Study on Capital Rationing and Managerial Retention

Description
Capital Rationing is the acquisition of new investments based some factors. These factors include: the recent performance of other capital investments

Capital rationing and managerial retention: the role of external capital?
Ying-Ju Chen† Mingcherng Deng‡

Abstract In modern businesses, ?rms face new challenges of managerial retention in capital budgeting process. We consider a model in which a manager privately observes the capital productivity of a project and has access to multiple outside ?nancing options. We show that if the manager can obtain funding from either internal or external capital (but not both), the ?rm may exclude highly pro?table investment projects but fund those projects that have moderate capital productivity, even when there is no limit on capital allocation. Furthermore, the ?rm may voluntarily impose capital rationing in order to keep the projects within the ?rm, even though it has su?cient capital to fund such pro?table projects. However, if the ?rm can utilize both the internal and external capital, highly pro?table projects are always retained and the voluntary capital rationing is not optimal. Our analysis identi?es testable empirical predictions on the association between capital budgeting and external capital. Keywords: capital rationing, external capital, agency theory, internal control

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Introduction

Capital allocation decisions are one of the most in?uential decisions in a ?rm’s long-term ?nancial health (Baldwin and Clark (1992) and Porter (1992)). Theory indicates that with a goal of maximizing its shareholders’ value, a ?rm accepts all pro?table projects and provide su?cient capital to utilize this opportunity. In practice, ?rms often operate under a capital constraint even though they have su?cient capital to fund all pro?table projects. This capital rationing process – allocating a limited amount of capital to pro?table capital budgeting projects – has been widely adopted in
?

We thank Ramji Balakrishnan (the Editor), Thomas Hemmer (the Associate Editor), and the reviewers for very

constructive and insightful suggestions that signi?cantly improved the quality of the paper. All remaining errors are our own. † University of California, 4121 Etcheverry Hall, Berkeley, CA 94720; e-mail: [email protected]. ‡ Corresponding author; University of Minnesota; 321 19th Ave S 3-131, Minneapolis, MN 55455; e-mail: [email protected].

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various ?rms across di?erent industries. In a survey by Mukherjee and Hingorani (1999), 64% of Fortune 500 ?rms in their sample frequently place a quantity limit on the internal capital available for investment plans. Of those ?rms that adopt capital rationing, 82% indicate that such rationing is made internally by executive managers rather than by external lenders. Senior managers argue that capital rationing is not irrational behavior but is instead a reaction to real problems. In particular, survey evidence shows that the main reason for capital rationing is the reluctance to borrow the external capital. This ?nding is puzzling, because if the expected project return is su?ciently high, then theoretically ?rms should still fund pro?table projects by utilizing the external capital. In this paper, we articulate potential economic consequences of utilizing external capital, thereby resolving the aforementioned capital rationing phenomenon. Intuitively, external capital allows a ?rm to exploit more growth opportunities when internal funds are insu?cient (Gutmann (1967)). However, the speed of technological change in recent years has increased both the importance and the di?culty of retaining managerial expertise. Now, it is the managers who identify pro?table projects, propose necessary ?nancial resources, and attract talents to execute the projects. Managers can easily walk away with their information and knowledge innovations and seek for funding with other employers or venture capitalists. Indeed, Gertner et al. (1994) points out that utilizing external capital may a?ect the residual rights of control over ?rms’ assets like intellectual capital, and consequently the choice of these capital resources has ?rst-order real e?ects on managers’ incentives. (Also see similar arguments by Dutta (2003) and the references therein). Thus, when making capital-budgeting decisions that involves external capital, ?rms need to consider an economic trade-o? between the exploitation of growth opportunities and potential cost of managerial retention and losing residual rights. We construct a principal-agent model in which an owner (principal) contracts with a manager (agent). The manager privately observes the capital productivity of an investment project and seeks to fund the project. The manager may request capital funding either from the owner or from an external market. When the project’s capital productivity increases, the owner can realize higher investment return, but the manager can also attract higher external capital. Through this framework, we intend to address the following research questions: What are the factors that deter ?rms from simply utilizing external capital and matching managers’ outside opportunities? How should ?rms design capital budgeting mechanism in the presence of such a retention problem? Could ?rms endogenize the external capital obtained by managers? We consider two scenarios in order to illustrate the e?ect of external capital. First, we consider the scenario in which the owner will lose the whole residual right of the investment project when the project is funded by the external capital. That is, if the manager (professor, product developer, or software designer), chooses to fund the project via external capital, she will walk away with relevant intellectual capital such as patents and the owner will not retain any investment return from the

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project. This scenario is labelled as exclusive capital budgeting. Indeed, Bhide (2000) reports that founders of more than 70 percent of the Inc. 500 ?rms admitted that they developed and marketed ideas identi?ed in their previous employment. Ittner et al. (2003) also argue that managerial retention becomes a more important issue when managers can walk away with information and technology. In the second scenario, the capital allocation may utilize a mix of both internal and external capital. We label this alternative scenario as centralized capital budgeting, because the owner here still retains partial residual right of the project, albeit funded by external capital. One possible example of this setting is that the owner may have a joint venture with external capital that the manager might obtain externally, thereby endogenizing the manager’s external opportunities. In the ?rst scenario, since the owner does not observe the capital productivity, it o?ers the manager a menu of contracts. If the manager chooses to fund the project internally, the owner funds the project via internal capital and requests a repayment from the manager after the fact. However, if the manager opts for external capital, the owner must forgo this investment opportunity. The owner here faces an economic trade-o?. On the one hand, in order to retain the manager, the owner must provide su?cient internal capital to match the manager’s external opportunities or reservation utility. On the other hand, since the owner cannot observe the capital productivity, it must design the menu appropriately to eliminate the manager’s incentive to overstate his capital productivity. To reduce this incentive for misreporting, the owner must pay the manager information rent, which consequently reduces the owner’s investment return from the project. In this exclusive capital budgeting case, we demonstrate that the owner may forgo highly pro?table projects but fund those projects that have moderate capital productivity (cf. Antle and Eppen (1985) and Dutta (2003)). Counterintuitive as it sounds, this result is actually quite natural. When the manager has high capital productivity, his external opportunity (reservation utility) is also high. Thus, if the owner would like to match the manager’s outside opportunity, it inevitably has to reduce the repayment signi?cantly. But a lower repayment may induce the ine?cient manager to mimic the e?cient one. As the owner is unable to observe the manager’s true capital productivity, the owner might further reduce the lower-type manager’s capital allocation and repayment to prohibit this temptation, but this will make the owner’s investment return signi?cantly lower. Thus the owner may choose to forgo these highly pro?table projects in order to balance between ex post investment e?ciency and ex ante incentive compatibility. Furthermore, our analysis also predicts that the ?rm may voluntarily impose capital rationing, even though it has su?cient capital to fund such pro?table projects. This strategy provides appropriate incentives for the manager to voluntarily disclose the pro?tability of the project and at the same time optimally balances the economic trade-o? between retaining the project and matching with the manager’s external opportunity. This result may help explain an empirical puzzle in which many ?rms voluntarily impose a quantity limit on capital spending, even though they have su?-

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cient capital to fund such pro?table projects. Thus, our analysis documents that the availability of external capital may give rise to severe ine?ciencies in capital budgeting process. Namely, the ?rm may forgo highly pro?table projects and impose capital rationing on pro?table projects. Our results may shed some light on the economic rationale of this market phenomenon. In the second scenario, the capital allocation may utilize a mix of both internal and external capital. We ?nd that (internal) voluntary capital rationing is never optimal, and the owner never forgoes any highly pro?table projects. Further, the owner may utilize external capital as an additional tool to reduce the information rent paid for the manager. The optimal incentive plan stipulates a mix of internal and external capital; it balances the cost of capital, the cost of inducing the manager’s truthful report, and the capital return from the investment project. We ?nd that the optimal amount of external capital increases in the manager’s capital productivity; however, because of the information asymmetry, the owner always distorts the allocation of external capital in order to limit the manager’s information rent. In this sense, the inability of retaining the high-type manager as well as the voluntary capital rationing vanish if the ?rm could have a better control over its ?nancing policies. The extant literature (e.g., Antle and Eppen (1985) and Dutta (2003)) has shown that the ?rm may o?er second-best capital allocation (or impose a higher hurdle rate) in the presence of information asymmetry. Capital is rationed because the allocation is lower than the ?rst-best level.1 But, it is never optimal to forgo pro?table investment projects (after being adjusted for information rent) in these models. Speci?cally, we document two new ?ndings that are absent in the extant literature. First, in the same setting as in Antle and Eppen (1985), we show that the ?rm may voluntarily impose capital rationing (not allocate the maximum capital) because of the presence of external capital. Moreover, the ?rm may forgo highly pro?table projects but fund only those projects that have moderate capital productivity. Second, if the ?rm can still retain partial residual right of the project, albeit funded by external capital, such capital rationing for highly pro?table projects can never be optimal. In fact, external capital can serve as an additional screening tool to reduce the information rent paid to the manager. Taken together, our analysis provides an economic rationale for capital rationing — ?rms may forgo highly pro?table projects only if utilizing external capital may lead to the loss of residual right of the investment projects. Conventional accounting wisdom argues that a ?rm has no bene?t to reject a pro?table project when there is su?cient capital to utilize this opportunity. However, Mukherjee and Hingorani (1999) shows that 64% of Fortune 500 ?rms in their sample frequently place a quantity limit on the internal capital available for investment plans. Capital rationing receives little attention despite its prevalence in practice. In a multi-divisional setting, Balakrishnan (1995) shows that an owner may ?nd it optimal to ration capacity allocation when divisions claim low realizations of productivity.
1

We thank an anonymous reviewer for pointing this out to us.

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Zhang (1997) and Paik and Sen (1995) demonstrate that when the manager’s e?ort and capital are substitutes, capital rationing should be adopted because of the moral hazard associated with the unobservable e?ort. In this paper, we show that capital rationing may result from the economic loss of external ?nancing. Optimal capital budgeting mechanisms are also investigated in the accounting literature. Most papers, stemming from Antle and Eppen (1985), focus on identifying the means by which ine?ciencies in capital allocation can be mitigated, such as the power of commitment in a repeated problem in Antle and Fellingham (1997), the introduction and design of information systems in Antle et al. (2001) and Arya et al. (2000). Balakrishnan (1991) shows that how information acquisition may a?ect the resource allocation decisions in the presence of information asymmetry. Another stream of research examines the reduction in ine?ciency by restructuring the resource allocation. Interestingly, Balakrishnan (1993) demonstrate that given a common resource, an owner may bene?t from ?xed cost allocations not only from inducing optimal utilization of available capacity but in deciding how much capacity to acquire. Antle et al. (1999) and Arya and Glover (2001) show how bundling projects and considering them simultaneously can ease incentive problems. In contrast, we do not propose a way to mitigate asymmetric information, but instead focus on analyzing the e?ect of external capital on existing capital budgeting mechanisms. In terms of modelling, this paper is related to the principal-agent problems in which the agent’s reservation utility is type-dependent as in Jullien (2000) and Maggi and Rodriguez-Clare (1995). Maggi and Rodriguez-Clare (1995) consider a regulation scenario in which a principal does not observe the agent’s private cost information and the agent has access to outside options. Jullien (2000) investigates a rather general and abstract setting in which outside options could refer to the available alternatives or could arise as a form of competition. In a similar vein, Dutta (2008) characterizes optimal pay-performance sensitivities of compensation contracts when a manager has outside employment opportunities correlated with his private information. However, these papers do not consider the possibility of allowing the agent to use both inside and outside options. We illustrate that the availability of external capital may naturally give rise to nonlinear reservation utility, which consequently results in very special incentive plans for managerial retention. In this regard, our analysis complements that of Dutta (2008), where reservation utility is assumed to be linear in the agent’s private information. The remainder of the paper proceeds as follows. Section 2 describes the economic setting. Section 3 examines capital allocation under exclusive capital budgeting. Section 4 investigates the centralized capital budgeting scenario. Section 5 provides an explanation for voluntary capital rationing. Section 6 concludes. All the proofs are in the appendix.

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2

Model

We consider a principal-agent model in which a ?rm’s risk-neutral owner (principal) contracts with its risk-neutral manager (agent). The manager owns an investment project, privately observes its marginal capital productivity (denoted by ? > 0), and may request capital funding either from the owner or from an external market. The owner has access to the maximum internal capital K and incurs a cost rI if it allocates capital I to the manager, where r is the cost of internal capital. The owner’s prior belief of ? is represented by the distribution function F (?), with f (?) being its density function and a monotone hazard rate: H (?) ?
1?F (?) f (? )

is decreasing in ?.

As a departure from the existing literature, the manager can access internal and external ?nancing where multiple options are available. Each outside ?nancing option is speci?ed by a pair (e, c(e)), where e is the amount of external capital raised, and c(e) with c(0) = 0 is the cost of external capital from the manager’s viewpoint. For example, the manager may access debentures, lines of credit from creditors, grants, venture capital, etc. In such cases, the cost of external capital c(·) may re?ect the investment return requested by a commercial bank or the risk of leveraging the budget across di?erent investment projects. There is a transaction cost associated with external capital (as in Stulz (1990)). Speci?cally, we assume the investment returns from external capital are given by ??e, where ? < 1 represents the extent to which potential agency con?ict may a?ect the e?ciency of external capital. Thus, the manager’s net payo? from external capital is represented by ??e ? c(e). In the university-professor example, (e, c(e)) may correspond to how costly for the professor to get his project funded via an alternative institution outside the university. Note that from this representation, the larger the parameter ?, the smaller the agency cost associated with external capital and the more e?cient external capital. As an extreme example, when ? = 0, the outside option degenerates and the capital allocation is determined a la Antle and Eppen (1985).2 We now derive the manager’s “reservation utility,” which is de?ned as the maximum payo? that he obtains from the available outside options. Suppose that the manager’s type is ?. After self-selecting his favorite external capital option e, the manager obtains the value of external capital E (?) = max {??e ? c(e)} ,
e? 0

(1)

where the maximum is assumed to exist (otherwise the manager would obtain an unbounded reservation payo?). The following lemma establishes the structural properties of E (?).
2

In certain scenarios, external capital may involve less transaction costs due to more generous decision rights and

less bureaucratic constraints. In this case, ? > 1 should be imposed alternatively. However, this alternative setting does not a?ect our main results regarding voluntary capital rationing, but will give rise to some di?erent empirical implications in the centralized capital budgeting (see Section 4 for details). We thank an anonymous referee for his insightful observations.

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Lemma 1. E (?) is increasingly convex in ?, and E (0) = 0. We make two points from Lemma 1. First, in the presence of external capital, the manager is better o? exploiting investment opportunities outside the ?rm. Second, as the manager’s capital productivity is higher, the investment project is more valuable, and thus he will ?nd it increasingly more attractive to utilize external capital. These two reasons therefore suggest that the manager’s outside opportunity is a convex function of his type. That is, a manager with higher marginal capital productivity also possesses higher reservation utility and that the marginal value of utilizing external capital increases in the capital productivity of the manager’s investment project.3 In what follows, we consider two di?erent scenarios to capture the economic e?ects of external ?nancing: 1) Exclusive capital budgeting: the investment project can be funded through either internal or external capital (not both), and the funding source is at the manager’s discretion. 2) Centralized capital budgeting: the owner has full control rights over capital allocation both internally and externally; thus, the amounts of internal capital and external capital are chosen by the owner as a bundle and the manager must follow this decision.

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Exclusive Capital Budgeting

In this section, we investigate a situation in which investment projects can be ?nanced through either internal capital or external capital, but not both. In particular, if the manager, (university professor, product developer and software designer), chooses to fund the project via external capital, he will walk away with relevant intellectual capital such as patents and the owner will not retain any investment return from the project. Let us start with the description of the incentive plans. The available menu speci?es {(Ie , te (Ie )}, where Ie is the internal capital and te (Ie ) is the corresponding repayment that the manager needs to return to the owner. By the Revelation Principle, we can further replace the above menu by the incentive plans {(Ie (?), te (?))}, where Ie (?) is the internal capital intended for type-? manager
3

It is worth emphasizing that one cannot directly apply the approach by Jullien (2000) to this capital rationing
F (? )? ? I f (? )

context. A crucial assumption in Jullien (2000) is the strict quasi-concavity of the “virtual surplus.” In our context this assumption requires that the function ?I + be strictly quasi-concave for all ? ? [0, 1]. One can easily

verify that this condition cannot be satis?ed universally in our context. Consider for example that ? is uniformly distributed over [0, 1]. The assumption of quasi-concavity requires that (2? ? ? )I2 > min{(2? ? ? )I1 , (2? ? ? )I3 }, ?I1 < I2 < I3 , ??, ?? ? [0, 1].
1 However, for every ? ? [0, 2 ], we can ?nd ? = 2? such that the above inequality fails. Thus, we need to consider an

alternative approach tailored to our speci?c context in order to characterize the optimal incentive plan.

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and te (?) is the corresponding repayment. The sequence of events is as follows. 1) The manager observes capital productivity of the investment project. 2) The owner announces the menu of incentive plans to the manager. 3) The manager selects an incentive plan and determines the amount of external capital if he is delegated to do so. 4) The investment return is realized, and the owner then obtains the promised repayment from the manager. To derive the optimal menu of incentive plans, we ?rst characterize the manager’s payo? upon ˆ and accepting a speci?c plan. A manager with capital productivity ? may pretend to be type ? ˆ), te (? ˆ)). In this case, the manager’s utility function is given by choose the incentive plan (Ie (? ˆ ?) = ?Ie (? ˆ) ? te (? ˆ). Ue (?, (2)

In contrast, if a type-? manager opts to not receive any internal capital, he should adopt the most pro?table outside option and obtain E (?). We use the following notation to characterize the manager’s utility. Let Me ? [?, ?] be the set of capital productivity in which the project is funded by internal capital, and [?, ?] \ Me is the set of capital productivity in which the manager opts out for external capital. Furthermore, let Ue (?) ? Ue (?, ?) represent the manager’s utility under truthful reporting. Under truth-telling of the manager, the optimal mechanism for the owner solves the following problem: ?
{Ie (·),te (·),Me }

max

? ?Me

[te (?) ? rIe (?)] dF (?)

s.t. (IC) (IR)

ˆ ?), ??, ? ˆ ? Me , Ue (?) ? Ue (?, ˆ ? Me , Ue (?) ? E (?), ??, ?

where the incentive compatibility ensures that the manager will truthfully report his capital productivity ? and the individual rationality constraint guarantees that the manager participates. We now characterize the optimal incentive plans. When external capital is accessible to the manager, the owner has to make two kinds of strategic decisions. First, how should the owner select the investment project funded by internal capital? That is, how should the owner select the appropriate set Me ? Second, provided that the set Me is determined, how should the owner determine the amount of internal capital for the manager, knowing that he has private information about capital productivity of the investment project? Proposition 1. Let ??? denote the unique solution to (??? ? r)E ? (??? ) ? E (??? ) = 0. Voluntary ¯ and 2) K > E ? (??? ). capital rationing emerges as the owner’s optimal incentive plan if 1) ??? ? ? ¯ In this case, there exists a hurdle rate ? and a corresponding rate ? where 0 < ??? < ? < ? ? ? such that

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1. The owner allocates the maximum internal capital K for capital productivity in the interval [?, ? ]. The manager whose capital productivity falls in [?, ? ] receives a positive information rent except for the boundary cases: ? = ? and ? = ? . 2. The manager with capital productivity ? ? [??? , ? ) su?ers from voluntary capital rationing. The optimal allocation of internal capital increases in the manager’s capital productivity and is determined by Ie (?) = E ? (?), and the manager is indi?erent between accepting interval capital from the owner and opting out for external capital. 3. The manager either with a su?ciently low capital productivity ? ? [0, ??? ) or with a su?¯] opts out for external capital and receives E (?) in ciently high capital productivity ? ? (?, ? equilibrium, i.e., Me = [??? , ? ]. We start with explaining the owner’s economic trade-o? between funding the investment project via internal capital or relinquishing it to external capital. In order to retain the manager, the owner must provide su?cient internal capital to match with his external opportunities or his reservation utility (i.e., Ue (?) ? E (?)). Thus, the presence of external capital reduces the owner’s net payo?. On the other hand, since the owner cannot observe how pro?table the project is, the owner needs to design appropriate incentive plans to eliminate the manager’s incentive to overstate the capital productivity and raise more internal capital. The solution to this problem exhibits four regimes and is illustrated in Figures 1 and 2. In the graph, the solid line denotes the manager’s utility from internal capital Ue (?) whereas the dotted line is his utility from external capital E (?). In the following we elaborate in detail on the optimal incentive plans, the manager’s utility, and the owner’s incentive. [Add Figure 1 here] [Add Figure 2 here] In the regime ? ? [?, ? ], the manager’s utility from internal capital is larger than that from external capital. When the manager’s capital productivity is larger than the hurdle rate ? , the owner allocates the maximum internal capital K to all types of managers above this cuto? level. The common capital K implies that the requested repayment, te , from the manager has to be identical. The manager obtains a positive utility (i.e., information rent) in this regime, and his utility is increasing in the capital productivity. Since the manager’s utility is larger than his external opportunities, the optimal internal capital allocation is not a?ected by this second-best contract; consequently, we obtain similar results as in Antle and Eppen (1985). In the regime ? ? [??? , ? ), voluntary capital rationing may emerge endogenously as the optimal 9

capital allocation. In this case, the manager’s utility from internal capital is matched with that from external capital. To understand the intuition, consider the manager whose type is slightly lower than ? . The owner intends to keep projects with lower capital productivity, but is still pro?table from the owner’s perspective. Because the internal capital allocation is at the maximum level at ? = ? , the remaining tool is to lower internal capital allocation and charge a lower repayment less e?cient managers ? ? [??? , ? ). This allocation serves two purposes: 1) it ensures that the manager receives his reservation utility; and 2) it also ensures that the manager with the highest capital productivity, who originally chooses the maximum internal capital, will not deviate to select the lower capital instead. The optimal internal capital is distorted from the maximum internal capital K ; thus, in the regime of [??? , ? ), voluntary capital rationing emerges as an optimal solution to keep the pro?table projects. These results are consistent with Maggi and Rodriguez-Clare (1995). When the managers have type-dependent reservation utilities, the optimal capital is driven by two factors separately: in one region, the capital is driven entirely by the local incentive compatibility, i.e., the “second-best” solution (which is K in our context from Antle and Eppen (1985)); in the other region, the optimal capital is designed such that the manager’s reservation utility is matched. Thus, naturally there are two regimes as speci?ed in our solution. The question is whether the owner wants to give the second-best solution (K ) to the relatively high types or relatively low types. Obviously, distorting the relatively low types results in a lower deadweight loss, because the low-type manager’s capital productivity is lower. Given that, the owner should match the reservation utilities of the relatively low types, rather than distorting the relatively high types. This explains why the maximum capital is o?ered to the relatively high types ? ? [?, ? ] while quantity rationing is given to less e?cient managers ? ? [??? , ? ). The optimal value of ? balances this trade-o?: when the owner increases ? , it gains from extracting the surplus (information rent) from more types of managers, but at the same time su?ers from the ine?ciency of o?ering lower capital to less e?cient managers. Surprisingly, the owner turns down the investment project when the manager is not only with relatively low capital productivity ? ? [0, ??? ), but also with relatively high capital productivity ¯]. Counterintuitive as it sounds, that the owner may forgo highly pro?table projects in ? ? (?, ? the presence of outside options is actually quite natural. When the manager has very high capital productivity, his reservation utility is also high. Thus, if the owner matches the manager’s outside opportunity with internal capital, it inevitably has to reduce the repayment signi?cantly, for the internal allocation is already capped by the maximum level K . Nevertheless, this may induce the managers that have a relatively lower capital productivity to choose these incentive plans intended for the higher-type manager, which forces the owner to forgo highly pro?table investment projects. It is worth emphasizing that this cut-o? point ??? is di?erent from the hurdle rate in Antle and Eppen (1985): There, the owner discards the projects with capital productivity less then ?? , the

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unique solution that solves ?? = r + H (?? ). In contrast with ?? , the threshold here ??? is a?ected by the transaction cost ?. This implies that the ine?ciency that arises from using external capital also a?ects the required rate of return for the internal capital. This is because the owner ?nds it optimal to design the internal incentive plans for the manager with relatively low capital productivity in order to match his reservation utility, and consequently the economic trade-o? is altered due to the existence of external capital. Therefore, we have identi?ed a previously ignored interplay between ¯, the ine?ciency of external capital and the internal incentive plan. Incidentally, when ??? ? ? if the owner insists on using capital rationing plan, the ?rm has to abandon all the investment projects in order to avoid the manager from misreporting his capital productivity. This is not a preferred solution, because the owner cannot materialize any investment opportunity due to the severe cannibalization problem. In this case, the owner would rather o?er a single incentive plan with an appropriately designed hurdle rate as suggested by Antle and Eppen (1985), as this single incentive plan avoids the sophisticated self-selection behavior from the manager. On the other hand, if the size of internal capital is relatively small (that is, K ? E (??? )), the owner cannot pool enough resources internally to match the manager’s bene?t from external capital. Thus, the owner is forced to provide the maximum internal capital once the manager’s investment project gets approved.4
?

4

Centralized Capital Budgeting

We have documented that when the owner forces the manager to use either the internal capital or external capital, the capital allocation is highly ine?cient because 1) the manager with high capital productivity may be turned down; and 2) even if the investment project is funded by the owner, the manager may still su?er from voluntary capital rationing. A natural question is whether the owner can achieve a higher payo? by utilizing both the internal and external capitals. One possible example of this setting is that the owner may have a joint venture with external capital that the manager might obtain externally, thereby endogenizing the manager’s external opportunities. In light of this practice, we investigate the scenario in which the owner may monitor both
4

We shall emphasize that this result is not driven by the assumption of linear cost of internal capital (i.e., rI

used in our model) or by the structural assumption on the cost of external capital c(e) (note that we do not impose any monotonicity or convexity on c(e) at this point). Rather, this result is driven by the economic tradeo? between satisfying the manager’s truthful-telling (IC) constraint and the outside opportunities (IR). The ?rm wants to keep all pro?table projects inside by allocating the maximum amount of internal capital; however, it cannot assign the same maximum amount of internal capital to all types of managers, because the manager may opt out for outside opportunities and/or misreport his type by selecting a project with lower capital allocation. We refer the allocation of internal capital to voluntary capital rationing, because it deviates from the maximum amount of internal capital.

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the internal and external capital. This scenario is labelled as the centralized capital budgeting, because it allows the owner to retain partial residual right of the project and endogenizes the manager’s outside opportunities. We consider a centralized capital budgeting setting in which the manager reports his private information to the owner, who then decides amount of capital allocation by utilizing a mix of internal and external capital. This arrangement represents the typical capital budgeting process within a centralized ?rm in which the owner makes the capital investment decisions based on the divisional managers’ proposals. In such a scenario, because the project is still partially funded by the ?rm, the principal (owner) can observe the actual amount of external capital borrowed by the manager. Conceptually, one can imagine that the owner speci?es a menu of incentive plans that indicate internal capital and how the ?rm will match the external capital that the manager brings in from outside sources. To overcome the asymmetric information problem vis-a-vis the manager, the owner o?ers a menu of ˆ), Ic (? ˆ), tc (? ˆ)) where incentive plans to the manager. This menu of incentive plans is speci?ed as (kc (? ˆ) = Ic (? ˆ) + ec (? ˆ) is the total capital intended for the type-? manager to use for the investment kc (? ˆ)) and externally (ec (? ˆ)), and tc (? ˆ) is the corresponding repayment. project from both internally (Ic (? ˆ) to the manager for implementing his investment project, and the The owner provides only Ic (? ˆ) from an external market. We assume that the cost of external capital manager needs to borrow ec (? is increasingly convex in the amount of the external capital e, that is, c? (e) > 0, and c (e) > 0. We now characterize the optimal incentive plans under centralized capital budgeting. If a ˆ, the owner shall select the incentive plans Ic (? ˆ), ec (? ˆ), and type-? manager reports to be type-? ˆ) for him. Thus, the expected utility of the manager is given by tc (? ˆ ?) = ?Ic (? ˆ) ? tc (? ˆ) + ??ec (? ˆ) ? c(ec (? ˆ)). Uc (?, Note that as long as the manager individual rationality constraint is satis?ed, the owner can make ˆ), contingent on both internal and external capital. De?ne the required repayment to the owner, tc (? Uc (?) ? Uc (?, ?) as the equilibrium payo? received by the type-? manager. We can write down the owner’s optimization problem as follows: ?
{kc (·), Ic (·),tc (·)} ¯ ?
??

max

[tc (?) ? rIc (?)] dF (?)

?

s.t. (IC) (IR)

ˆ ?), ??, ? ˆ ? [?, ?], Uc (?) ? Uc (?, Uc (?) ? 0, ?? ? [?, ?],

where the two inequalities are respectively the incentive compatibility and the individual rationality constraint. The optimal incentive plans can then be characterized as follows. Proposition 2. Under centralized capital budgeting,

12

1. The owner allocates the maximum internal capital, K , to the manager if and only if ? ? ?? , where ?? is the unique solution that satis?es ?? = r + H (?? ). 2. The owner allocates a positive amount of external capital if and only if ? ? ?c where ??c ? H (?c ) = 0. The optimal amount of external capital ec (?) is determined by c? (ec (?)) = ?? ? H (?) for all ? ? ?c , and ?ec (?)/?? > 0, ??. The hurdle rate for external capital ?c decreases in the e?ciency of external capital (??c /?? < 0). 3. The optimal amount of external capital increases in capital productivity of the project (?ec (?)/?? > 0). Moreover, it is smaller than the e?cient (?rst-best) amount (i.e., ec (?) < eF B (?)), where eF B (?) is the unique solution to ?? ? c? (eF B (?)) = 0. 4. There exists a cut-o? point ?c such that ?c ? ?? if and only if ? ? ?c . Proposition 2 summarizes the key results under centralized capital budgeting. First, despite the presence of external capital, the hurdle rate for internal capital ?? under centralized capital budgeting is identical to that in Antle and Eppen (1985), and voluntary capital rationing as documented in Proposition 1 disappears. When the owner makes the investment decision, he simply allocates the maximum internal capital K whenever the manager’s capital productivity is larger than the hurdle rate (? ? ?? ). Now the owner has two contracting parameters: kc (?) and Ic (?). Consequently, the owner ?nds it optimal to o?er the manager the e?cient internal capital K , but screen the manager with di?erent capital productivity by requesting for di?erent amounts of external capital. We also ?nd that the owner now distorts downwards the allocation for external capital in order to re?ect the cost of inducing the manager to report truthfully. Thus, when the manager has incentives to under-report capital productivity, the owner faces a trade-o? between inducing the manager’s truthful report and achieving the ex post investment e?ciency. Proposition 2 shows that the owner allocates a positive amount of external capital if and only if capital productivity is larger than a threshold ?c > 0. This implies that the hurdle rate rationing introduced in Antle and Eppen (1985) appears in the allocation of external capital when centralized capital budgeting is invoked. This hurdle rate decreases when the e?ciency of external capital is improved, i.e., d?c /d? < 0, because the owner expects to receive higher return from the investment project.Finally, the optimal incentive plan stipulates a mix of internal and external capital, and balances the cost of capital, the cost of inducing the manager’s truthful report, and the capital return from the investment project. For example, when external capital is as e?cient as internal capital (? = 1), the owner simply compares the marginal costs between two sources of capital. If there is a cap on the amount of total investment in the project, the owner then would prefer to

13

borrow from the source of capital that is of lower marginal capital costs.5 [Add Figure 3 here] [Add Figure 4 here] Figures 3 and 4 present the optimal incentive plans under centralized capital budgeting for cases when ?c ? ?? and ?c > ?? , respectively and the comparative statics of the cut-o? threshold ?c on the e?ciency of external capital ?. Consider a simpli?ed case in which there are two sources
c and ? c of external capital with di?erent levels of e?ciency (?H and ?L , and ?H > ?L ). Let ?H L

correspond to the hurdle rates of ?H and ?L , respectively. First, when external capital is relatively e?cient (? > ?c ), the hurdle rate for external capital is lower than that for internal capital ?? . In this case, the investment projects with relatively lower capital productivity ? ? [?c , ?? ] will be funded solely by external capital rather than internal capital. But the investment projects with higher capital productivity ? ? [?? , ?] will be funded both the maximum internal capital K and external capital ec . Second, when the e?ciency of external capital is low (? < ?c ), the owner may fund the investment project only by internal capital and no external capital is utilized unless capital productivity is high enough. As external capital becomes very ine?cient, the capital allocation in our setting resembles that in Antle and Eppen (1985). Third, when external capital becomes more e?cient (? increases), the hurdle rate for external capital ?c is lower and the total capital allocation kc is higher. This relation is demonstrated by the dotted line in the ?gure in which ?H indicates the case of relatively higher e?ciency of external capital. Taken together, these results predict that there may exist a pecking order, as a function of the e?ciency of external capital, for capital allocation in capital budgeting.

5

Discussions and Extensions

In this section, we ?rst articulate the di?erence between our basic model (the exclusive capital budgeting scenario) with the existing literature. Following this, we evaluate whether the aforementioned ine?ciencies can be eliminated under an alternative ?nancing policy.
5

When the e?ciency of external capital is very high ? > 1, the hurdle rate ?c is lower, deviating further from

?? and the external capital ec is higher. In the extreme case, the owner would never discard the projects (?c ? ?), because he can always subsidize the ine?ciency of internal capital by highly e?cient external capital. We thank an anonymous referee for this insight.

14

5.1

Voluntary Capital Rationing

Why doesn’t voluntary capital rationing on highly pro?table projects emerge as the optimal solution in the earlier literature such as Antle and Eppen (1985) and Dutta (2003)? In their models, the owner allocates the maximum (zero) capital when the capital productivity is above (below) than the hurdle rate. In this case, the owner must set the repayment such that the cuto? type of manager receives just his reservation utility. The question arising from this setting is whether the owner may gain from retaining lower-type managers by o?ering them lower capital allocation. Antle and Eppen (1985) assume that all types of managers have an identical reservation utility (normalized to zero) and thus the owner does not gain from retaining lower-type manager. Dutta (2003) assumes that the manager’s reservation utility increases linearly in the manager’s types. Under this assumption, the owner ?nds that the loss from matching the lower-type manager’s outside opportunities is larger than the gain from employing lower-type mangers. Consequently, when the managers’ reservation utilities are su?ciently homogeneous as in Antle and Eppen (1985) and Dutta (2003), it is suboptimal to o?er any additional incentive plan for lower-type managers. In contrast, our model assumes that the manager has multiple investment opportunities available from external capital. As shown in Lemma 1, this assumption implies that the manager’s reservation utility is a convex function of her capital productivity; the marginal increase of the manager’s reservation utility decreases as the manager’s capital productivity decreases. The owner again faces the same trade-o? between retaining less e?cient managers and matching their outside opportunities. Nevertheless, provided that the manager’s reservation utility decreases at a faster rate than his capital productivity, the owner now ?nds it optimal to retain less e?cient managers. In the meantime, the owner sets a relatively high repayment for this new plan (but still lower than the repayment for the original plan). Because this repayment is relatively high, if those managers who originally choose the maximum capital select the new plan instead, they would obtain a lower payo? than their original ones. Therefore, the owner is able to o?er multiple allocation plans to the manager while still maintaining the incentive compatibility.6
6

We abstract from explicitly modelling the e?ect of information asymmetry on external capital markets. However,

this model is ?exible enough to add some insight into this problem. To elaborate, consider a simple case where there exits only one external capital supplier. This supplier, like the owner, faces the adverse selection problem and ˆ), c(e(? ˆ)))} for the manager to self-select, where e is the external capital and c(e) is the would design a menu {(e(? corresponding repayment (similar to the menu {(Ie , te (Ie )} o?ered by the owner). To induce truth-telling, the external ˆ)) will not be the actual capital cost supplier may need to incorporate the cost of asymmetric information and c(e(? for the external capital supplier. By the revelation principle, we can limit attention to the truth-telling mechanism in which the manager reports her private information to external capital suppliers {(e(?), c(e(?))}. Because our analysis holds for any possible allocation {(e(?), c(e(?))}, it is clear that the main economic trade-o?s herein are not sensitive to this assumption. Moreover, one may extend this observation to the case of multiple external capital suppliers

15

5.2

Opportunity Cost of Quitting

What is the role of an opportunity cost of quitting in the contracting relationship?7 To explore this issue, let us assume that when the owner decides to exclude the manager, he has to incur an opportunity cost Co ? 0. This quitting cost may be lower for an established ?rm but higher for a startup. In contrast, the manager’s opportunity cost of quitting is Cm + ??, where Cm represents a common ?xed cost and ? ? 0 is the time value of searching for external capital. This implies that an e?cient manager will ?nd it more costly to leave the ?rm and ?nd a new source of capital externally. As a result, the owner reduces the hurdle rate and the project is more likely to be funded by internal capital. The following proposition characterizes how the hurdle rate ??? changes in response to these opportunity costs.8 Proposition 3. The hurdle rate ??? is decreasing in Co , Cm , and ?. The above proposition has an intuitive interpretation. As either the owner or the manager incur a higher opportunity cost, it is relatively more pro?table to fund the project. Consequently, the owner intends to exclude fewer types of managers. An empirical implication is that compared with an established ?rm, a startup ?rm is more likely to set a lower hurdle rate and to retain the manager, as the opportunity cost of quitting is higher for the startup ?rm.

6

Conclusion

In this paper, we argue that the owner will lose the whole residual right of the investment project when the project is funded by the external capital. To avoid potential economic loss, the investment projects can be ?nanced through either internal capital or external capital, but not both. We show that under this exclusive capital budgeting scenario, the ?rm may forgo highly pro?table
ˆ), ci (ei (? ˆ)))} to the manager. In this case, what really matters to and each individual supplier i o?ers a menu {(ei (? the manager is the collective menu available to her rather than each individual contract. Thus, we could conveniently ˆ), c1 (e1 (? ˆ)))}, ..., {(en (? ˆ), cn (en (? ˆ)))} from all external capital suppliers as the collection summarize all menus {(e1 (? ˆ), c(e(? ˆ)))}. Thus, our model could be interpreted as if the owner (the internal (combination) of these n menus {(e(? capital supplier) designs the menu {(Ie (?), te (?))} for the privately-informed manager, while facing the threat of multiple external capital suppliers. From the ?rm’s perspective, the threat is summarized by the available options o?ered to the manager {(e, c(e))}. 7 We thank an anonymous referee for inspiring this extension. 8 We also introduce the personal risk for the manager to join the external party (mean variance speci?cation). However, because both the owner and the entrepreneur are risk-neutral, the main economic trade-o?s are not sensitive to this assumption.

16

projects but fund those projects that have moderate capital productivity. Furthermore, the ?rm may voluntarily impose capital rationing, even though it has su?cient capital to fund such profitable projects. This solution mainly results from the economic trade-o? between matching with the manager’s external opportunities and inducing the manager’s truthful telling. This result provides another theoretical explanation for voluntary capital rationing we observe in practice. Under centralized capital budgeting, losing residual right of the project is not a concern for the owner. We show that if the owner can incorporate external capital in the contract with the manager, highly pro?table projects are funded by the ?rm and the capital rationing for highly pro?table projects is no longer optimal. The optimal amount of external capital increases in the manager’s capital productivity; however, because of the information asymmetry, the owner always distorts the allocation of external capital in order to limit the manager’s information rent. Our analysis highlights the role of external capital in capital budgeting process and identi?es opportunities for managers to enhance ?rm pro?tability through employing external capital. The study can be extended into several directions. One may extend our analysis into the setting in which the owner oversees multiple internal divisions. In this case, the allocation mechanism within a ?rm cannot leave managers’ outside opportunities aside, and sometimes it might be pro?table to “subsidize” less e?cient divisions more internal capital and retain more pro?table projects within a ?rm. This remains a research priority. Another possible direction is to study the the interaction between external capital and internal capital. If the investment project could result in negative pro?ts with some probability, thereby triggering a binding debt covenant, the owner may have a stronger incentive to utilize internal capital than external capital. In such a scenario, the interaction between external capital and internal capital may come into play. Finally, a simple “type-dependent” agency model will not su?ce to fully address the issues related to managerial retention. Many important aspects in labor markets (such as managerial recruiting, sorting, training, and promotion) are inevitably intertwined with managerial retention. An important topic for future research would be to build a multi-period model in which the managerial retention decision is dynamically a?ected by these aspects (see, e.g., Lazear and Oyer (2004)).

Appendix. Proofs
Proof of Lemma 1. Consider two types ?1 and ?2 . Assume that the maximizer for type-?1 manager is e1 ? 0, i.e., r(?1 ) = ??1 e1 ? c(e1 ). We have E (?2 ) = max {??2 e ? c(e)} ? ??2 e1 ? c(e1 ) = ??1 e1 ? c(e1 ) + ?e1 (?2 ? ?1 ) = E (?1 ) + ?e1 (?2 ? ?1 ),
e? 0

and therefore E (?2 ) ? E (?1 )+ ?e1 (?2 ? ?1 ), ??1 , ?2 . Note that the above inequality holds for arbitrary pair of types ?1 and ?2 . Thus, E (?) is convex in ?. To prove the monotonicity of E (?), consider a 17

pair (?1 , ?2 ) and without loss of generality we assume ?1 ? ?2 . The optimality condition leads to E (?2 ) ? ??2 e1 ? c(e1 ) = ??1 e1 ? c(e1 ) + ?e1 (?2 ? ?1 ) ? E (?1 ), where the last inequality follows from e1 being nonnegative. Therefore, E (?) is increasing. Finally, E (0) = maxe?0 {?c(e)} = 0. Proof of Proposition 1. Following the approach established in Jullien (2000), we can restrict our attention to the case in which the owner o?ers internal incentive plans to only the manager whose capital productivity falls into an interval. Therefore, we will assume that 1) The owner o?ers a menu of incentive plans to the manager with ? ? [?, ? ); 2) The manager with ? ? [?, ? ] accepts the same internal incentive plan with the maximum capital, K , and an associated promised return te (? ), regardless of his capital productivity,; 3) The manager with capital productivity that falls in ¯] is excluded, where 0 ? ? ? ? ? ? ? ? ¯. [0, ?) ? (?, ? We further assume that by accepting (K, te (? )), the type-? manager receives exactly his reservation utility, i.e., te (? ) = ? K ? E (? ). We will verify later that this is a necessary condition for ¯, which optimality. Note that we do not exclude the possibilities of ? = ?, ? = ? , ? = ? , or ? = ? represent respectively the cases when no manager is excluded, only a single internal incentive plan is o?ered, exactly one type of manager receives the maximum capital, and no manager with a relatively high capital productivity is excluded. Given all the above assumptions, the owner’s problem becomes the following: } { ? ? (te (?) ? rIe (? ))f (?)d? max (te (? ) ? rIe (? ))(F (? ) ? F (? )) +
{Ie (·),te (·)} ?

(3)

s.t.

(IC-1) ? ? argmaxz ?[?,? ) ?Ie (z ) ? te (z ), ?? ? [?, ? ), (IC-2) Ue (?) ? max Ue (z, ?), ?? ? [?, ? ],
z ?[?,? )

(IC-3) E (?) ? max Ue (z, ?), ?? ? [0, ?),
z ?[?,? ]

¯], (IC-4) E (?) ? max Ue (z, ?), ?? ? [?, ?
z ?[?,? ]

(IR-1) Ue (?) ? E (?), ?? ? [?, ? ), (IR-2) Ue (?) ? E (?), ?? ? [?, ? ]. In Eq. (3), the ?rst four inequalities are incentive compatibility (IC) conditions: (IC-1) guarantees that a manager that receives an incentive plan speci?c for himself does not attempt to choose some other incentive plans, (IC-2) ensures that the manager that receives the maximum capital K is willing to accept it, and (IC-3) and (IC-4) are for respectively the manager who is excluded from below and above. The last two inequalities in Eq. (3) represent the (IR) conditions. Since ¯] obtains his reservation utilities, his (IR) the manager with capital productivity in [0, ?) ? [?, ? condition is automatically satis?ed. In the following we ?rst focus on the case when ? > ?? and 18

K > E (?? ). Our strategy is to ?rst ignore the (IC) and (IR) conditions for the manager whose type falls outside the interval [?, ? ], i.e., (IC-2), (IC-3), (IC-4), and (IR-2), and then verify that they are satis?ed under our proposed menu. For this subproblem, we can replace (IC-1) in (3) by the following local incentive compatibility (LOIC) condition: Ue (?) = Ie (?), ? ? [?, ? ), and that Ie (?) is increasing in ?. Note that from the de?nition of Ue (?), we have te (?) = ?Ie (?) ? Ue (?). Observing that the ?rst term in the objective function in Eq. (3) is independent of the choice of (Ie (?), te (?)), ?? ? [?, ? ), we can ignore it for a moment and solve the optimization problem with respect to {(Ie (?), te (?)), ? ? [?, ? )}. Replacing te (?) by ?Ie (?) ? Ue (?), we can show the owner’s objective function as ? ? max [(? ? r)Ie (?) ? Ue (?)]f (?)d?,
{Ie (·)}
?

?

(4)

s.t. (LOIC) (MON) (IR)

? ? Ue (?)
?

= Ie (?), ? ? [?, ? ),

Ie (?) ? 0, ?? ? [?, ? ), Ue (?) ? E (?), ?? ? [?, ? ).
?

The solution to this subproblem can be easily characterized. First choose Ie (?) = E (?), which is increasing in ? by the convexity of E (?). Given this level of investment, we can select the promised return te (?) so that the manager receives the same utility Ue (?) as from the reservation utility: te (?) = ?E (?) ? E (?), ?? ? [?, ? ). Under this menu of incentive plans, the manager’s (IR) condition is binding, and therefore this menu achieves the upper bound of the optimization problem (4) and thus is optimal. Having characterized the optimal menu for the subproblem (4), we now determine the promised return te (? ). Note that a jump in the internal capital may occur at ? = ? if E (? ) < K . However, the received utility should be continuous (otherwise a manager with at ? = ? would deviate to choose another plan that gives him a strictly positive utility). Thus, we have that Ue (? ) = ? K ? te (? ) = E (? ) ? te (? ) = ? K ? E (? ). ¯ and the solution to the equality ?K ? The upper bound ? is determined by the minimum of ? (? K ? E (? )) = E (? ), which gives rise to the cuto? type of manager who is indi?erent between accepting the internal capital and using the external capital. From the convexity of E (·) and that E (? ) < K , there exists a unique ? for any given K and ? . When E (? ) = K , [?, ? ] degenerates. We can then verify that all the incentive compatibility and individual rationality constraints are satis?ed if this proposed menu is used. Since the manager receives a nonnegative payo?, the (IR) condition is automatically satis?ed. (IC-1) follows directly from the local incentive compatibility condition. Now consider (IC-2). Given the menu, (IC-2) becomes Ue (?) ? ?E (z ) ? zE (z ) + E (z ), ?? ? [?, ? ], ?z ? [?, ? ]. 19
? ? ? ? ? ?

Note that Ue (?) ? E (?) from (IR-2). Therefore, since E (?) ? E (z ) + (? ? z )E (z ), ??, ?z by the convexity of E (?), we obtain that Ue (?) ? E (?) ? ?E (z ) ? zE (z ) + E (z ), ?? ? [?, ? ], ?z ? [?, ? ]. Thus, (IC-2) is settled. Likewise, we can verify (IC-3) and (IC-4) based on the convexity of E (?). We now consider the optimal choice of ? and ? . The threshold ? is determined once we have ?xed ? . Let us ?rst determine ?, the cuto? threshold from below. The owner is willing to o?er internal capital to a manager if and only if it obtains a positive pro?t from him. When ? ? [?, ? ], the owner’s net pro?t from o?ering to a type-? manager is te (?) ? rIe (?) = (? ? r)E (?) ? E (?) ? 0, where the left-hand side of the inequality can be shown to be strictly increasing due to the convexity of E (?). Therefore, there exists a unique threshold ??? such that (??? ? r)E (??? ) ? E (??? ) = 0, and at optimality the owner should set ? = ??? . Note that this is independent of the choice of ? . The choice of ? can then be determined. Given the menu of incentive plans, the owner’s expected payo? is as follows: ? max[(? ? r)K ? E (? )][F (? ) ? F (? )] +
? ?
? ? ? ?

?

???

((? ? r)E (?) ? E (?))f (?)d?.

?

¯ or The optimal can be obtained easily by the ?rst-order condition. Finally, when either ??? > ? K < E (??? ), the region in which voluntary capital rationing degenerates. Proof of Proposition 3. With the opportunity costs, we can rewrite the owner’s problem Eq. (3) as follows: max { } ? ? (te (? ) ? rIe (? ))(F (? ) ? F (? )) + (te (?) ? rIe (? ))f (?)d? ? Co F (?) (5)
?
?

{Ie (·),te (·)}

s.t.

(IC-1) ? ? argmaxz ?[?,? ) ?Ie (z ) ? te (z ), ?? ? [?, ? ), (IC-2) Ue (?) ? max Ue (z, ?), ?? ? [?, ? ],
z ?[?,? )

(IC-3) E (?) ? Cm ? ?? ? max Ue (z, ?), ?? ? [0, ?),
z ?[?,? ]

¯], (IC-4) E (?) ? Cm ? ?? ? max Ue (z, ?), ?? ? [?, ?
z ?[?,? ]

(IR-1) Ue (?) ? E (?) ? Cm ? ??, ?? ? [?, ? ), (IR-2) Ue (?) ? E (?) ? Cm ? ??, ?? ? [?, ? ]. We can then follow the proof of Proposition 1 and ?nd that the optimal hurdle rate is determined by
d the marginal cost-bene?t analysis as follows: (? ? r) d? [E (?) ? Cm ? ??] ? [E (?) ? Cm ? ??] = ?Co .

De?ne [ ? ] ? ?(?) = (? ? r) E (?) ? ? ? [E (?) ? Cm ? ??] ? Co = (? ? r)E (?) ? E (?) + Cm + r? + Co .

20

We then obtain that ? (?) = (? ? r)E (?), which is positive when ? ? r due to the convexity of E (?). Note that we only need to consider the case ? ? r because it is obvious that the hurdle rate is greater than r. Since ?(?) is increasing when ? ? r, the critical point for which ?(?) = 0 is therefore decreasing in Cm + r? + Co ; therefore, we conclude that the proposition holds. Proof of Proposition 2. We again adopt the standard approach to obtain the optimal incentive plans for the owner. We ?rst replace the incentive compatibility condition by Uc (?) ? Ic (?) + ec (?), ?? ? [?, ?], and the monotonicity condition: Ic (?) + ec (?) is increasing in ?. Recall that Uc (?) = ?Ic (?) ? tc (?) + ??ec (?) ? c(ec (?)). We can then represent tc (?) by ?Ic (?) + ??ec (?) ? c(ec (?)) ? Uc (?). After these substitutions, the owner’s problem becomes ?
{tc (·), Ic (·),kc (·)} ¯ ?
?

?

??

max

[(? ? r)Ic (?) + ??ec (?) ? c(ec (?)) ? Uc (?)] dF (?)

(6)

?

s.t.

Uc (?) ? Ic (?) + ?ec (?), ?? ? [?, ?], Ic (?) + ?ec (?) ? 0, ?? ? [?, ?], Uc (?) ? 0, ?? ? [?, ?].
? ?

?

Let us ignore the monotonicity condition and apply optimal control theory to solve (6). De?ne Ic and ec as the control variables and let Uc be the state variable. The Hamiltonian of (6) is Lc (Ic , ec , ?; ?) = [(? ? r)Ic + ??ec ? c(ec ) ? Uc ]f (?) + ?c (Ic + ?ec ), where ?c is the multiplier for the local incentive compatibility. Di?erentiating the Hamiltonian with respect to the control and state variables, we obtain that ?Lc ?Lc ?Lc ? ? c = f (?). = (? ? r)f (?) + ?c , = (?? ? c (ec ))f (?) + ?c , and ? =? ?Ic ?ec ?Uc ¯ implies that ?c = F (?) ? 1. Moreover, since the Hamiltonian The transversality condition at ? = ? is linear in Ic , the owner should set In = K if ? ? r ? H (?) ? 0, and set In = 0 otherwise. The optimal level of external capital ec follows from the equation c? (ec ) = ?? ? H (?), and the secondorder condition is satis?ed due to the convexity of c(·). The global incentive compatibility for the manager under this optimal menu of incentive plans can be veri?ed similar to that of Proposition 1 and thus is omitted. We now show that the optimal amount of external capital ec (?) is ine?cient. Without information asymmetry, the ?rst-best external capital eF B (?), is characterized by the ?rst-order condition: ?? ? c? (eF B (?)) = 0. The uniqueness is ensured by the convexity of c(e). From the de?nition of ec (?), we obtain from the implicit function theorem that Since c? (ec ) is always positive, there must exist a lower bound
?ec ??

=

[??H ? (?)] ? c?? (ec ) > 0, given that H (? ) < 0. ??? such that the ?rst order condition

is valid, where the cut-o? point ?c is given by ??c ? H (?c ) = 0. The cuto? point ?? is determined 21

such that ?? ? r ? H (?? ) = 0. When ? = 1, we can easily verify ?c < ?? . As ? decreases, the hurdle rate for external capital ?c is higher. Thus there exists a cut-o? point for ?c such that ?c ? ?? if and only if ? ? ?c . In contrast, when ? > 1, the hurdle rate ?c is lower, deviating further from ?? and the external capital ec is higher. In the extreme case, the owner would never discard the projects (?c ? ?), because he can always subsidize the ine?ciency of internal capital.

References
Antle, R., P. Bogetoft, and A. Stark (1999). Selection among mutually exclusive investments with managerial private information and moral hazard. Contemporary Accounting Research 16, 397–418. Antle, R., P. Bogetoft, and A. Stark (2001). Information systems, incentives and the timing of investments. Journal of Accounting and Public Policy 20 (4-5), 267–297. Antle, R. and G. Eppen (1985). Capital rationing and organizational slack in capital budgeting. Management Science 31, 163–174. Antle, R. and J. Fellingham (1997). Models of capital investments with private information and incentives: A selective review. Journal of Business Finance and Accounting 24, 887–908. Arya, A., J. Fellingham, J. Glover, and K. Sivaramakrishnan (2000). Capital budgeting, the hold-up problem, and information system design. Management Science 46 (2), 205–216. Arya, A. and J. Glover (2001). Option value to waiting created by a control problem. Journal of Accounting Research 39 (3), 405–415. Baldwin, C. and K. B. Clark (1992). Capabilities and capital investment: new perspective on capital budgeting. Journal of Applied Corporate Finance 5 (2), 67–82. Bhide, A. (2000). The Origin and Evolution of New Businesses. New York, NY: Oxford University Press. Dutta, S. (2003). Capital budgeting and managerial compensation: Incentive and retention e?ects. The Accounting Review 78, 71–93. Dutta, S. (2008). Managerial expertise, private information and pay-performance sensitivity. Management Science 54 (3), 429–442. Gertner, R., D. Scharfstein, and J. Stein (1994). Internal versus external capital markets. The Quarterly Journal of Economics 109 (4), 1211–30. Gutmann, P. (1967). External ?nancing and the rate of economic growth. American Economic Review 57 (4), 864–869. Ittner, R., R. Lambert, and D. Larcker (2003). The structure and performance consequences of 22

equity grants to employees of new economy ?rms. Journal of Accounting Economics 34, 89– 127. Jullien, B. (2000). Participation constraints in adverse selection models. Journal of Economic Theory 93 (1), 1–47. Lazear, E. and P. Oyer (2004). Internal and external labor markets: a personnel economics approach. Labour Economics 11 (5), 527–554. Maggi, G. and A. Rodriguez-Clare (1995). On countervailing incentives. Journal of Economic Theory 66, 238–263. Mukherjee, T. and V. Hingorani (1999). Capital-rationing decisions of fortune 500 ?rms: A survey. Financial Practice and Education 1, 7–15. Paik, T. and P. Sen (1995). Project evaluation and control in decentralized ?rms: Is capital rationing always optimal? Management Science 41 (8), 1404–1414. Porter, M. (1992). Capital choices: Changing the way America invests in industry. Journal of Applied Corporate Finance 5 (2), 4–16. Stulz, R. (1990). Managerial discretion and optimal ?nancing policies. Journal of Financial Economics 26 (1), 3–27. Zhang, G. (1997). Moral hazard in corporate investment and the disciplinary role of voluntary rationing. Management Science 43, 737–750.

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