Capital budgeting and delegation

Journal of Financial Economics 50 (1998) 259—289

Capital budgeting and delegation1

Milton Harris!,*, Artur Raviv"

! Graduate School of Business, University of Chicago, Chicago, IL 60637, USA" Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208, USA

Received 26 June 1997; received in revised form 14 January 1998

Abstract

As part of our ongoing research into capital budgeting processes as responses todecentralized information and incentive problems, we focus in this paper on when a levelof a managerial hierarchy will delegate the allocation of capital across projects and timeto the level below it. In our model, delegation is a way to save on costly investigation ofproposed projects. Therefore, it is more extensive the larger are the costs of suchinvestigations. This delegation takes advantage of the fact that the lower-level manager’spreferences are assumed to be similar (though not identical) to those of the higherlevel. ( 1998 Elsevier Science S.A. All rights reserved.

JEL classification: G31

Keywords: Capital budgeting; Delegation; Budget rollovers

1. Introduction

In most large corporations, considerable attention is paid to designing oftenelaborate capital budgeting systems. These systems allow for decentralizeddecision making, taking care to provide incentives for agents at various levels ofthe organization to make value-maximizing choices. Despite the large invest-ment firms make in designing and implementing these schemes, relatively littleacademic research has been undertaken to understand what determines their

*Corresponding author. Tel.: 773/702-2549; fax: 773/702-3195; e-mail: [email protected].

1We thank Bhagwan Chowdhry, Yaniv Grinstein, an anonymous referee, and seminar partici-pants at Columbia, Duke, Hong Kong University of Science and Technology, Indiana, Tel Aviv,Vanderbilt, and the European Finance Association Meetings (Vienna, 1997).

0304-405X/98/$— see front matter ( 1998 Elsevier Science S.A. All rights reservedPII: S 0 3 0 4 - 4 0 5 X ( 9 8 ) 0 0 0 3 8 - 5

nature. In our previous work, Harris and Raviv (1996), we attempt to explainsome features of capital budgeting schemes. Our hypothesis is that theseschemes are designed to deal with problems of private information and manage-rial preferences for ‘empire’, i.e., larger capital allocations. The firm’s ownerswant to use the superior but private information of managers to allocate capitalmore efficiently. But managerial preference for empire makes it necessary forowners to provide incentives for truthful revelation. This model is capable ofexplaining the widespread use of capital spending limits and the procedures bywhich these limits can be relaxed. The present paper adopts the same frameworkbut extends the model to address issues related to the allocation of capital acrossmultiple projects. In particular, we consider the question of when one wouldexpect owners to delegate the allocation across projects of a given amount ofcapital, what form this delegation will take, and under what circumstancesallocations for a given period can be ‘rolled over’ to a subsequent period.

Our model is quite similar to that of Harris and Raviv (1996). There are twoagents: headquarters, representing the interests of the shareholders, and a divis-ion manager. The division manager has access to two projects whose only inputsare capital. The fact that the division has access to more than one project isthe essential point of departure from the model of Harris and Raviv (1996).(Section 7 addresses the extent to which some of the specific assumptions of ourmodel, such as the existence of only two projects, can be relaxed.) The divisionmanager must obtain capital for the two projects from headquarters.2 Eachproject can have either high or low marginal productivity of capital, but theprojects are otherwise identical (i.e., two projects with the same productivity areidentical). The reason for decentralization in our model is that the divisionmanager is assumed to know these productivities while headquarters can learnthem only at a cost through a detailed audit. To prevent the trivial solution inwhich the manager voluntarily reveals his information when asked, we postulatethat the division manager prefers larger capital allocations to smaller ones, otherthings equal. This preference could reflect managerial utility for being in chargeof larger enterprises (i.e., a preference for a larger empire) or the fact that largercapital allocations result in greater managerial perquisite consumption. Inaddition, we assume that the manager’s marginal utility for capital allocated toa given project increases with the project’s productivity.

Headquarters seeks to design an incentive-compatible capital allocationscheme to trade off the distortion due to decentralized information and

2As in Harris and Raviv (1996), we do not address the issue of whether the division would bebetter off as a stand alone entity. Two recent papers that provide a rationale for including variousdivisions within a single corporate structure are Stein (1997) and Thakor (1990). Nor do we addressthe issue of why divisions are generally not allowed to seek capital directly from the capital market.For some recent literature on this point, see Gertner et al. (1994) and Stein (1997).

260 M. Harris, A. Raviv/Journal of Financial Economics 50 (1998) 259–289

managerial preference for empire against the cost of auditing. In particular,headquarters chooses an audit strategy and capital allocations, as functions ofthe manager’s request for capital, to maximize the value of the residual claim,given the constraints implied by private information and the manager’s prefer-ence for empire. The focus of the analysis is to uncover the conditions underwhich headquarters chooses to delegate the ‘project allocation’, i.e., the distribu-tion of capital across projects, to the division manager.

We show that an optimal scheme involves an initial, aggregate capital spend-ing limit (i.e., total for both projects). The manager can either accept this amountor, in some cases, request additional capital above the spending limit. Inparticular, we show that the extent to which the manager can negotiate hisinitial budget depends on whether audit costs are low, medium, or high. For lowaudit costs, the manager can request one of two larger amounts of capital; forintermediate audit costs, he can request only one larger amount; and for highaudit costs, the manager is held to the initial spending limit. When audit costsare low, the manager will request additional capital as long as at least one of theprojects has high productivity. For intermediate audit costs, he will requestmore only if both projects have high productivity. And for high audit costs, themanager receives his initial budget regardless of true productivity levels. Thus,the sensitivity of the aggregate capital allocation to actual productivitiesdeclines with audit costs. We refer to the degree of sensitivity as the ‘flexibility’ ofthe allocation process.

If the manager requests a larger amount, headquarters can either allocatea compromise level of capital (between the initial spending limit and the amountrequested) or it can audit the proposed projects and discover the true productiv-ity of capital in each. Audit probabilities decrease with audit costs and increasewith the amount of capital requested. If the audit reveals that the projects’productivity levels justify the higher level of capital requested, this amount isallocated. Otherwise, no capital is allocated to either project.3 The compromisecapital allocation also decreases with audit costs and increases with the reques-ted allocation. These results show that the optimal capital budgeting schemewhen there are two projects determines the aggregate capital allocation in thesame way as the optimal scheme in the single-project case described in Harrisand Raviv (1996).

Our results also help explain the extent to which the project allocation isdelegated to the division manager. The main result is that headquarters willdelegate this decision over the set of realizations of productivities in which the

3This helps ensure that the manager never ‘cheats’, i.e., requests more capital than the projectsjustify. In fact, we need not assume that the projects are passed up in this case; all that is required isthat this particular manager is not in charge of them and, hence, obtains no utility from the capitalallocation.

M. Harris, A. Raviv/Journal of Financial Economics 50 (1998) 259–289 261

same total capital is allocated. When audit costs are low, the only productivityrealizations for which the same total capital is allocated are those for which oneproject has high productivity and the other has low productivity. In this case,the manager is allowed to allocate the total budget to the projects subject to theconstraint that he invest at least a certain minimum amount in each. Forintermediate audit costs, the set of realizations that result in the same totalallocation includes the case in which both projects have low productivity as wellas the case in which exactly one project has high productivity. The initialspending limit applies in all three situations. Again, the manager chooses theproject allocation subject to a minimum investment constraint, so the projectallocation is delegated except when both projects have high productivity. Forhigh audit costs, the manager’s total budget is the initial spending limit regard-less of productivities, and the manager is allowed to choose the project alloca-tion as before. For high audit costs, therefore, the project allocation is alwaysdelegated, regardless of the true productivities. Thus, the extent of delegationincreases with audit costs. Since the flexibility of the allocation process decreaseswith audit costs as explained above, the extent of delegation and the flexibility ofthe capital budgeting system are inversely related.

If we interpret the two projects as occurring sequentially rather than simulta-neously, the results on delegation of the project allocation have implications forwhen one would expect managers to be allowed to roll over allocations from oneperiod to the next. In particular, delegating the project allocation when theprojects occur sequentially is equivalent to allowing the manager to roll over hisbudget subject to some constraints. For example, suppose one project has highproductivity and the other low. One interpretation of the optimal capitalbudgeting scheme is that headquarters allocates the total capital appropriate forthis situation in the first period. Headquarters then requires the manager to rollover at least the amount appropriate for the least productive project but notmore than the amount appropriate for the more productive project.

The intuition for these results follows from the fact that headquarters’ prob-lem is to prevent overinvestment due to exaggerated claims about the productiv-ity of capital. This leads to a capital spending limit. A completely rigid limit,however, fails to take advantage of the manager’s private information. There-fore, when audit costs are not ‘too high’, it is efficient for headquarters to allowmanagers to request larger allocations. Headquarters employs several devices to‘keep the manager honest’. First, it audits the division manager’s capital requestand penalizes him if his request is unjustified. Auditing occurs with a probabilitythat increases with the manager’s capital request, but, to save audit costs, theprobability is always less than one. Second, the initial spending limit is ‘gener-ous’ relative to the first-best level of investment when both projects have lowproductivity. Third, the compromise allocation for an unaudited request ofcapital in excess of the initial spending limit is ‘stingy’ relative to first-best. Theseadjustments encourage the manager not to request more capital when in fact


both projects have low productivity. The results on delegation of the projectallocation follow mainly from the fact that the manager’s preferences are similarto those of headquarters (i.e., he prefers to allocate more capital to moreproductive projects). Therefore, there is no need to use a costly audit todistinguish situations across which the total allocation is the same.

The literature on this topic is surveyed in Harris and Raviv (1996), so a briefoverview will suffice. The empirical literature is mainly survey research (Mao,1970; Gittman and Forrester, 1977; Schall et al., 1978; Scott and Petty, 1984;Stanley and Block, 1984; Ross, 1986). The ‘typical’ capital budgeting procedurethat emerges from these studies and articles by corporate executives is describedin Taggart (1987) and is similar to the optimal scheme described above. Inparticular, divisions are assigned an initial spending limit and can increase theircapital allocations, but this typically requires a major convincing effort and usesup ‘credit’ with corporate headquarters. This is consistent with our view ofcostly monitoring as the means by which capital spending limits are exceeded.Some related theory papers are Lambert (1986), which examines incentives forinformation gathering and investment choice, Baiman and Rajan (1994), whichexamines the issue of centralized (owner) vs. decentralized (manager) investmentchoice, Harris et al. (1982), which analyzes transfer pricing in the presence ofboth information and incentive problems, Antle and Eppen (1985), whichattempts to explain ‘organizational slack’ (allocation of more capital thanis actually used in a project), capital rationing, and underinvestment, andHolmstrom and Ricart i Costa (1986), which derives capital rationing as a re-sponse to managerial overinvestment caused by the fact that an optimal wagecontract insures the manager against the possibility that an investment willreveal him to be of low ability.

The remainder of the paper is organized as follows. The model is presentedformally in Section 2. This is followed by careful statements of and motivationfor our results in Section 3. In Section 4, we reinterpret the optimal schemediscussed in Section 3 to relate it better to observed practice. Section 5 exploresour results on delegation of the project allocation. Section 6 applies our modelto consider when one would expect divisions to be allowed to roll over part oftheir allocation to the next period. Section 7 considers the implications ofmodifying some of our assumptions, and Section 8 concludes.

2. Model

We model a firm with a headquarters and one division. The division is headedby a risk-neutral manager with access to two projects. The manager has privateinformation regarding the productivity of capital in each project. Since we areinterested in the internal allocation of capital, we assume the manager canobtain capital only from headquarters. Headquarters represents the interests of


shareholders and is assumed to behave as if shareholders are risk neutral.Headquarters would like to use the manager’s private information to allocatecapital to the projects, but the manager is assumed to have preferences thatprevent him from revealing this information truthfully unless he is provided withincentives to do so. Our objective is to characterize an optimal capital budgetingscheme. We focus on the extent to which the allocation of capital is delegated tothe division manager.

Each project, p3M1, 2N,P, has two possible technologies, tp3M1, 2N,¹.

Thus, the pair of projects is characterized by a technology profile denotedt"(t

1, t

2)3¹2 (generally, boldface denotes a vector). The division manager is

assumed to observe the technology profile privately before any capital isallocated. Without a costly audit, headquarters knows only the distribution ofpossible technology profiles, denoted n, i.e., n(t) is the probability of technologyprofile t. We assume that n(t)'0 for all t3¹2. If headquarters pays a cost q'0,it can discover the true technology profile.4

A project’s technology determines its net present value (NPV) as a function ofthe amount of capital allocated to that project. To simplify the analysis, weassume that a project’s NPV depends only on the technology of that project andthe capital allocated to it. In particular, the NPV is assumed not to depend onthe project itself. We denote by v(t, k) the NPV of either project with technologyt3¹ and capital input k. Let �(t, k)"(v(t

1, k

1), v(t

2, k

2)) be the profile of net

present values of the two projects, given the technology profile t and the capitalallocation k. We assume that v(t, 0)"0 and v@(t, 0)"R for all t, where v@ is thederivative with respect to k, and that, for all t, v is strictly concave in k and hasa unique maximum at k*(t)'0. Let v*(t)"v(t, k*(t)) be the maximal NPVobtainable from technology t. Assume technology 2 is more productive thantechnology 1, both in total product and in marginal product, i.e., both v and v@increase with t. Consequently, k* also increases with t. Finally, we need oneadditional assumption regarding the function g, defined by

g(t, k)"v(t, k)!v@(t, k)k.

We can interpret g/k as average NPV minus marginal NPV. Concavity of vin k implies that g is nonnegative and increases with k. It is easy to check thatg(t, k*(t))"v*(t). To simplify the analysis, we assume that g decreases with t andis convex in k.5

4Townsend (1979) was one of the first models to use the idea of costly verification. A model moresimilar in spirit to ours is that of Border and Sobel (1987).

5A sufficient condition for g to decrease with t is that vA increases with t, i.e., that the moreproductive technology is less concave. A sufficient condition for g to be convex in k is that v@ isconcave, i.e., that marginal NPV decreases at an increasing rate with k, or, equivalently, concavity ofNPV increases with k.


As mentioned above, headquarters and the division manager have divergentpreferences. Headquarters wants to maximize shareholder wealth, defined as theexpected sum of project net present values net of audit costs. Note that head-quarters is not concerned with managerial compensation. This reflects twoassumptions. First, and most critically, we assume that managerial compensa-tion cannot be used as an incentive device to obtain truthful reporting of theproject technologies. We justify this by arguing that managerial compensationschemes cannot fully solve the incentive/information problems in this model.Since analyzing incentive compensation schemes and capital budgeting schemessimultaneously seems intractable and since relatively more attention has beenpaid to compensation, we choose to abstract from such considerations.6 Second,we assume the manager’s utility from running this division exceeds his oppor-tunity cost (even if no capital is allocated to the division).7 Thus, even a fixedsalary, independent of managerial reports or performance, is unnecessary (nega-tive salary is not allowed).

The manager has a taste for size (capital) even at the expense of net presentvalue. His utility from running the division is assumed to increase with theamount of capital allocated to each project, and his marginal utility for capitalalso increases with the productivity of the capital. Thus, the manager, otherthings equal, wants to allocate more capital to the more productive project. Infact, we assume that the manager’s preferences are linear, i.e., his marginal utilityfor capital is independent of the amount of capital. Marginal utility for capital,b(t), is assumed to increase with the project’s productivity, t, but is independentof the project (again this is to simplify the analysis). Thus, the manager’s utilityfor a capital allocation k when the technology profile is t is given byb(t) ) k"b(t

1)k

1#b(t

2)k

2(the dot in the expression on the left hand side of this

equality indicates inner product).To characterize an optimal capital budgeting scheme, we apply the Revelation

Principle, which states that any scheme is equivalent to a revelation game inwhich the manager reports a technology profile, t, and is given incentives toreport truthfully (see, e.g., Harris, 1987). The manager’s report results in auditprobabilities and a capital allocation. An optimal revelation game is one inwhich these probabilities and allocations are chosen to maximize the expectedNPV of the firm net of expected audit costs. More precisely, headquarters’

6For a more detailed discussion, see Harris and Raviv (1996). For an analysis of a situationinvolving project selection and managerial compensation that focuses exclusively on the role ofcompensation in creating incentives, see Hirshleifer and Suh (1992).

7Equivalently, we could assume that there is some exogenously specified fixed salary by simplyredefining the manager’s opportunity cost to be net of this salary. This assumption simplifies theanalysis by guaranteeing that the manager’s ‘participation constraints’ are not binding. It does notaffect the qualitative results of the model.


choice variables are the probability of auditing, a(t), the capital allocationif there is no audit, k(n, t) (‘n’ stands for ‘no audit’), and the capital allocation ifthere is an audit, as functions of the reported technology profile t. The capitalallocated if there is an audit can also depend on the results of the audit. Wedenote by k(a, t) (‘a’ stands for ‘audit’) the capital allocated after an audit if themanager reports t and the true technology profile is t. We do not need notationfor the capital allocated after an audit when the report is not truthful, since it iseasy to show that in any optimal solution, this capital allocation is zero (or,equivalently, the projects are assigned to a different manager). Intuitively, thisallocation never occurs in equilibrium. Therefore, there is no cost to head-quarters of inflicting the maximum punishment on the manager for lying.Because of the nonnegativity constraints, the maximum punishment in this caseis to allocate no capital (or replace the manager). As is common in models likeours, we assume that headquarters can commit to allocating zero capital in theout-of-equilibrium event that the manager lies. Similarly, we assume that head-quarters can commit to auditing with the chosen probability, even though this iscostly and headquarters knows that the manager is telling the truth. Sucha commitment can stem from reputational considerations that we do not modelhere. Headquarters’ problem is as follows:8

maxa,k

+t

n(t)M(1!a(t))�(t, k(n, t)) ) 1#a(t)[!q#�(t, k(a, t)) ) 1]N (1)

subject to

(1!a(t))b(t) ) k(n, t)#a(t)b(t) ) k(a, t)*(1!a(r))b(t) ) k(n, r), ∀rOt, (2)

1*a(t)*0, k(n, t)*0, k(a, t),*0 ∀t. (3)

The objective function (1) is the expected net present value of the sum of the twoprojects minus expected audit costs. For each technology profile, t, the first termin braces is the probability of not auditing multiplied by the sum of the netpresent values of the two projects given t and the capital allocation if there is noaudit. The second term in braces is the probability of an audit multiplied by thesum of the NPVs of the projects net of audit costs. The constraints in Eq. (2) arethe incentive compatibility constraints. The left side is the division manager’sexpected utility if the true technology profile is t, and the manager reports t. Theright side is his expected utility if the truth is t, but he reports r. This expressionreflects the fact that if the audit reveals the manager is lying, no capital is

8For two vectors x and y in R/, the notation, x*y means that xi*y

ifor all i, x'y means that

x*y and xOy, and xAy means that xi'y

ifor all i. Also 1"(1, 1), and 0"(0, 0).


allocated. The constraints in Eq. (3) simply guarantee that a and 1!a arewell-defined probabilities, and the capital allocations are nonnegative.

3. Results

In this section, we characterize an optimal solution of headquarters’ problem(1)—(3). We start by proving that it is never optimal to audit with probability oneas long as the audit is costly. The intuition is that if headquarters audits for sure,the manager’s utility if he lies is zero, whereas if he tells the truth, his utility ispositive. Thus auditing with probability one provides ‘too much’ incentive notto cheat in the sense that, in this case, the incentive constraints (2) are notbinding. headquarters can reduce expected audit costs without changing theallocation by reducing the audit probability to at least slightly below one.Formally, we have

¹heorem 1. If q'0, then a(t)(1 for all t.

The proofs of this and all other results are in the appendix.Next, we write the Lagrangian for headquarters’ problem as

L(a, k, l)"+t

n(t)M(1!a(t))�(t, k(n, t)) ) 1#a(t)[!q#�(t, k(a, t)) ) 1]N

# +r,t,rEt

ktrb(t) ) M(1!a(t))k(n, t)#a(t)k(a, t)!(1!a(r))k(n, r)N,

where ktr is the Lagrange multiplier associated with the incentive constraint (2)trfor which t is the true technology profile and r is the reported profile. Thefirst-order conditions for the capital allocation are then

n(t)�@(t, k(n, t))"+rEt

b(r)krt!b(t)+rEt

ktr (4)

and

n(t)�@(t, k(a, t))"!b(t)+rEt

ktr, if a(t)'0. (5)

Note that we have divided Eq. (4) by 1!a(t), since by Theorem 1, 1!a(t)'0.Summing Eq. (4) over t gives

+t | T2

n(t)�@(t, k(n, t))"0. (6)


We also write the coefficient of a(t) in the Lagrangian as

c(t)"n(t)[!q#[�(t, k(a, t))!�(t, k(n, t))] ) 1]#k(n, t) +rEt

b(r)krt

#b(t) ) [k(a, t)!k(n, t)]+rEt

ktr. (7)

Since a(t)(1 by Theorem 1, necessary conditions for a maximum require that ifa(t)'0, c(t)"0, and if a(t)"0, c(t))0. For t such that a(t)'0, we cansubstitute from the first-order conditions (4) and (5) to obtain

c(t)n(t)

"!q#[u(t, k(a, t))!u(t, k(n, t))] )1"0 (8)

where u(t, k)"(g1(t1, k

1), g

2(t2, k

2)).

We proceed by conjecturing a solution to headquarters’ problem and thenproving that this solution satisfies the necessary conditions for an optimum. Theconjecture is based on the following intuition. First, since NPV and the marginalutility of the manager do not depend directly on the project (except through thetechnology), it is sensible that the solution exhibit symmetry with respect to theprojects, i.e.,

k1(i, t, t)"k

2(i, t, t) for t 3 M1, 2N,

k1(i, 1, 2)"k

2(i, 2, 1),

k1(i, 2, 1)"k

2(i, 1, 2), for i 3 Mn, aN (9)

anda(1, 2)"a(2, 1).

It is convenient, therefore, to define the following notation for the total capitalallocated to both projects and the allocation to the less productive project whenthere is no audit:

Kt(q)"k(n, t) ) 1 and k(q)"k1(n, 1, 2)"k

2(n, 2, 1). (10)

In this notation, we emphasize the dependence of the capital allocation on auditcosts, q. Note that the symmetry conditions (9) imply that K

12"K

21and

Kt"2k1(n, t) for t3M(1, 1), (2, 2)N.

Second, we conjecture that the incentive constraints are binding only whenthe true technology t"(1, 1). That is, if the capital budgeting scheme motivatesthe manager to tell the truth when the technology is the worst possible, it willalso provide sufficient incentives for truth-telling when the technology is better.The complementary slackness conditions then imply that for all tO(1, 1), ktr"0for all rOt. Using this fact and the new notation, the first-order conditions (4)are equivalent to

b(1)k1,1,t

"n(t)v@(t1, k

1(n, t))"n(t)v@(t

2, k

2(n, t)) ∀t. (11)


The right equality in Eq. (11) is the familiar condition that marginal productsshould be equalized across projects. Using the notation in Eq. (10), Eq. (11) fort"(1, 2) then implies

v@(1, k(q))"v@(2, K12

(q)!k(q)). (12)

Now Eq. (6) implies

F(K11

(q), k(q), K22

(q))"0, (13)

where

F(x1, x

2, x

3)"n(1, 1)v@A1,

x12 B#[n(1, 2)#n(2, 1)]v@(1, x

2)

#n(2, 2)v@A2,x32 B.

Note that F is strictly monotone-decreasing in all three arguments. Next, ifa(2, 2)'0, then Eq. (8) must hold, or in the new notation,

q"2[v*(2)!g(2, K22

(q)/2)]. (14)

Similarly, if a(1, 2)"a(2, 1)'0, then we must have

q"v*(1)#v*(2)!g(1, k(q))!g(2, K12

(q)!k(q)). (15)

Third, we conjecture that, since the manager has no reason to claim the worsttechnology when in fact the technology is not the worst, there is no need to audita claim that t"(1, 1), i.e., we conjecture a(1, 1)"0. Whether the other auditprobabilities are positive depends on the cost of auditing, all else equal. Weconjecture that there are cutoff levels of audit costs, 0(q

2(q

3, such that for

q3(0, q2), a(1, 2)"a(2, 1)'0 and a(2, 2)'0, for q3[q

2, q

3), a(1, 2)"a(2, 1)"0

and a(2, 2)'0, and for q*q3, a(1, 2)"a(2, 1)"a(2, 2)"0.

Fourth, since we conjecture that the incentive constraint (2) is binding whenthe true technology is (1, 1), it follows from a(1, 1)"0 and (2)

1,1,tthat

a(1, 2)"a(2, 1)"1!K

11(q)

K12

(q),

and

a(2, 2)"1!K

11(q)

K22

(q). (16)

Note that our conjecture that a(1, 2)"a(2, 1)"0 for q*q2

implies that wemust have K

11(q)"K

12(q) for q*q

2, and our conjecture that a(1, 2)"a(2, 1)

"a(2, 2)"0 for q*q3

implies that K11

(q)"K12

(q)"K22

(q) for q*q3.

We summarize the conjectured solution thus far in Table 1.


Table 1Conjectured solution for a(t) and k(n, t).

q range K11

(q), K12

(q), K22

(q), k(q) satisfy a(1, 1) a(1, 2)"a(2, 1) a(2, 2)

(0, q2) (12)—(15) 0 1!K

11/K

121!K

11/K

22[q

2, q

3) (12)—(14) and K

11"K

120 0 1!K

11/K

22[q

3, R) (12)—(13) and K

11"K

12"K

220 0 0

Notice that for each range of q, the capital allocation consists of fourvariables that solve four equations. To complete the characterization of theconjectured solution, it remains only to define q

2and q

3precisely and

to specify k(a,t) for all t. Since q+

( j"2, 3) is at the boundary between tworegions, we define q

jto be such that, together with the four capital allocation

variables, K11

, K12

, K22

, and k, all conditions for both regions are satisfied.Thus, q

2, K

11(q

2), K

12(q

2), K

22(q

2), and k (q

2) satisfy Eqs. (12)—(15) and

K11

(q2)"K

12(q

2). Similarly, q

3, K

11(q

3), K

12(q

3), K

22(q

3), and k (q

3) satisfy

Eqs. (12)—(14) and K11

(q3)"K

12(q

3)"K

22(q

3).

Finally, after an audit in which the manager is found to have reportedtruthfully, it is sensible to allocate the NPV-maximizing capital, i.e., k(a, t)"k*(t). Since in the conjectured solution headquarters is certain not to auditif the manager reports the worst technology, k(a, 1, 1) is irrelevant. It willbe convenient to define k(a, 1, 1)"k(n, 1, 1). For tO(1, 1), we conjecture k(a, t)"k*(t) as mentioned.

In the next theorem, we show that the conjectured solution is indeed optimaland establish some of its properties.

¹heorem 2. ¹he following solution is optimal:

f q2, K

11(q

2), K

12(q

2), K

22(q

2), and k(q

2) satisfy Eqs. (12)—(15)) and

K11

(q2)"K

12(q

2); q

3,K

11(q

3), K

12(q

3), K

22(q

3), and k(q

3) satisfy

Eqs. (12)—(14) and K11

(q3)"K

12(q

3)"K

22(q

3);

f kp(n, 1, 1)"K

11(q)/2 and k

p(n, 2, 2)"K

22(q)/2 for p3M1, 2N; k

1(n, 1, 2)"k

2(n,

2, 1)"k(q) and k1(n, 2, 1)"k

2(n, 1, 2)"K

12(q)!k(q), where K

11, K

12, K

22,

and k are defined in Table 1;f For tO(1, 1), k(a, t)"k*(t) and k(a, 1, 1)"k(n, 1, 1);f a(t) is as defined in Table 1.

This solution has the following properties:

(a) 0(q2(q

3;

(b) k(q)(K12

(q)/2(K12

(q)!k(q), for all q'0;


(c) for q3(0, q2), K

11(q) increases with q, K

12(q), k(q), K

12(q)!k(q), and

K22

(q) decrease with q; for q3[q2, q

3), K

11(q)"K

12(q) and k(q) increase

with q and K22

(q) decreases with q, and for q*q3, K

11"K

12"K

22and

k are independent of q;(d) K

11(0)"2k*(1), K

12(0)"k*(1)#k*(2), K

22(0)"2k*(2), k (0)"k*(1);

K11

(q)'2k*(1) for all q'0, K12

(q)(k*(1)#k*(2) for q)q2, and

K22

(q)(2k*(2), for all q'0;(e) for q3(0, q

2), K

11(q)(K

12(q)(K

22(q); for q3[q

2, q

3), K

11(q)"K

12(q)

(K22

(q);(f) for q"0, a(1, 2)"a(2, 1)"1!2k*(1)/[k*(1)#k*(2)], a(2, 2)"1!k*(1)/

k*(2); a(1, 2)"a(2, 1) decreases with q for q3(0, q2) and is identically zero

for q*q2; a(2, 2)'a(1, 2) decreases with q for q3(0, q

3) and is identically

zero for q*q3.

The audit probabilities and aggregate capital allocations described inTheorem 2, parts (a) — through (f), are depicted in Figs. 1 and 2, respectively. Thefact that the audit probabilities decrease to zero with audit costs accords withsimple economic intuition. If headquarters is certain not to audit a claim ofhigher productivity than the worst case, (1, 1), the total capital allocated must bethe same as that allocated when the worst case is claimed. Thus when a(t)"0,K

11"Kt. It is important to note that this logic does not apply to the allocation

of capital across projects. The reason is that even if headquarters is certain notto audit, the allocation across projects can be delegated to some extent since themanager’s preferences for allocating capital across projects align qualitativelywith the marginal products of capital across projects. For example, suppose

Fig. 1. Audit probabilities as functions of audit cost, generated using the following example:v(t,k)"k!1

2htk2, where h

1"2 and h

2"1 and with n(t)"0.25 for all t.


Fig. 2. Aggregate capital allocations as functions of audit cost, generated using the followingexample: v(t, k)"k!1

2htk2, where h

1"2 and h

2"1 and with n(t)"0.25 for all t.

headquarters chooses not to audit claims of t"(1, 2) and t"(2, 1). It can stillspecify that more capital be allocated to project 2 in the former case and more toproject 1 in the latter, and the manager will still report truthfully as long as thetotal capital is the same in both states and incentives not to claim that t"(2, 2)are present. Thus k

1(n, 1, 2)(k

2(n, 1, 2) and k

1(n, 2, 1)'k

2(n, 2, 1) even when

a(1, 2)"a(2, 1)"0.When audit probabilities are positive, aggregate capital allocations can differ.

When audit costs are sufficiently low, it is worthwhile undertaking audits withpositive probability since this allows larger capital allocations when productiv-ity is higher. It is, therefore, not surprising that K

22'K

12when a(2, 2)'0

(i.e., q(q3). To preserve incentives for truth-telling, however, one must have

a(2, 2)'a(1, 2). Similarly, K12'K

11when a(1, 2)'0 (i.e., q(q

2).

As audit costs increase and audit probabilities decline, incentive compatibilityrequires that capital allocations converge. Thus K

22(q) declines and

K12

(q)"K11

(q) rises as q increases from q2

to q3; similarly, K

22(q) and K

12(q)

decline and K11

(q) rises as q increases from zero to q2.

4. The optimal capital budgeting scheme

To relate the results of the previous section to observed capital budgetingpractices, we reinterpret the optimal revelation game defined by Theorem 2. Thereinterpretation is simply another mechanism that is equivalent to the optimalrevelation game in the sense that the new mechanism has exactly the sameequilibrium outcomes as the revelation game. Since the new mechanism is


equivalent to the optimal revelation game, it is also optimal. In addition, itresembles observed capital budgeting schemes.

In our reinterpretation of the optimal capital budgeting scheme, the divisionmanager receives an initial aggregate budget. In some cases the manager can askfor additional capital and has some choice of the allocation of the total acrossprojects. Requests for additional capital can result in an increase in the capitalallocated without an audit or may result in an audit. If a requested increase isaudited and found to be justified, the manager will be allocated the requestedincrease. In either case, the manager can have some discretion in allocatingcapital across projects depending on audit costs and the requested increase. Ifthe request is found not to be justified, no capital is allocated. The followingtheorem and Table 2 describe the optimal capital budgeting mechanism in moredetail.

Table 2Optimal capital budgeting scheme

(Shaded areas indicate situations across which the project allocation is delegated to the divisionmanager)

t Request Audit Cost

q(q2

q2)q(q

3q3)q

No audit Audit! No audit Audit! No audit Audit!

(1, 1) None K11

/2allocatedto eachproject

Probability"0

(1, 2)

K*(1, 2)

K12

allocated,at least kto eachproject

Probability"a(1, 2);K*(1, 2)allocated,at leastk*(1)to eachproject

K11

allocated,at least kto eachproject

Probability"0 K

11allocated,at least kto eachproject

Probability"0

(2, 1)

(2, 2) K*(2, 2) K22


Probability"a(2, 2);K*(2, 2)/2allocatedto eachproject

K22


Probability"a(2, 2);K*(2, 2)/2allocatedto eachproject

!Allocations assume audit reveals that request is justified; otherwise, no capital is allocated to thisdivision.


¹heorem 3. In an optimal capital budgeting scheme,

f ¹he division manager receives an initial total budget of K11

(q) but can ask formore.

f If audit costs are low (q)q2), the manager can ask for K*(1, 2)"k*(1)#k*(2)

or K*(2, 2)"2k*(2). If he does not ask for more, there is no audit, and he mustsplit the total K

11(q) evenly between the two projects. If he asks for K*(1, 2), he

is audited with probability a(1, 2)"a(2, 1), which are functions of q. If there isno audit, he receives K

12(q) and must allocate at least k(q) to each project. If

there is an audit and t is found to be in M(1, 2), (2, 1)N, the manager receiveshis request and must allocate at least k*(1) to each project. If there is an auditand t is found not to be in M(1, 2), (2, 1)N, the manager receives no allocation. If themanager requests K*(2, 2), he is audited with probability a(2, 2). If there is noaudit, he receives K

22(q) which he must allocate evenly between the projects. If

there is an audit and t"(2, 2), he receives his request which he must allocateevenly. If there is an audit and t is found not to be (2, 2), the manager receives noallocation.

f For intermediate audit costs (q2)q(q

3), the manager can ask for K*(2, 2);

requests for less than K*(2, 2) are ignored. If he does not ask for more, thereis no audit, and he must allocate at least k(q) to each project. If he asks formore, he is audited with probability a(2, 2). If there is no audit, he receivesK

22(q) which he must allocate evenly between the projects. If there is an

audit and t"(2, 2), he receives his request which he must allocate evenly.If there is an audit and t is found not to be (2, 2), the manager receives noallocation.

f If audit costs are high (q*q3), the manager receives only the initial allocation,

K11

(q), regardless of what he asks for. He must allocate at least k(q) to eachproject. ¹here is no audit.

The correspondence between the scheme described in Theorem 3 and theoptimal revelation game of Section 3 is as follows. The manager’s failureto ask for additional capital is interpreted as a declaration that t"(1, 1).Requests for K*(1, 2) and K*(2, 2) correspond to declarations that t3M(1, 2),(2, 1)N and t"(2, 2), respectively. For low audit costs (below q

2), the

manager can request either increase. For intermediate audit costs (betweenq2

and q3), he can request only K*(2, 2) since, in the revelation game,

K11"K

12for q in this range. For high audit costs (above q

3), the manager

cannot request additional capital (in the revelation game, K11"K

12"K

22for

q in this range).In the revelation game, if audit costs are low (below q

2) and the manager

declares t"(1, 2) or (2, 1), he is audited with probability a(1, 2)"a(2, 1). If thereis no audit, he is allocated k(q) for the less productive project and K

12(q)!k(q)

for the other project. If there is an audit, the manager is allocated k*(1) for the


less productive project and K*(1, 2)!k*(1)"k*(2) for the other project. In thecurrent interpretation, when t is either (1, 2) or (2, 1), the manager requestsK*(1, 2), i.e., his request reveals only that t3M(1, 2), (2, 1)N. As in the revelationgame, he is audited with probability a(1, 2)"a(2, 1). Nevertheless, the fact thatthe manager’s marginal utility for capital allocated to the more productiveproject is higher results in the same allocation as in the revelation game.In particular, the manager prefers to invest as much as possible in the moreproductive project because b(1)(b(2). In the current interpretation, however,the manager must invest at least k(q) (if he is not audited) or k*(1) (if he isaudited) in each project. Given this constraint, he will allocate k(q) or k*(1) to theless productive project and the rest to the more productive project, resultingin the same allocation as in the revelation game. If the manager declarest"(2, 2) in the revelation game, he is audited with probability a(2, 2). If he is notaudited, he is allocated K

22(q)/2 for each project, and if he is audited (and found

to have declared truthfully), he is allocated K*(2, 2)/2"k*(2) for each project.This is exactly what happens if the manager requests K*(2, 2) in the currentinterpretation.

For intermediate audit costs (between q2

and q3), the manager’s failure to

request K*(2, 2) is interpreted as a declaration that t3M(1, 1), (1, 2), (2, 1)N. In therevelation game, if the manager declares any of these three situations, he isallocated K

11(q) in total and there is no audit, as in the current interpretation.

In the revelation game, however, he must allocate half to each project ifhe declares (1, 1), k(q) to project 1 and K

11(q)!k(q) to project 2 if he declares

(1, 2), and the reverse if he declares (2, 1). The scheme of Theorem 3 results inexactly the same allocation. If t"(1, 1), the manager is indifferent among allproject allocations, so we can assume he will allocate half to each project.If t is (1, 2) or (2, 1), the manager will allocate k(q) to the least productive projectand the rest to the other as argued above. If the manager requests K*(2, 2)(declares t"(2, 2) in the revelation game), he is audited with probabilitya(2, 2) and allocated K*(2, 2) if the audit reveals that t"(2, 2) which he mustallocate equally across projects in either interpretation. If there is no audit, ineither interpretation the manager is allocated K

22(q) which he must split

equally.For high audit costs (above q

3), there is no audit in either interpretation and

the manager is allocated a total of K11

(q) regardless of t. In the revelation game,both projects receive the same allocation if t is (1, 1) or (2, 2) and the leastproductive project receives k(q) if t is either (1, 2) or (2, 1). The same allocationresults in the current interpretation.

The capital budgeting schemes described above resemble actual capitalbudgeting schemes. In particular, it is common for managers to be given initialspending limits that are, to some extent, negotiable. Our results suggest that thelimits and the extent to which these limits are negotiable depend importantly onaudit costs. When audit costs are high, initial spending limits are also high, but


there is little scope for negotiation to increase them. In the extreme case of veryhigh audit costs (above q

3), the scheme is totally inflexible in that no increase of

the initial allocation is possible.9 On the other hand, the less flexible is theaggregate capital allocation, the more discretion is given to the manager toallocate capital across projects. This is discussed in more detail in the nextsection.

5. Delegation of the project allocation

In this section, we focus on what determines the extent to which the projectallocation is delegated to the division manager. By delegation, we mean a situ-ation in which headquarters allows the division manager a choice of projectallocations and cannot predict with certainty which allocation the manager willchoose. This last point is important. If headquarters can predict with certaintywhat the manager will choose, it may as well simply impose this allocation onthe division; there is no reasonable sense in which delegation occurs. Given thisnotion of delegation, we must discuss whether it occurs not for a particularrealization of productivities but rather over a set of possible realizations. Byextent of delegation, we mean the probability that the manager has a nontrivialchoice in allocating his overall budget. Thus, if the probability of a nontrivialchoice by the manager is high in repeated situations, delegation is more exten-sive in the sense that one would expect to observe delegation more often,other things equal. In our model, there are two projects, each with two possiblecapital productivities. Therefore, four possible situations can occur. Whichsituations admit delegation and their probabilities determine the extent ofdelegation. For example, if the manager has a nontrivial choice only when oneof the projects has high productivity and the other has low productivity, thenthe extent of delegation is given by n(1, 2)#n(2, 1). From the previous section, itis clear that the extent to which firms use a capital budgeting scheme thatexhibits more extensive delegation of the project allocation depends on auditcosts.

Firms with low audit costs (less than q2) will delegate with probability

n(1, 2)#n(2, 1), i.e., only when the two projects have different productivities. Insuch firms, if the manager asks for no increase in capital or asks for themaximum increase (K*(2, 2)), he is required to allocate capital equally across the

9A number of other comparative statics results relating aggregate allocations and audit probabil-ities to requests and exogenous variables other than audit costs are possible. In fact, one can showthat the comparative statics results in Harris and Raviv (1996) derived for the single-project caseapply to aggregate allocations in the multiproject case. Consequently, we do not pursue these indetail here.


projects.10 When the manager requests K*(1, 2), headquarters knows that one ofthe projects has high productivity and the other low productivity. Withoutknowing which is which, headquarters can rely on the fact that the manager’smarginal utilities are such that he will want to allocate more capital to the moreproductive project. In this case, delegation of the allocation across projectsoccurs only when t3M(1, 2), (2, 1)N, i.e., with probability n(1, 2)#n(2, 1). Whenaudit costs are low, headquarters relies more on investigation of projects (or thethreat of investigation) and less on delegation. The high investigation probabil-ity allows the aggregate allocation to be fully sensitive to the manager’sinformation.

Firms with intermediate audit costs (between q2

and q3) will delegate with

probability n(1, 1)#n(1, 2)#n(2, 1), i.e., only when the two projects havedifferent productivities or both have low productivity. If the manager asksfor an increase, he must ask for K*(2, 2) which he must allocate equally. Thus,only if the manager does not ask for an increase is he allowed to choose.Since the probability of delegation in this case is n(1, 1)#n(1, 2)#n(2, 1),delegation is more extensive than when audit costs are low. The aggregateallocation is, however, less sensitive to the manager’s information, since onlyif both projects have high productivity will the total exceed the initial spendinglimit.

Firms with high audit costs (above q3) will always delegate the project

allocation. In such situations, however, division managers simply cannot negoti-ate an increase in their overall budgets, i.e., the aggregate allocation is com-pletely insensitive to true productivity.

It should be clear from this discussion that there is a tradeoff betweenthe flexibility of the capital budgeting system with respect to overall spendinglimits and the extent of delegation of the allocation of capital across projects. Asaudit costs increase, the system allows less adaptation of the total capitalallocated to actual conditions and more delegation of the project allocation.This potentially testable implication of the theory is summarized in thefollowing theorem.

¹heorem 4. As audit costs increase, the extent of delegation increases, and theflexibility of the capital budgeting scheme decreases.

10An equivalent mechanism would allow the manager to choose the allocation across projectssubject to a lower bound on the allocation to each project and with an expressed preference byheadquarters that the allocation be equal, even when the manager does not request an increase orrequests K*(2, 2). There is no real delegation in these two cases, however, since, given the manager’srequest, headquarters knows that the manager will choose equal allocation (a small monetaryincentive could be added to insure this).


6. Sequential projects and ‘rollovers’

Our model can be used to understand some aspects of ‘rollover’ policy, i.e.,when would one expect managers to be allowed to roll over to the next periodsome or all of their allocation for the current period.11 In particular, allowinga rollover amounts to delegating the project allocation to the manager (subject,perhaps, to constraints) when projects arrive sequentially instead of simulta-neously. There are two ways one can modify our model to consider rollovers.Both involve allowing for two dates, 0 and 1, and assume that project 1 isinitiated at date 0 and project 2 is initiated at date 1. The difference betweenthe two cases is in what one assumes about the manager’s information at date 0.In the ‘simultaneous knowledge’ case, information about both projects is avail-able to the manager at date 0. In the ‘sequential knowledge’ case, the managerlearns about the productivity of project 1 at date 0 but does not learn aboutproject 2 until date 1. We argue below that some form of rollover is optimal inboth cases.

The simultaneous knowledge case is formally identical to the model we havebeen analyzing. As long as the manager knows the productivity of capital inboth projects at the outset, the fact that the second project is available only atdate 1 does not affect the analysis. It is simply a reinterpretation of the samemodel. As mentioned, delegation of the project allocation corresponds toallowing the division to roll over unused budget from one period to the nextwhen projects are available sequentially. For example, suppose audit costs arehigh so that the total allocation is K

11, independent of true productivities. In

this case, headquarters could allocate K11

to the first period and stipulate that atleast k and at most K

11!k be rolled over to the second period. In general, our

results from Section 5 imply that the probability of rollovers increases withaudit costs.

The sequential knowledge case is formally different from the current model.To analyze this case in full generality is difficult and beyond the scope ofthis paper. Instead, we simplify by focusing on the case in which audit costsare sufficiently high that audit probabilities are zero and analyze the revelationgame between headquarters and the manager at date 0. In this game, themanager reports the productivity of project 1, say r3M1, 2N, resulting in a capitalallocation for both periods, k(r). Without threat of audit and with only onecapital allocation remaining at date 1, incentive compatibility requires that thecapital allocation at date 1 be independent of any report of the manager at date1. The task of headquarters is to specify this allocation so as to maximize

11Two other papers that address intrafirm resource allocation in a multiperiod framework areAntle and Fellingham (1990) and Fellingham and Young (1990).


expected total value subject to the incentive conditions that the managerprefers to report the true productivity of project 1. Formally, headquarters’problem is

maxk1( > )w0,k2( > )w0

E[v(tI1, k

1(tI1)]#E[v(tI

2, k

2(tI1)] (17)

subject to

b(t)k1(t)#E[b(tI

2)Dt

1"t]k

2(t)*b(t)k

1(r)#E[b(tI

2)Dt

1"t]k

2(r)

∀ t, r 3 M1, 2N with tOr. (18)

If the solution to this problem involves two different allocations, i.e., k(1)Ok(2),then delegation is optimal in the sense that the manager is given a non-trivial choice for allocating capital across the two periods. Moreover, it isclear that incentive compatibility implies that it cannot be that one alloca-tion provides more capital in both periods than the other. Thus, when theallocation depends on the manager’s reported productivity, the manager’schoice involves sacrificing capital in one period to obtain more capital in theother. Such choice can be interpreted as allowing the manager discretion in howmuch capital to roll over from date 0 to date 1. In fact, we show in the nexttheorem that rollovers are optimal unless productivity is perfectly correlatedover time.

¹heorem 5. In the absence of auditing, allowing the manager discretion in rollingover capital across periods is optimal if and only if productivity is not perfectlycorrelated over time.

The intuition for this result is essentially the same as before. The manager hasinformation that headquarters can use to improve the capital allocation. Be-cause the manager prefers to allocate more capital to the more productiveproject, it is possible to give the manager incentives to reveal the informationtruthfully, even when there is no threat of audit. This is accomplished by givingthe manager a choice between more capital now and less later or the reverse. Ifthe first project has low productivity, it is in the manager’s interest to accepta lower allocation initially in return for a larger allocation later since, unlessproductivity is perfectly correlated, productivity (and the manager’s marginalutility for capital) will be higher on average in the second period. Thus theoptimality of discretionary rollovers follows from the similarity of preferencesbetween headquarters and the division manager. Note that if headquarters andthe division manager have opposite preferences, offering the manager a tradeoffbetween capital now and capital later will result in the perverse allocation, i.e.,the manager will choose to allocate the larger budget to the less productiveproject.


7. Robustness

We made number of simplifying assumptions in deriving the results ofSection. In this section, we discuss the likely effects of modifying some ofthese.

7.1. Imperfect auditing

The analysis above considers only two possibilities, perfect auditing atcost q or no auditing (at zero cost). In some cases, however, the capitalbudgeting scheme does not take advantage of all the information revealedby the perfect audit. In such cases, if an auditing scheme that provides onlythe information actually used is cheaper than the full information system,it is optimal to use it instead. There are two such cases. First, it is clearthat headquarters never uses its ability to distinguish which project is moreproductive, given that their productivities differ. Suppose an imperfectauditing scheme exists that can distinguish only whether productivitiesdiffer and, if not, whether both are more or less productive, i.e., a schemethat results in the partition MM(1, 1)N, M(1, 2), (2, 1)N, M(2, 2)NN of the set ofsituations. If such a scheme costs less than q, headquarters will prefer touse it. The rest of our results will go through exactly as before except thatq would now be interpreted as the cost of the imperfect auditing system.Second, when q3[q

2, q

3), the only information headquarters actually

uses is whether t"(2, 2) or not. Therefore, if a scheme that can only distin-guish this case is cheaper than q, headquarters prefers to use that schemeinstead of the perfect auditing scheme. Moreover, even if q*q

3, headquarters

prefers to use the scheme that can only distinguish (2, 2) if it is cheaperthan q

3.

Aside from the obvious result that coarser auditing schemes will be used ifthey are sufficiently cheap, it is difficult to say anything definite about otherforms of imperfect auditing. Given the intuition above that delegation substi-tutes for auditing, it is likely that delegation is more extensive when less preciseauditing schemes are used.

7.2. Variations on managerial preferences for capital

The essential intuition regarding delegation is that delegation is useful whenthe manager’s preferences are similar to those of headquarters. By similar, wemean that the manager’s marginal utility for the more productive projectexceeds that for the less productive project, and when the productivities are thesame, the manager is indifferent among all project allocations. In this case, it isoptimal to delegate the project allocation across situations in which the totalcapital allocated is the same. This should be true even if the manager’s


preferences are nonlinear.12 If the manager has a strict preference for one of theprojects when their productivities are equal (but still prefers the more productiveproject when they differ), however, delegation will not be optimal when productivi-ties are equal and possibly not even when productivities differ if headquarterscannot distinguish this. For example, when q is high and the total allocation is thesame for all technology profiles, a strict preference for project 1, other things equal,will always lead the manager to choose to allocate the most possible to project1 whenever the productivities are the same. In this case, it can be suboptimal toallow any delegation, even though the manager will choose correctly when produc-tivities differ. Next, suppose the manager prefers the least productive project. In thiscase, it seems apparent that delegation is not optimal. Finally, suppose thatheadquarters is not sure whether the manager’s preferences are similar to those ofheadquarters. If the manager’s marginal utilities are sufficiently highly correlatedwith productivity (and independent of the project), however, it seems likely thatour results will go through. Thus, delegation appears to result from similarity inpreferences between headquarters and division managers. Given the analysis ofour model, it seems obvious that the division manager will never be givencomplete discretion in choosing the project allocation unless the manager’spreferences in this regard coincide precisely with those of headquarters.

7.3. More projects and/or more productivity levels

The main results of the current model will continue to hold even if there aremore projects or more productivity levels, with one minor modification. If thereare more than two projects and more than two productivity levels, the man-ager’s choice set when there is delegation cannot be of the form ‘allocate at leastk(q) to each project’. Instead, the manager’s choice set would include exactly onechoice for each possible technology profile over which headquarters is delega-ting the project allocation.13

7.4. Asymmetric projects

When projects are asymmetric, it is difficult to generalize unless the manager’spreferences for allocating a given total amount of capital coincide with those of

12Note that our results depend only on the monotonicity of b(t), not its specific values. This isa result of our assumption that the manager’s participation constraints are not binding. Results fromthe single-project case suggest, however, that, although the initial aggregate spending limit, K

11,

would depend on b if one of the participation constraints were binding, the qualitative nature of thesolution would not.

13Of course, the scheme described in Theorem 3 can be recast in this fashion as well. For example,when audit costs are low, the manager requests K*(1, 2), and there is no audit, the manager could begiven the choice set M (k, K

12!k), (K

12!k, k)N instead of ‘allocate at least k to each project’.


headquarters. If preferences in this regard are aligned, then one would expectdelegation to occur across situations in which the total allocation is the same.

8. Conclusions

This paper explains various features of observed capital budgeting schemes.In particular, we focus on the extent to which headquarters, representingowners, delegates the allocation of a given amount of capital across projects toa division manager. We find that an optimal scheme involves an initial capitalspending limit that can be negotiated upward. Requests by managers foradditional capital can be either ignored, partially granted without investigatingthe manager’s projects, or granted fully after a careful audit that shows therequest to be justified. When requests are ignored, the allocation of the givenamount of capital across projects is delegated, to some extent, to the manager.Whether requests are ignored, partially granted, or fully granted depends on thecost of auditing the projects. As audit costs increase, the aggregate capitalallocation is less sensitive to actual conditions in that requests are more oftenignored and less often fully granted. At the same time, as audit costs increase, theallocation of capital across projects is more likely to be delegated to themanager. Thus the model predicts a tradeoff between sensitivity of the aggregateallocation to project characteristics and the extent of delegation of the allocationacross projects. The intuition derived from the model suggests that delegation isdriven by similarity of preferences between headquarters and the manager.

If projects are carried out sequentially, delegation of the allocation acrossprojects can be interpreted as allowing managers to roll over capital from oneperiod to the next. When the division manager has all available information atthe outset, the extent to which rollovers are allowed depends negatively on auditcosts. When the division manager learns about the second project only in thesecond period and audit costs are high, rollovers are essentially always optimal.

A great deal of work remains to be done. We have abstracted from problemsassociated with multiple divisions, transfer pricing, and incentive compensationschemes. These issues must be addressed before we have a reasonable under-standing of capital budgeting processes in firms.

Appendix A.

¹heorem 1. If q'0, then a(t)(1 for all t.

Proof. Suppose t is such that a(t)"1. Define a new solution, (a@, k@), as follows:for any rOt, a@(r)"a(r) and k@(i, r)"k(i, r), for i3Mn, aN;a@(t)"1!e, k@(i, t)"k(a, t), for i3Mn, aN,


where e'0 will be defined below.It is easy to check that the objective function of headquarters’ problem at

(a@, k@) exceeds that at (a, k) by n(t)qe'0. Also, the left-hand sides of Eq. (2)sr arethe same at (a@, k@) as at (a, k) for all s and r. So are the right-hand sides of Eq. (2)srfor rOt. Therefore, (a@, k@) satisfies Eq. (2)sr for rOt. Now, the right-hand side ofEq. (2)st at (a@, k@)"eb(s) ) k(n, t). Since the left-hand side of Eq. (2)st at (a@, k@) isstrictly positive, (a@, k@) satisfies Eq. (2)st for e sufficiently small. Thus we haveshown that (a@, k@) satisfies all the constraints and increases the objective functionof headquarters’ problem, contradicting the fact that (a, k) is an optimalsolution. Q.E.D.

¸emma 1. (a) v@(1, x)"v@(2, y) implies either x"k*(1) and y"k*(2) or x(k*(1)and y(k*(2) or x'k*(1) and y'k*(2);

(b) g(1, x)#g(2, y!x)*2g(2, y/2).

Proof. Part (a) follows trivially from Eq. (12). For part (b), we have

g(1, x)#g(2, y!x)*g(2, x)#g(2, y!x)*2g(2, y/2),

where the first inequality follows from the assumption that g decreases with t,and the second follows from the assumption that g is convex in k. Q.E.D.

In what follows, we will make use of the following trivial result:

¸emma 3. If x and y are any two-dimensional vectors with x2*x

1and y

2*y

1or

x2)x

1and y

2)y

1, then x · y*(1 ) x)(1 ) y)/2. If, instead, x

2)x

1and y

2*y

1or

x2*x

1and y

2)y

1, then x ) y)(1 ) x)(1 ) y)/2. If any inequality is not strict, then

x ) y"(1 ) x)(1 ) y)/2.

¹heorem 2. ¹he following solution is optimal:

f q2, K

11(q

2), K

12(q

2), K

22(q

2), and k(q

2) satisfy Eqs. (12)—(15) and

K11

(q2)"K

12(q

2); q

3, K

11(q

3), K

12(q

3), K

22(q

3), and k(q

3) satisfy

Eqs. (12)—(14) and K11

(q3)"K

12(q

3)"K

22(q

3);

f kp(n, 1, 1)"K

11(q)/2 and k

1(n, 2, 2)"K

22(q)/2 for p3M1, 2N; k

1(n, 1, 2)"

k2(n, 2, 1)"k(q) and k

1(n, 2, 1)"k

2(n, 1, 2)"K

12(q)!k(q), where K

11, K

12,

K22

, and k are defined in Table 1;f For tO(1, 1), k(a, t)"k*(t) and k(a, 1, 1)"k(n, 1, 1);f a(t) is as defined in Table 1.

¹his solution has the following properties:

(a) 0(q2(q

3;

(b) k(q)(K12

(q)/2(K12

(q)!k(q), for all q'0;


(c) for q3(0, q2), K

11(q) increases with q, and K

12(q), k(q), K

12(q)!k(q), and

K22

(q) decrease with q;for q3[q

2, q

3), K

11(q)"K

12(q) and k(q) increase with q and K

22(q) decreases

with q;for q*q

3, K

11"K

12"K

22and k are independent of q.

(d) K11

(0)"2k*(1), K12

(0)"k*(1)#k*(2), K22

(0)"2k*(2), k(0)"k*(1);K

11(q)'2k*(1) for all q'0, K

12(q)(k*(1)#k*(2) for q)q

2and

K22

(q)(2k*(2), for all q'0;(e) for q3(0, q

2), K

11(q)(K

12(q)(K

22(q);

for q3[q2, q

3), K

11(q)"K

12(q)(K

22(q);

(f) For q"0, a(1, 2)"a(2, 1)"1!2k*(1)/[k*(1)#k*(2)], a(2, 2)"1!k*(1)/k*(2); a(1, 2)"a(2, 1) decreases with q for q3(0, q

2) and is identically

zero for q*q2; a(2, 2)'a(1, 2) decreases with q for q3(0, q

3) and is

identically zero for q*q3.

Proof. We start by proving the properties (a)—(f).

(a) First suppose q2)0. Then, by Eq. (15) and Lemma 1(a), k(q

2)*k*(1) and

K12

(q2)!k(q

2)*k*(2). Therefore K

11(q

2)"K

12(q

2)*k*(1)#k*(2)

'2k*(1). But Eq. (14) implies that K22

(q2)*2k*(2). This and the mono-

tonicity of F contradicts (13). Now suppose q2*q

3. Then Eq. (14) implies

that K22

(q2))K

22(q

3). Suppose K

11(q

2)(K

11(q

3). Then Eq. (12) implies

that k(q2)(k(q

3). But monotonicity of F implies that Eq. (13) cannot hold

at both q2

and q3. Therefore K

11(q

2)*K

11(q

3). This implies

K12

(q2)*K

12(q

3) since K

11(q

2)"K

12(q

2), and k(q

2)*k(q

3) and

K12

(q2)!k(q

2)*K

12(q

3)!k(q

3) from Eq. (12). Therefore,

q2)v*(1)#v*(2)!g(1, k(q

3))!g(2, K

12(q

3)!k(q

3) )

(2v*(2)!2g(2, K12

(q3)/2)"q

3,

where the second inequality follows from v*(1)(v*(2) and Lemma 1(b),and the equality follows from K

12(q

3)"K

22(q

3) and Eq. (14).

(b) This follows trivially from Eq. (12) and monotonicity of v@ in t.(c) First, Eq. (12) implies that K

12(q), k(q) and K

12(q)!k(q) all move in the

same direction with q. Then Eq. (15) implies that K12

(q), k(q), andK

12(q)!k(q) decrease with q for q(q

2. Also, Eq. (14) implies that K

22(q)

decreases with q for q(q3. Therefore Eq. (13) implies that K

11(q) in-

creases with q for q(q2. For q3[q

2, q

3), the facts that K

22(q) decreases

with q, K11,K

12, K

12and k move in the same direction with q, and

Eq. (13) imply that K11

(q)"K12

(q) and k(q) increase with q. Next, weshow that for q*q

3, K

11"K

12"K

22are independent of q. Suppose

not, i.e., suppose that for some qA, q@*q3, K

12(qA)'K

12(q@). Then Eq. (12)

implies that k(qA)'k(q@). Now, the assumption that K11,K

12,K

22and monotonicity of F in all three arguments implies that Eq. (13) cannot


hold at both qA and q@. Therefore, we must have K11"K

12"K

22inde-

pendent of q for q*q3. Finally, since K

12and k move in the same

direction with q, k is independent of q for q*q3.

(d) The fact that g(t, k*(t))"v*(t) and Eq. (14) imply K22

(0)"2k*(2). SinceK

22(q) decreases with q for q(q

3and K

22is independent of q for q*q

3,

K22

(q)(2k*(2) for all q'0. Next, Lemma 1(a) and Eq. (15) implyk(0)"k*(1) and K

12(0)!k(0)"k*(2). Adding these two inequalities

implies that K12

(0)"k*(1)#k*(2). Now Eq. (13) implies thatK

11(0)"2k*(1). Then, from part (c), k(q)(k*(1) and K

12(q)(k*(1)

#k*(2) for q)q2

and K11

(q)'2k*(1) for all q'0.(e) From (d), K

11(0)(K

12(0). Since K

11and K

12are monotone on (0,q

2) and

K11

(q2)"K

12(q

2), it must be that K

11(K

12on (0, q

2). Also, Eq. (14) and

(15), and Lemma 1(b) imply that K12

(q)(K22

(q) for q)q2. Now mono-

tonicity of K12

and K22

on [q2, q

3) and K

12(q

3)"K

22(q

3) imply that

K12(K

22on [q

2, q

3).

(f) This follows from Eq. (16) and the previous parts of this Theorem.

We now show that the solution is optimal. The solution is defined to satisfy thefirst-order conditions. The regularity conditions on v and v@ insure that therelevant equations as specified in Table 1 have a nonnegative solution. More-over, part (e) above and the definitions of a and the aggregate capital allocationsin Table 1 guarantee that a(t)3[0, 1). Therefore the technological constraints (3)are satisfied. Consequently, for optimality it suffices to check that the incentiveconstraints (2) and the complementary slackness conditions hold.14 First con-sider incentive constraint (2)

1,1,r:

left-hand side of (2)1,1,r

"b(1)K11

(q)"Right-hand side of (2)1,1,r

.

Thus Eq. (2)1,1,r

is binding. Since ktr'0 only for t"(1, 1), the complementaryslackness conditions hold.

Using the fact that b is increasing, Lemma 3, the definitions of a(1, 2) anda(1, 1), and k(q)(K

12(q)!k(q), we have the


*1 ) b(1, 2)[K11

(q)#a(1, 2)1 ) k*(1, 2)]/2

*1 ) b(1, 2)K11

(q)/2,

14This assumes that the first-order conditions are sufficient for an optimum. Although we canshow that the second-order necessary conditions hold (i.e., the bordered Hessian is positivesemidefinite), the strict positive definiteness of the bordered Hessian which would guaranteeoptimality is not satisfied. If one could show that all optimal solutions of headquarters’ probleminvolve a(1, 1)"0, it would follow that, at the conjectured solution, the bordered Hessian of thereduced problem, with 0 substituted for a(1, 1), is positive definite.


and the

right-hand side of (2)1,2,1,1

"1 )b(1, 2)K11

(q)/2,


"[1!a(2, 1)][b(1)(K12

(q)!k(q))#b(2)k(q)]

)1 )b(1, 2)K11

(q)/2,


"1 )b(1, 2)K11

(q)/2.

This shows that Eq. (2)1,2,r

holds. Since Eq. (2)2,1,r

is the same as Eq. (2)1,2,r

bysymmetry, we have that (2)

2,1,ralso holds. Finally, the


"b(2)[K11

(q)#2a(2, 2)k*(2)]*b(2)K11

(q)

"right-hand side of (2)2,2,r

,

so Eq. (2)2,2,r

holds. Q.E.D.To prove Theorem 5, we need some additional notation. Let

n1(t)"n(t, 1)#n(t, 2),

n2(t)"n(1, t)#n(2, t),

b(t)"E[b(tI2)Dt

1"t]/b(t),

andMkM

jN"argmax

kw0

Ev(tIj, k), for j 3 M1, 2N.

Note that, since b is strictly increasing, b(2))1)b(1) (with equality if andonly if tI

1and tI

2are perfectly positively correlated). From the definition of kM

j,

nj(1)v@(1, kM

j)#n

j(2)v@(2, kM

j)"0 for j 3 M1, 2N. (A.1)

It then follows from the fact that v@(1, k)(v@(2, k) for all k, that v@(2, kMj)'0, and

v@(1, kMj)(0, for j3M1, 2N.

Using the above notation, we can rewrite the incentive constraints as

b(1)[k2(1)!k

2(2)]*k

1(2)!k

1(1) (A.2)

andk1(2)!k

1(1)*b(2)[k

2(1)!k

2(2)]. (A.3)

Note that since b(2))b(1), the constraint implies that k2(2))k

2(1) and

k1(2)*k

1(1). Let j

1*0 be the Lagrange multiplier for constraint (A.2) and

j2*0 be the multiplier for constraint (A.3). The Kuhn—Tucker conditions for

this problem (in addition to complementary slackness) are

n1(1)v@(1, k

1(1))"j

2!j

1, (A.4)

n1(2)v@(2, k

1(2))"j

1!j

2, (A.5)

n(1, 1)v@(1, k2(1))#n(1, 2)v@(2, k

2(1))"b(2)j

2!b(1)j

1, (A.6)


and

n(2, 1)v@(1, k2(2))#n(2, 2)v@(2, k

2(2))"b(1)j

1!b(2)j

2. (A.7)

¹heorem 5. In the absence of auditing, allowing the manager discretion in rollingover capital across periods is optimal if and only if productivity is not perfectlycorrelated over time.

Proof. Let k(t) be an optimal solution to headquarters’ problem in this case.First, suppose k(1)"k(2). This, together with the Kuhn—Tucker conditions,requires that k

+(t)"kM

jfor all j, t3M1, 2N. We refer to this solution as the pooling

solution. Solving Eqs. (A.5) and (A.7) for j2

and imposing j2*0 implies

n(2, 1)v@(1, kM2)#n(2, 2)v@(2, kM

2)*n

1(2)b(1)v@(2, kM

1). (A.8)

We claim that this implies that kM1*kM

2, with equality if and only if productiv-

ity is perfectly correlated over time. To see this, suppose to the contrary thatkM1(kM

2. Then

b(1)v@(2, kM1)*v@(2, kM

1)'v@(2, kM

2)'v@(1, kM

2),

where the first inequality follows from b(1)*1, the second follows from the factthat v is strictly concave, and the third follows from our assumption thatv@(1, k)(v@(2, k) for all k. Since the left-hand side of Eq. (A.8) is a weightedaverage of v@(2, kM

2) and v@(1, kM

2), we have a contradiction. Therefore kM

1*kM

2.

Now suppose kM1"kM

2. This implies, from Eq. (A.1), that n

1(1)"n

2(1), which,

in turn, implies that n(2, 1)"n(1, 2). The above inequalities still hold exceptthat the middle one is now an equality. Now (A.8) holds if and only ifn(2, 2)"n

1(2)"n

1(2) b(1). Given n(2, 1)"n(1, 2), n(2, 2)"n

1(2) if and only

if productivity is perfectly correlated over time. Also, b(1)"1 if and only ifproductivity is perfectly correlated over time. This proves the claim.

Since kM1*kM

2, v@(1, k) is strictly decreasing, and v@(1, kM

j)(0, v@(1, kM

1)/ v@(1, kM

2)

*1 (with equality if and only if productivity is perfectly correlated over time).Now, solve Eq. (A.1) for v@(2, kM

j) in terms of v@(1, kM

j) for j3M1, 2N and substitute

into Eq. (A.8). After some manipulation, using the fact that v@(1, kMj)(0, we

obtain

n(2, 2)n2(1)!n(2, 1)n

2(2)*

v@(1, kM1)

v@(1, kM2)b(1)n

1(1)n

2(2)*n

1(1)n

2(2), (A.9)

with equality if and only if productivity is perfectly correlated over time (we haveused the facts that b(1)*1 and v@(1, kM

1)/ v@(1, kM

2)*1, with equality if and only if

productivity is perfectly correlated over time). The outer inequality in Eq. (A.9)simplifies to

!n(2, 1)n(1, 2)*n(1, 2)[1!n(2, 1)], (A.10)


with equality if and only if productivity is perfectly correlated over time. ButEq. (A.10) can hold only as an equality and then if and only if n(1, 2)"0.Therefore, Eq. (A.8) holds if and only if productivity is perfectly correlated overtime. This shows that pooling is not optimal if productivities are not perfectlycorrelated.

Now suppose productivity is perfectly correlated over time. In that case it isclear that (kM

1, kM

2) satisfy Eqs. (A.4), (A.5), (A.6) and (A.7) for j

2"0 and

j1"n

1(2)v@(2, kM

1)'0. Moreover, the complementary slackness conditions hold

since both constraints are binding for the pooling solution. Therefore, assumingthat the Kuhn—Tucker conditions are sufficient for an optimum, pooling isoptimal if productivity is perfectly correlated over time. Q.E.D.

References

Antle, R., Eppen, G., 1985. Capital rationing and organizational slack in capital budgeting.Management Science 31, 163—174.

Antle, R., Fellingham, J., 1990. Resource rationing and organizational slack in a two-period model.Journal of Accounting Research 28, 1—24.

Baiman, S., Rajan, M., 1994. Organization design for capital investment decisions. Unpublishedworking paper. University of Pennsylvania, Philadelphia.

Border, K., Sobel, J., 1987. Samurai accountant: a theory of auditing and plunder. Review ofEconomic Studies 54, 525—540.

Fellingham, J., Young, R., 1990. The value of self-reported costs in repeated investment decisions.The Accounting Review 65, 837—856.

Gertner, R., Scharfstein, D., Stein, J., 1994. Internal versus external capital markets. QuarterlyJournal of Economics 109, 1211—1230.

Gittman, L., Forrester, J. Jr., 1977. A survey of capital budgeting techniques used by major US.firms. Financial Management 6, 66—71.

Harris, M., 1987. Dynamic Economic Analysis. Oxford University Press, New York.Harris, M., Kriebel, C., Raviv, A., 1982. Asymmetric information, incentives, and intrafirm resource

allocation. Management Science 28, 604—620.Harris, M., Raviv, A., 1996. The capital budgeting process: incentives and information. Journal of

Finance 51, 1139—1174.Hirshleifer, D., Suh, Y., 1992. Risk, managerial effort, and project choice. Journal of Financial

Intermediation 2, 308—345.Holmstrom, B., Ricart i Costa, J., 1986. Managerial incentives and capital management. Quarterly

Journal of Economics 101, 835—860.Lambert, R., 1986. Executive effort and selection of risky projects. RAND Journal of Economics 17,

77—88.Mao, J., 1970. Survey of capital budgeting: theory and practice. Journal of Finance 25,

349—360.Ross, M., 1986. Capital budgeting practices of twelve large manufacturers. Financial Management

15, 15—22.Schall, L., Sundem, G., Geijsbeek, W. Jr., 1978. Survey and analysis of capital budgeting methods.

Journal of Finance 33, 281—287.Scott, D., Jr., J. Petty, II, 1984. Capital budgeting practices in large American firms: a retrospective

analysis and synthesis. Financial Review 19, 111—123.


Stanley, M., Block, S., 1984. A survey of multinational capital budgeting. Financial Review 19,36—54.

Stein, J., 1997. Internal capital markets and the competition for corporate resources. Journal ofFinance 52, 111—133.

Taggart, R. Jr., 1987. Allocating capital among a firm’s divisions: hurdle rates vs. budgets. Journal ofFinancial Research 10, 177—190.

Thakor, A., 1990. Investment ‘myopia’ and the internal organization of capital allocation decisions.Journal of Law, Economics, and Organization 6, 129—154.

Townsend, R., 1979. Optimal contracts and competitive markets with costly state verification.Journal of Economic Theory 21, 265—293.


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