decisionI∗1 Indicator variable denoting optimal date 1 invest-

ment decisioni Index for divisions

N Number of divisionsPVi Present value of divisions i’s cash flows, discounted

to period t = 1R1 Return rate of continuation project at date 1r Firm’s cost of capitalrk Firmwide capital charge rate applied to date k

investment costT Number of time periodst Index for time periods

xit Commonly observed intertemporal growth profileof division i at date t

zit4r5 Intertemporal cost-allocation rule given a capitalcharge rate, r

zRBit 4r5 Relative benefit cost-allocation rule given a capitalcharge rate, r

�i Division i’s privately observed productivity param-eter (realized at date 1)

�i Division i’s report about its private information �i

�∗i Division i’s critical profitability parameter (absent

adverse selection)�i Intertemporal growth parameter for division i

under geometric cash flows� Common intertemporal growth parameter for all

divisions under geometric cash flows� Firm’s discount factor equal to 1/41 + r5�i Division i’s privately observed productivity param-

eter (realized at date 0)�i Division i’s report about its private information �i�ki Share of date k investment cost allocated to division

i (asset-sharing rule)�it Division i’s performance measure at date t

Additional Variables Introduced in §4(Adverse Selection Setting)

ait Hidden managerial effort in division i at date tci Target cash flows for manager i at date tc†i Target cash flows for manager i at date t under

second-best solutionHi4�i5 Inverse hazard rate for distribution of �i

I†0 Indicator variable denoting optimal date 0 invest-ment decision

I†1 Indicator variable denoting optimal date 1 invest-ment decision

sit Manager i’s compensation payment at date ts†it Manager i’s compensation payment at date t

under second-best solutionUi Manager i’s utility functionvit Manager i’s disutility of effort at date t�†i Division i’s critical profitability parameter under

the second-best solution

�it Bonus coefficient in manager i’s compensationpayment at date t

�iHi4�i5 Present value of division i’s virtual costs, dis-counted to t = 1

�it Fixed portion of manager i’s compensation pay-ment at date t

Appendix B. Proofs

Proof of Proposition 1. As shown in the text, the cap-ital budgeting mechanism is robust. Now, we prove neces-sity. We consider all performance measures of the form

�it = cit4�i1�i1 I11 I05

− 4zit4r15�1i4�1 �5b1I1 + zit4r05�0i4�5b05I00 (B1)

We fix i and t. Using the relation (recall that zRBit 4r5 =

xit/∑T

t=2 xit�t−1)

PVi4�i1�i5I1I0 =

T∑

t=2

xit4�i +�i5�t−1I1I0

= cit4�i1�i1 I11 I05/zRBit 4r51 (B2)

we can restate the residual income measure equivalently as

�it = zRBit 4r5 ·(

PVi4�i1�i5I1I0 − zRBit 4r5−1

· 4zit4r15�1i4�1 �5b1I1 + zit4r05�0i4�5b05I0

)

0 (B3)

As outlined, the firmwide objective function is given by

((

PVi4�i1�i5+N∑

j 6=i

PVj4�j1�j5− b1

)

I1 − b0

)

I00 (B4)

To induce the optimal investment decision I1, (B3) and(B4) must have the same roots for all 4I11 �i1�i5. (See Foot-note 17.) Noting that any optimal metrics must induce truthtelling, this condition can only be satisfied if

zit4r15�1i4�1�5b1I1 = zRBit 4r5

(

b1 −

N∑

j 6=i

PVj4�j1�j5

)

I10 (B5)

Similarly, to get the optimal investment decision I0,(B3) and (B4) must have the same roots for all 4I0, �i5. Asbefore, this condition can only be satisfied if

4zit4r15�1i4�1�5b1I1 + zit4r05�0i4�5b05I0

= zRBit 4r5

((

b1 −

N∑

j 6=i

PVj4�j1�j5

)

I1 + b0

)

I01 (B6)

leading to (12): zit4r05�0i4�5b0 = zRBit 4r5b00Now we turn our attention to (B5) and (B6), which yields

for t, t = 21 0 0 0 1 T :

zit4r05

zit4r05=

zRBit4r5

zRBit4r5

=xitxit

andzit4r15

zit4r15=

zRBit4r5

zRBit4r5

=xitxit

0 (B7)

Hence, we get zit4rk5 = 4xit/xit5zit4rk5 for k = 011. Accord-ing to the clean surplus relationship, the intertemporalcost-allocation rules satisfy the conservation property with

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respect to the associated capital charge rate,∑T

t=2 zit4rk5 ·

41 + rk5−4t−15 = 1. Hence, we get

T∑

t=2

xitzit4rk5

xit41 + rk5

−4t−15= 1 → zit4rk5

=xit

∑Tt=2 xit41 + rk5

−4t−15= zRB

it4rk51 (B8)

and, thus, zit4rk5 = xit/∑T

t=2 xit41 + rk5−4t−15 = zRBit 4rk5.

Using the comprehensive income requirement,∑N

i=1 �0i =∑N

i=1 �1i = 1, shows the result. �

Proof of Corollary 1. (i) For T = 2, the asset-sharing rule reduces to �0i = 41 + r5/41 + r05 = 1/N forall i. The latter condition follows from the requirement∑N

i=1 �0i = 1. (ii) Using∑N

i=1 �0i = 11 we get �0j =∑T

t=2 kij ·

xit41 + r05−4t−15/

∑Tt=2 kijxit41 + r5−4t−15 = �0i = 1/N 0 �

Proof of Corollary 2. The asset-sharing rule �0i isstrictly decreasing in r0. Noting that �0i = 1 for r0 = r estab-lishes the result. �

Proof of Corollary 3. See proofs of Corollaries 1 and 2in Baldenius et al. (2007). �

Proof of Corollary 4. As outlined in §3.2. It remainsto show the N = 1 case. For N = 1 it is sufficient to set�0i4 · 5= 1 and �1i4 · 5= 1 and the capital charge rate equal tor to satisfy (17). �

Derivation of the Unconstrained Optimization Problem (21).Throughout this proof, we assume the firm induces themaximum effort level; i.e., ait4�1�5 = ait for all 4�1�1 i1 t5.We first prove the following lemma, which shows how toreplace the two incentive constraints, 4IC05 and 4IC151 bytheir local counterparts.

Lemma 1. We obtain the following results:(i) The incentive constraints 4IC15 can be replaced by the fol-

lowing condition:

Ui4�1�i1 �−i � �i1�i5 =

T∑

t=2

�t−1vitxit

∫ �i

¯�i

I14�1 qi1 �−i5I04�5 dqi

+Ui4�1¯�i1 �−i � �i1

¯�i5 (B9)

∀ 4�i1 �1�i1 �−i5, if the function qi → I14�1 qi1 �−i5I04�5 is non-decreasing for all 4�1 �−i5.

(ii) The incentive constraints 4IC05 can be replaced by the fol-lowing condition:

E�6Ui4�i1 �−i1� � �i1�i57

=

T∑

t=2

�t−1vitxitE�

[

∫ �i

¯�i

I14qi1 �−i1�5I04qi1 �−i5 dqi

]

+E�6Ui4

¯�i1 �−i1� �

¯�i1�i57 (B10)

∀ 4�i1 �−i5, if the function qi → E�6I14qi1 �−i1�5I04qi1 �−i57 isnondecreasing for all �−i.

Proof of Lemma 1. As is standard, we replace thetwo incentive constraints 4IC15 and 4IC05 by their localcounterparts and determine each manager’s expected infor-mation premium.

(i) Invoking the envelope theorem yields

d

d�i

Ui4�1 � � �i1 �i5

∣

∣

∣

∣

�i=�i

=¡

¡�i

Ui4�1 � � �i1�i5

∣

∣

∣

∣

�i=�i

=

T∑

t=2

�t−1vitxitI14�1�i1 �−i5I04�50 (B11)

Integrating the above equation leads to

Ui4�1�i1 �−i � �i1�i5 =

T∑

t=2

�t−1vitxit

∫ �i

¯�i

I14�1 qi1 �−i5I04�5 dqi

+Ui4�1¯�i1 �−i � �i1

¯�i5 (B12)

∀ 4�1�i1 �−i5. It is standard to show that the above equa-tion implies the incentive compatibility constraints 4IC15,if the function qi → I14�1 qi1 �−i5I04�5 is nondecreasing for all4�1 �−i50

(ii) Invoking the envelope theorem yields:

d

d�iE�6Ui4�1� � �i1�i57

∣

∣

∣

∣

�i=�i

=¡

¡�iE�6Ui4�1� � �i1�i57

∣

∣

∣

∣

�i=�i

=

T∑

t=2

�t−1vitxitE�6I14�i1 �−i1�5I04�570 (B13)

Integration leads to

E�6Ui4�i1 �−i1� � �i1�i57

=

T∑

t=2

�t−1vitxitE�

[

∫ �i

¯�i

I14qi1 �−i1�5I04qi1 �−i5 dqi

]

+E�6Ui4¯�i1 �−i1� �

¯�i1�i57 (B14)

for all 4�i1 �−i5.Finally, assuming without loss of generality (w.l.o.g.) �i <

�i and E�6Ui4�1� � ·57 > E�6Ui4�i1 �−i1� � ·57 impliesT∑

t=2

�t−1vitxit

∫ �i

�i

6E�6I14�i1 �−i1�5I04�i1 �−i57

−E�6I14qi1 �−i1�5I04qi1 �−i577 dqi > 00 (B15)

This is a contradiction, if qi −→ E�6I14qi1 �−i1�5I04qi1 �−i57 isnondecreasing. Thus, (ii) and Lemma 1 is proven.

Next, we show that the sequential adverse selection prob-lem can be restated equivalently as the unconstrained opti-mization problem (21). Setting E

�6Ui4

¯�i1 �−i1� �

¯�i1�i57= 0

minimizes manager i’s expected information rents whilesatisfying the participation constraints, 4PC5. UsingLemma 1(ii) and Ui4sit4 · 51 ait5 =

∑Tt=2 �

t−14sit4 · 5 − vit ait5yields

E�

[ T∑

t=2

�t−1sit4 · 5

]

=

T∑

t=2

�t−1vitxitE�

[

∫ �i

¯�i

I14qi1 �−i1�5 I04qi1 �−i5 dqi

]

+

T∑

t=2

�t−1vit ait 0 (B16)

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Changing the order of integration and using integration byparts yields

E�−i

[

∫ �i

¯�i

E�

[

∫ �i

¯�i

I14qi1 �−i1�5I04qi1 �−i5 dqi

]

fi4�i5 d�i

]

= E�1�6Hi4�i5I14�1�5I04�570 (B17)

Using Equations (B16) and (B17) in the principal’sobjective function yields the unconstrained optimizationproblem (21).

Finally, for sake of completeness, setting Ui4�1¯�i1 �−i �

�i1¯�i5 = E

�6Ui4�i1 �−i1� � �i1�i57 − E�i

6∑T

t=2 �t−1vitxit ·

∫ �i

¯�i

I14�1 qi1 �−i5I04�5 dqi7 ensures that Lemmas 1 (i) and(ii) are satisfied (and thus the incentive constraints 4IC05and 4IC15). �

Proof of Observation 1. A comparison of (22), (23),(4), and (5) yields Observation 1. �

Proof of Proposition 2. As outlined in the text, the per-formance measure is given by

�it = zRBit 4r5

((


j 6=i

4PVj4�j1 �j5

−�jHj4�j55− b1

)

I1 − b0

)

I0 + ait 0 (B18)

Noting that manager i’s decisions, I0 and I1, depend on histrue productivity and the reports, manager i will chooseI14�i1 �1�i1 �5= 1 if and only if


j 6=i

4PVj4�j1 �j5−�jHj4�j55≥ b1 (B19)

and I04�i1 �5 = 1 if and only if (anticipating truthful report-ing at t = 1)

E�

[(


j 6=i

4PVj4�j1�j5−�jHj4�j55− b1

)

· I14�i1 �1�i1�5

]

≥ b00 (B20)

The first equation shows that I14�i1 �1�i1 �5 is nondecreas-ing in �i and �. The second equation and the mono-tonicity of I14�i1 �1�i1�5 reveal that I04�i1 �5 is nonde-creasing in �i and �. Hence, �i → I14qi1 �i1 �−i1�i1�5 ·

I04qi1 �i1 �−i5 is nondecreasing.It remains to show that manager i reports �i truthfully

in dominant strategies. Calculating E6�it4 · 57, we find thatmanager i’s expected utility is an affine function of (omit-ting the fixed payment):

(

E�

[(


j 6=i

VPVj4�j1�j5− b1

)

· I14�i1 �1�i1�5

]

− b0

)

I04�i1 �51 (B21)

where VPVj4�j1�j5 = PVj4�j1�j5 − �jHj4�j5. The incentiveconstraints imply (invoking the envelope theorem and

Leibniz’s rule, noting that 4¡/¡�i5I04�i1 �5��i=�i= 0 almost

everywhere (a.e.) and 4¡/¡�i5I04�i1 �5= 0 a.e. ∀ � and settingE�6Ui4

¯�i1 �−i1� �

¯�i1�i57= 0):

E�6Ui4�1� � �i1�i57 =

T∑

t=2

�t−1vitxitE�

[

∫ �i

¯�i

I14qi1 qi1 �−i1�i1�5

· I04qi1 qi1 �−i5 dqi

]

∀�0 (B22)

Next, we show that the contract induces truthful reporting.This follows by contradiction. Assuming w.l.o.g. �i < �i andE�6Ui4�i1 �−i1� � �i1�i57 > E�6Ui4�1� � �i1�i57, we get

∫ �i

�i

E�6I14qi1 �i1 �−i1�i1�5I04qi1 �i1 �−i5

− I14qi1 qi1 �−i1�i1�5I04qi1 qi1 �−i57 dqi > 00 (B23)

This contradicts that �i → I14qi1 �i1 �−i1�i1�5I04qi1 �i1 �−i5is nondecreasing. Equation (B22) shows that the man-ager receives the same information premium as under theoptimal centralized mechanism. Hence, the investment deci-sions I04 · 5 and I14 · 5 coincide with those of the optimalcentralized mechanism. Thus, according to the revenueequivalence theorem, the residual-income-based contractmust be optimal. The remainder of the proof follows alongthe lines of Proposition 1. �

Proof of Corollary 5. Follows directly from (26). �

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multistage capital budgeting for shared investments

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