Description
A budget is a quantitative expression of a plan for a defined period of time. It may include planned sales volumes and revenues, resource quantities
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Journal of Accounting and Economics 41 (2006) 257–270 www.elsevier.com/locate/jae
Capital budgeting for new projects: On the role of auditing in information acquisition$
Doyoung KimÃ
College of Business and Economics, University of Idaho, Moscow, ID 83844-3161, USA Received 24 February 2005; received in revised form 5 March 2006; accepted 15 March 2006 Available online 9 June 2006
Abstract This article studies capital budgeting for new projects in which information is acquired by managers. When information acquisition costs are small, optimal capital budgeting is not qualitatively different from that for routine projects where managers have pre-existing information. However, the need to provide incentives to acquire information results in more intensive auditing and further distortions in capital allocations. When information acquisition costs are large, optimal capital budgeting differs from that for routine projects. To provide strong incentives for information acquisition, auditing becomes more extensive, and more than the ?rst best amount of capital is allocated whenever auditing occurs. r 2006 Elsevier B.V. All rights reserved.
JEL classi?cation: D82; D83; G31; M42 Keywords: Capital allocation; Auditing; Information acquisition
1. Introduction Capital budgeting may be the most important decision made by corporations. While the net present value rule has long been a standard prescription, it does not guide the practical details of internal capital allocation (Harris and Raviv, 1996). In response to this, researchers have paid attention to asymmetric information in internal capital markets. In
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I thank Rick Antle (the referee) and Ross Watts (the editor) for helpful comments.
E-mail address: [email protected]. 0165-4101/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jacceco.2006.03.001
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particular, they posit that division managers have better information than headquarters about the prospects of projects at the outset.1 Due to unaligned interests, information asymmetry creates agency problems. Capital budgeting is then optimally structured to provide managerial incentives for information revelation. While the assumption of endowed information is a reasonable approximation in some settings, managers may not be omniscient from the outset in many relevant settings (Lewis and Sappington, 1997). In particular, while managers could have information about the ongoing projects they have routinely managed, they might not have superior information about new or non-routine projects. For the successful implementation of such projects, it is important to motivate managers to acquire valuable information about the projects. However, there has been scant attention to this issue in the literature on capital budgeting. This article studies optimal capital budgeting from the standpoint of ef?ciently motivating managers to acquire and reveal information. The paper ?rst discovers an important, but unexplored, role of auditing. For routine projects where information is given to a manager at the outset, auditing is simply used to discipline the manager’s incentive to lie about his pre-existing private information. However, for new projects where information should be acquired by the manager, auditing is used to motivate him to acquire that information. More intensive and extensive auditing encourages the manager to acquire information by decreasing the opportunity cost of information acquisition. In addition to the disciplinary role the literature has presumed, this paper thus emphasizes the positive incentive role of auditing. Next, it ?nds that the optimal allocation of capital for new projects is qualitatively different from that for routine projects. The ef?cient provision of incentives to acquire information results in the distortion of resource allocations beyond that involved in the creation of incentives to disclose pre-existing private information. Formally, this study extends a standard capital budgeting model to consider a project where information is endogenously acquired by a manager. In particular, at the outset neither the manager nor headquarters has information about the productivity of capital, which will determine the net present value of the project. The manager, who has a preference for empire building, can acquire the information by incurring the personal costs of the information acquisition effort. Headquarters audits the manager’s report on the productivity of capital to verify the truthfulness of the report.2 Optimal capital budgeting can be characterized according to information acquisition costs. When these costs are small, headquarters audits (randomly) only if the manager reports that the productivity of capital is high. To discipline the manager’s incentive to overstate capital productivity, headquarters allocates less than the ?rst best amount of capital for the high-productivity state and more than the ?rst best amount of capital for the low-productivity state. Since these results hold unless information acquisition costs are large, they can also characterize the capital budgeting process for routine projects where information acquisition costs are zero, or equivalently where information is given to the manager at the outset. A contribution of the paper is then to show that headquarters
1 See for instance Harris et al. (1982), Antle and Eppen (1985), Holmstro ¨ m and Ricart I Costa (1986), Harris and Raviv (1996, 1998), Zhang (1997), Bernardo et al. (2001, 2004), Dutta and Reichelstein (2002), Dutta (2003), Baldenius (2003), Berkovitch and Israel (2004), and Marino and Matsusaka (2005). 2 The model follows Fama and Jensen (1983) in that the manager is responsible for project initiation (by information acquisition) and implementation, and headquarters is in charge of project rati?cation and auditing.
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distorts capital allocations and auditing intensities from their optimal levels for routine projects. In particular, more capital is allocated for the low-productivity state, less capital is allocated for the high-productivity state, and auditing occurs with higher probabilities. These distortions arise in order to motivate the manager to acquire information. Due to the manager’s preference for empire building, he has an incentive to request more capital by overstating the productivity of capital even without acquiring information. A more intensive audit of the manager’s capital request and a more stringent allocation of capital for the high-productivity state then discourage the manager from pretending to be informed and requesting more capital. That is, they reduce the opportunity cost of becoming informed. When information acquisition costs are large, optimal capital budgeting for new projects is qualitatively different from that for routine projects. With large information acquisition costs, the provision of incentives to acquire information becomes a serious issue. To provide strong incentives, headquarters broadens its audit such that it investigates the manager’s report even if the report says that the productivity of capital is low. A more extensive audit provides better incentives to acquire information by reducing the opportunity cost of becoming informed. In addition, for any productivity state the manager reports, headquarters allocates more than the ?rst best amount of capital when it audits the manager’s report, and more capital when it audits than when it does not. Linking more capital to the audit that veri?es the manager’s report increases the bene?t of becoming informed and, at the same time, decreases the opportunity cost of becoming informed. This article is closely related to the paper by Harris and Raviv (1996) as it extends their model.3 While they consider a project where information is given at the outset, this article focuses on a project where information must be acquired by a manager. With this difference, it shows that the capital budgeting process identi?ed in their paper may not always be optimal and that auditing is used as an incentive device to motivate a manager to acquire information. There are a few papers that model information acquisition in capital budgeting.4 Arya et al. (2000) study the effect of information systems (early vs. late information) on a manager’s incentive to exert research effort on the choice of projects.5 They show that information systems can serve as a commitment device to allow the manager to enjoy information rents that motivate him to exert research effort. Laux (2001) studies who (a division manager vs. a budget center) should be assigned the task of information acquisition. He shows that it depends on expected wage costs in inducing information acquisition. Stein (2002) studies which type of organization, a small decentralized ?rm or a large hierarchical ?rm, better motivates a manager to acquire information. He shows that it depends on the veri?ability of information. While the papers
The paper is also broadly related to the literature on capital budgeting for a single division under asymmetric information. The list includes Harris et al. (1982), Antle and Eppen (1985), Holmstro ¨ m and Ricart I Costa (1986), Zhang (1997), Bernardo et al. (2001), Dutta and Reichelstein (2002), Dutta (2003), and Baldenius (2003). In these papers, information is given at the outset. See also a review by Lambert (2001). 4 In a project choice model, Lambert (1986) pioneered the modeling of information acquisition. See also Lewis ´ mer et al. (1998), and Kessler (1998) for the modeling of information acquisition in and Sappington (1997), Cre procurement contracts, and Aghion and Tirole (1997) and Dewatripont and Tirole (1999) in the allocation of authority and organizational design. 5 Antle and Fellingham (1995) also study the choice of (public) information systems. But they focus on the effect of information systems on a manager’s activities that in?uence the ?rm’s choice of information systems.
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mentioned above take an organizational focus, this article examines the distortions in resource allocation and auditing involved in the provision of incentives to acquire information. The rest of the paper is organized as follows. Section 2 presents the model of capital budgeting with information acquisition. Section 3 characterizes optimal capital budgeting with regard to ef?ciently motivating a manager to acquire and reveal information. Section 4 concludes the paper. All proofs are relegated to the Appendix A. 2. Model The model consists of a risk-neutral headquarters and a risk-neutral division manager. Headquarters has access to capital k, which will be allocated to the manager for the implementation of a new or non-routine project.6 The state of nature, indexed by i 2 f1; 2g, determines the net present value of the project Vi(k), where Vi( Á ) is a strict concave function with V i ð0Þ ¼ 0 and V 0i ð0Þ ¼ 1, and has unique value-maximizing capital kFB i such FB that V 0i ðkFB Þ ¼ 0. k is then the ?rst best amount of capital that headquarters would i i allocate if it were informed about the state of the project. The productivity of capital is higher under i ¼ 2 than under i ¼ 1. That is, V 01 ðkÞoV 02 ðkÞ for all k, implying that FB kFB 1 ok2 . At the outset, neither headquarters nor the manager has information about the state of the project or, equivalently, the productivity of capital. Instead, they share the common belief that state i will be realized with probability pi 2 ð0; 1Þ, where p1 þ p2 ¼ 1. It is essential for headquarters to know the productivity of capital since capital can then be more ef?ciently allocated according to its productivity. Headquarters lets the manager acquire information about the productivity of capital, which can be justi?ed by the fact that corporate managers are responsible for the planning of projects in which information acquisition may be a main task. In other words, the manager’s experience in project planning (his human capital) makes it much less costly for headquarters to have the manager acquire information than to undertake its own information acquisition effort. The manager can acquire information about the productivity of capital by incurring a cost R, which can be seen as the manager’s personal, unobservable cost of research effort. This is the only chance to learn the productivity of capital, so capital allocation and project implementation will be done without information if the manager chooses not to acquire information. To make information acquisition relevant in the model, it is assumed that the information acquisition cost R is not too large, so that headquarters prefers to induce the manager to acquire information. As is standard in the literature, it is assumed that information acquisition is a private action by the manager and that the information acquired is private and unveri?able. Headquarters is then unaware of not only whether the manager acquired the information, but also which state he discovered.7 The manager has a preference for empire building.8 He derives higher utility from managing more capital and bigger projects. This may be due to the fact that a manager’s
It is thereby assumed that the manager has no resources of his own and that all resources must come from headquarters. 7 ´ mer et al. For this modeling of endogenous information acquisition, see Lewis and Sappington (1997) and Cre (1998). 8 For discussions about corporate managers’ preference for empire building and its modeling, see for instance Harris and Raviv (1996, 1998), Bernardo et al. (2001, 2004), and Baldenius (2003).
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reputation is enhanced by the size of the projects or divisions he has managed. It may also be the case that managers have more opportunities to enjoy perquisite consumption when they manage more capital. To model this, let the manager’s utility from managing capital be bk, where b is the manager’s marginal bene?t associated with additional capital. To simplify the analysis, monetary compensation is assumed away from the manager’s utility function, re?ecting the case where headquarters’ capital allocation decision is independent of its decision about managerial compensation. Technically, this simpli?cation implies that capital allocations and auditing are the only incentive devices available to headquarters. However, it must be noted that optimal capital budgeting as characterized in this simple model would still qualitatively survive even in a model that explicitly considers monetary incentives.9 The manager reports information about the productivity of capital to headquarters. Notice that the manager’s report does not have to be truthful since it is not veri?able. With the manager’s preference for empire building, this creates an agency problem, which in turn creates a need for monitoring. Headquarters therefore audits the manager’s report. Given that the manager reports state i, headquarters can choose auditing intensity ai 2 ½0; 1? by incurring a cost of C(ai), where C( Á ) is a strict convex function with C ð0Þ ¼ 0 and C 0 ð0Þ ¼ 0. In particular, with probability ai, headquarters audits the manager’s report and discovers whether the manager has reported truthfully or not. However, with probability (1–ai), headquarters does not audit and thereby receives no information about the veracity of the manager’s report. Based on the outcome of the audit, headquarters allocates capital to the manager for the implementation of the project. There are four possible capital allocations in equilibrium: kni and kai for all i 2 f1; 2g. kni is the amount of capital allocated when auditing does not occur for the manager’s report i, and kai is the amount of capital allocated when auditing reveals that the manager’s report i is true. Notice that there is no need for the introduction of capital allocations when auditing reveals that the manager’s report i is untruthful as these allocations never occur in equilibrium due to the direct revelation mechanism employed in solving the model. Out-of-equilibrium capital allocations in such a case are optimally zero. Intuitively, headquarters would not receive any gains from allowing the manager to lie. Since headquarters would not bear any cost of imposing a maximum penalty for the manager’s lie, it would allocate zero capital—the maximum penalty in the model—when it discovers the manager’s lie.10 The interaction between headquarters and the manager can be summarized as follows. Headquarters announces a capital budgeting scheme consisting of capital allocations and auditing intensities: {kni, kai, ai}. As is standard in the literature, it is assumed that
As will be shown later, headquarters distorts capital allocations to motivate the manager to acquire and reveal information. Headquarters can also provide such an incentive by giving the manager information rents in the form of monetary compensation. However, notice that the manager cannot take or use capital for his personal purposes in the model. Then, as is standard in hidden information models, monetary compensation in?icts a ?rstorder loss while a distortion in capital allocations in?icts a second-order loss to headquarters. Hence, in a model with monetary compensation, headquarters ?rst distorts capital allocations and then gives information rents as the distortion gets worse, though headquarters ?nally use both in equilibrium in order to minimize the loss. It implies that the optimal capital allocations are qualitatively the same regardless of the inclusion of monetary compensation in the model. 10 See Baron and Besanko (1984) for the principle of maximum penalties.
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headquarters can commit to this scheme.11 Given this scheme, the manager acquires information about the productivity of capital. The manager then reports this information. Headquarters audits the manager’s report i with auditing intensity ai as announced at the beginning. Headquarters allocates capital to the manager who then implements the project and renders the net present value of the project to headquarters. 3. Results 3.1. Headquarters’ problem Since information is essential for the successful implementation of the project, headquarters lets the manager acquire information. However, notice that information acquisition is not observable, which may imply that headquarters faces the manager’s moral hazard problem in the task of information acquisition. The manager in fact has an incentive not to acquire information in order to save his personal acquisition cost R, but to nonetheless pretend that he did acquire information. Headquarters then has to motivate the manager to acquire information. It can do so by making the manager’s expected payoff when he becomes informed at least as large as his expected payoff when he remains uninformed, as shown in the following information acquisition constraints: ðIAi Þ p1 ½ð1 À a1 Þbkn1 þ a1 bka1 ? þ p2 ½ð1 À a2 Þbkn2 þ a2 bka2 ? À RXð1 À ai Þbkni þ pi ai bkai 8i. The LHS is the manager’s expected payoff when he acquires information, and the RHS is his expected payoff when he is uninformed and claims that he discovered state i. Even without acquiring information, he can claim that he discovered state i since headquarters cannot distinguish between an informed and uninformed manager. If he reports state i, with probability (1–ai), no auditing occurs and he receives kni. With probability ai, auditing occurs and reveals whether the manager’s report is correct or not, and the manager receives kai if his report is correct (with probability pi) and no capital allocation otherwise. After inducing information acquisition, headquarters faces another agency problem. The information acquired is private to the manager. Given his preference for empire building, the manager lies about the true productivity of capital in a way to maximize his payoff. To discipline this incentive, headquarters’ decision on capital budgeting must be subject to the following incentive compatibility constraints: ðIC i Þ ð1 À ai Þbkni þ ai bkai Xð1 À aj Þbknj 8iaj 2 f1; 2g.
These constraints guarantee that the manager is better off by reporting truthfully.
11 Whether headquarters can commit to auditing may be an issue since the manager will be induced to report truthfully in equilibrium, which in turn makes auditing useless ex post. Without a commitment to auditing, a question is then whether there exists a mixed strategy equilibrium where the manager lies with some probabilities as in Khalil (1997) who studies auditing in a procurement model. A necessary condition for the existence of the mixed strategy equilibrium is that headquarters receives ex-post gains by letting the manager lie since it creates a trade-off between the manager’s truth-telling and lying from headquarters’ perspective. However, one can easily see that this condition does not hold in the model since headquarters does not receive any ex-post gains by letting the manager lie and then punishing him for his lie. Thus, the mixed strategy equilibrium would not arise in the model considered here.
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With the information acquisition and incentive compatibility constraints, headquarters’ optimal decision on {kni, kai, ai} is max
2 P i ¼1
pi ½ð1 À ai ÞV i ðkni Þ þ ai V i ðkai Þ À C ðai Þ? 8i:
ðPH Þ
s:t: ðIAi Þ; ðIC i Þ; 0pai p1; kni X0; kai X0
Notice that capital allocations are restricted to be non-negative, making the individual rationality conditions redundant by guaranteeing non-negative payoffs to the manager. 3.2. Capital budgeting when information acquisition costs are small The solution to headquarters’ problem (PH) above depends on information acquisition costs. The following proposition characterizes the optimal capital budgeting scheme, à à denoted by kà ni ; kai and ai , when information acquisition costs are small. ¯ , the optimal auditing intensities and capital allocations are Proposition 1. For RoR à FB à FB à FB FB à à ¯ characterized by a1 ¼ 0, aà 2 40, k n1 4k 1 , k n2 ok2 , and k a2 ¼ k 2 , where R p1 p2 a2 bk 2 . Headquarters chooses to audit only if the manager reports that the productivity of à capital is high ðaà 1 ¼ 0; a2 40Þ. For the manager’s report of high capital productivity, headquarters allocates the ?rst best amount of capital if it audits the manager’s report ex FB à FB post ðkà a2 ¼ k 2 Þ and less than the ?rst best amount otherwise ðk n2 ok2 Þ. When the manager reports that the productivity of capital is low, headquarters allocates more than FB à the ?rst best amount without auditing ðkà n1 4k 1 Þ. As a1 ¼ 0, ka1 is irrelevant. Since the manager with a preference for empire building would not understate the productivity of capital, there is no need for an audit when he reports that the productivity of capital is low ðaà 1 ¼ 0Þ. However, an audit is necessary when he reports that the productivity of capital is high since otherwise he would always overstate the productivity of capital regardless of whether or not he is informed about the productivity ðaà 2 4 0Þ . Headquarters distorts capital allocations to discipline the manager’s incentive to lie when there is no audit. In particular, headquarters provides the manager with an incentive not to overstate capital productivity by increasing the amount of capital allocated for the FB low-productivity state above the ?rst best level ðkà n1 4k 1 Þ, and discourages the manager from overstating capital productivity by decreasing the amount of capital allocated for the FB high-productivity state below the ?rst best level ðkà n2 ok2 Þ. Of course, there is no need for a distortion in capital allocations when an audit reveals that the manager’s report is FB truthful ðkà a2 ¼ k 2 Þ. To summarize, over-investment occurs when the productivity of FB capital is low ðkà n1 4k 1 Þ, whereas under-investment occurs when the productivity of à FB FB 12 capital is high ðkn2 ok2 ; kà a2 ¼ k 2 Þ.
12 Following Harris and Raviv (1996), one can argue that the optimal capital budgeting scheme described in Proposition 1 can be also implemented by a typical capital budgeting process consisting of an initial spending limit and additional capital allocations upon approval (see Fremgen (1973), Gitman and Forrester (1977), Taggart (1987), and Ferreira and Brooks (1988) for a survey of this capital budgeting process in corporations). Headquarters sets an initial spending limit equal to kà n1 . When the manager discovers that capital productivity is à high, he requests more capital. Then headquarters allocates additional capital equal to ðkà a2 À kn1 Þ when it audits à à and veri?es the manager’s request, and ðkn2 À kn1 Þ otherwise.
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Since these results hold unless information acquisition costs are large, they prevail even in routine projects where information is freely given at the outset, i.e., where information acquisition costs are zero in the model. A question then arises as to the role of endogenous information acquisition in optimal capital budgeting. To explore this issue, let the information acquisition cost R increase above zero. A comparative static analysis with respect to R, summarized in the following proposition, reveals the role of endogenous information acquisition.
à à ¯ , aà Proposition 2. For RoR 2 and kn1 increase with R, whereas k n2 decreases with R.
The probability of auditing ðaà 2 Þ increases with the cost of information acquisition. When there is no audit, the amount of capital allocated for the low-productivity state ðkà n1 Þ increases, while the amount of capital allocated for the high-productivity state ðkà n2 Þ decreases with the cost of information acquisition. If the cost of information acquisition is zero, headquarters deals with the manager only for information revelation. However, as the cost increases above zero, the manager’s moral hazard in information acquisition becomes an issue, and headquarters has to motivate the manager to acquire information in the ?rst place. To do so, as shown above, headquarters distorts capital allocations further from the optimal levels for routine projects. The intuition behind these distortions is straightforward. The manager acquires information if the bene?t of becoming informed, i.e., the expected payoff from becoming informed, is larger than the opportunity cost of becoming informed, i.e., the expected payoff when he remains uninformed and pretends to be informed. To motivate the manager to acquire information, headquarters has to increase the bene?t of becoming informed, while decreasing the opportunity cost of becoming informed. To increase the bene?t of becoming informed, headquarters may have to increase the amount of capital allocated to the à à manager, kà n1 or kn2 or both. However, an increase in k n2 increases the opportunity cost of becoming informed as well. With a preference for empire building, the uninformed manager would claim to have discovered the high-productivity state in order to receive à more capital, kà n2 , which constitutes the opportunity cost. Indeed, as k n2 increases, the opportunity cost increases more than the bene?t of becoming informed. Thus, while à headquarters increases kà n1 to improve the bene?t of becoming informed, it decreases k n2 to reduce the opportunity cost of becoming informed. An interesting result here is that, to motivate the manager to acquire information, headquarters increases the probability of auditing above the optimal level for routine projects. Again, since the uninformed manager would claim to have discovered the highproductivity state, a more intensive audit of the manager’s report of the high-productivity state, a2, discourages him from pretending to be informed and thereby decreases the opportunity cost of becoming informed.
3.3. Capital budgeting when information acquisition costs are large As shown below, optimal capital budgeting is qualitatively different when information acquisition costs are large as compared to when they are small. The following proposition characterizes the solution to headquarters’ capital budgeting problem (PH), denoted by ÃÃ ÃÃ kÃÃ ni ; kai and ai , when information acquisition is costly.
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ÃÃ Proposition 3. The optimal auditing intensities are characterized by aÃÃ 1 X0 and a2 40 for FB ¯ ¯ ÃÃ ÃÃ ¯ , and a1 40 for R4R, where R p1 p2 bk2 . Given that a1 40, the optimal capital RXR ÃÃ ÃÃ ÃÃ ÃÃ FB ÃÃ FB allocations are characterized by kÃÃ n1 oka1 , kn2 oka2 ; ka1 4k 1 , and ka2 4k 2 .
¯ Þ, headquarters may choose to When information acquisition costs are not small ðRXR audit the manager’s report of the low-productivity state ðaÃÃ 1 X0Þ. As information ¯ Þ, headquarters certainly chooses to audit the acquisition costs become larger ðR4R manager’s report of the low-productivity state ðaÃÃ 1 40Þ. In such a case, headquarters allocates a larger amount of capital when it audits, ex post, the manager’s report than ÃÃ ÃÃ ÃÃ when it does not ðkÃÃ n1 oka1 ; kn2 oka2 Þ, and more than the ?rst best amount of capital when FB ÃÃ FB it audits the manager’s report ðkÃÃ a1 4k 1 ; ka2 4k 2 Þ. Of course, headquarters always chooses to audit the manager’s report of the high-productivity state regardless of information acquisition costs ðaÃÃ 2 40Þ. If headquarters chooses, ex ante, not to audit the manager’s report of the low-productivity state ðaÃÃ 1 ¼ 0Þ, the optimal capital allocations are qualitatively the same as those when information acquisition costs are small. In contrast to the case where information acquisition costs are small, an interesting result here is that headquarters audits the manager’s report even if the report says that the productivity of capital is low ðaÃÃ 1 40Þ. As shown in Proposition 2, when information acquisition costs are small, the amount of capital allocated for the low-productivity state, kn1, increases as R increases in order to motivate the manager to acquire information. As R increases further so that the amount of capital allocated for the low-productivity state becomes larger, the uninformed manager now develops an incentive to understate the productivity of capital as well. This is because given that there is no audit when he understates (a1 ¼ 0), he can secure a non-trivial amount of capital with certainty. In other words, the distortion of capital allocations to motivate the manager to acquire information creates an incentive to understate the productivity of capital even without acquiring information. To counter this, headquarters also audits the manager’s report of low capital productivity to decrease his opportunity cost of becoming informed. Another contrasting result is that headquarters over-invests in capital whenever it audits FB the manager’s report ðkÃÃ ai 4ki Þ. As R increases, in order to motivate the manager to acquire information, headquarters has to increase the manager’s bene?t of becoming informed, while decreasing the opportunity cost of becoming informed. When information acquisition costs are small, this is accomplished by an increase in kn1 and a decrease in kn2 as shown in Proposition 2. However, when information acquisition costs are large, an increase in kn1 increases the opportunity cost of becoming informed since the uninformed manager now has an incentive to claim that he has learned that capital productivity is low. Headquarters then has to lower kn1, which decreases the opportunity cost, while setting ka1 above the ?rst best level, which increases the bene?t of becoming informed. Similarly, headquarters allows ka2 to be above the ?rst best level to increase the bene?t of becoming informed. In sum, when information acquisition costs are large, the creation of incentives for the manager to acquire information becomes a serious issue. To strengthen incentives, headquarters audits more extensively (such that auditing occurs in every state reported) and allocates more capital when it audits, ex post, the manager’s report than when it does not. By doing so, headquarters can increase the bene?t of becoming informed and, at the same time, decrease the opportunity cost of becoming informed. The analysis has shown that optimal capital budgeting varies with information acquisition costs. In practice, these costs depend on managerial, project, division, and ?rm
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characteristics.13 Including the case of pre-existing information, i.e., zero cost of information acquisition, certainly makes the capital budgeting scheme characterized in Proposition 1 plausible. A question then arises as to how large information acquisition costs must be for headquarters to choose the capital budgeting scheme characterized in ¯ becomes largest ð0:25bkFB Þ. Proposition 3. In the extreme case where p1 ¼ p2 ¼ 0:5, R 2 However, this is still only 25% of the manager’s personal bene?ts when he manages the ?rst best amount of capital for the high-productivity state. It is therefore likely that information acquisition costs can be larger than this amount in many cases, making the capital budgeting scheme characterized in Proposition 3 plausible. 4. Conclusion Corporate managers may not have superior information about the prospects of new or non-routine projects. Instead, they acquire information in connection with normal project initiation or planning tasks. It is then important for headquarters to motivate them to acquire and reveal that information. This paper studies the optimal capital budgeting process that provides such incentives. This article shows that the characteristics of optimal capital budgeting vary with information acquisition costs. For routine projects where information acquisitions costs are zero, headquarters audits the manager’s report only if the manager reports that the productivity of capital is high. When the manager reports a high-productivity state, headquarters allocates the ?rst best amount of capital if it audits the report, and less than the ?rst best amount if it does not audit. When the manager reports a low-productivity state, headquarters allocates more than the ?rst best amount without auditing. Unless information costs are large, this characterization still prevails for new, nonroutine projects where information acquisitions costs are positive. However, there are further distortions in capital allocations and auditing intensities from their optimal levels for routine projects. In particular, optimal capital budgeting consists of relatively larger amounts of capital for the low-productivity state, relatively smaller amounts of capital for the high-productivity state, and more intensive auditing. These distortions occur in order to motivate the manager to acquire information. When information acquisition costs are large, optimal capital budgeting is qualitatively different from that for routine projects. As information acquisitions costs increase, the provision of incentives to acquire information becomes a serious issue. To strengthen incentives, headquarters audits the manager’s report regardless of the productivity state reported therein and allocates more capital when it audits the manager’s report than when it does not. Furthermore, for any productivity state the manager reports, headquarters allocates more than the ?rst best amount of capital when it audits. To conclude, this article offers two important economic insights. First, it suggests that auditing is a robust incentive device for motivating managers to acquire information. Second, it shows that the ef?cient provision of incentives to acquire information involves the distortion of resource allocations beyond that required creating incentives to disclose pre-existing private information.
13 For instance, one can argue that information acquisition is more important and costly in R&D, marketing, and new product divisions. One may also identify new economy ?rms and ?rms in emerging markets as the ?rms where information acquisition is important (see Ittner et al. (2003) and Murphy (2003) for the de?nition of new economy ?rms).
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Appendix A Proof of proposition 1. Simplifying the information acquisition constraints gives ðIAi Þ ð1 À aj Þbknj þ aj bkaj Xð1 À ai Þbkni þ R pj 8iaj .
From these, it is clear that (IAi) implies (ICj) for all iaj , making both (IC) constraints redundant. Let li be the Lagrange multiplier for (IAi), then the ?rst-order conditions for interior solutions for kni, kai, and ai are given by p1 V 01 ðkn1 Þ ¼ l1 À l2 , p1 V 01 ðka1 Þ ¼ Àl2 , p2 V 02 ðkn2 Þ ¼ Àl1 þ l2 , p2 V 02 ðka2 Þ ¼ Àl1 , p1 C 0 ða1 Þ ¼ p1 ½V 1 ðka1 Þ À V 1 ðkn1 Þ? þ l2 ðka1 À kn1 Þ þ l1 kn1 , p2 C 0 ða2 Þ ¼ p2 ½V 2 ðka2 Þ À V 2 ðkn2 Þ? þ l1 ðka2 À kn2 Þ þ l2 kn2 . The following lemmas are used for the rest of the proof. Lemma 1. ai ¼ 0 if and only if li ¼ 0 for all i. Proof. To show ?rst that ai ¼ 0 if li ¼ 0, suppose to the contrary that ai 40 and li ¼ 0. From the ?rst-order conditions stated in (A.1)–(A.4), li being zero implies that kai ¼ kni . Then, since C 0 ð0Þ ¼ 0 and C 0 ðai Þ40 for ai 40, the ?rst-order conditions for ai stated in (A.5) and (A.6) imply that ai ¼ 0, which is a contradiction. Next, to show that li ¼ 0 if ai ¼ 0, suppose to the contrary that li 40 and ai ¼ 0. From the ?rst-order conditions stated in (A.1)–(A.4), li being positive implies that kai 4kni . Then, since C 0 ð0Þ ¼ 0 and C 0 ðai Þ40 for ai 40, the ?rst-order conditions for ai stated in (A.5) and (A.6) imply that ai 40, which is a contradiction. & Corollary 1. ai 40 if and only if li 40 for all i. (A.1) (A.2) (A.3) (A.4) (A.5) (A.6)
Proof. It immediately follows from Lemma 1. Lemma 2. l2 40.
&
Proof. Suppose to the contrary that l2 ¼ 0. The ?rst-order conditions stated in (A.1) and (A.2) imply that kn1 pka1 ¼ kFB 1 , and the ?rst-order conditions stated in (A.3) and (A.4) FB imply that kn2 ¼ ka2 XkFB . Given that kFB 2 1 ok2 , the following inequality holds: ð1 À a1 Þbkn1 þ a1 bka1 obkFB 2 pð1 À a2 Þbk n2 þ R , p1
where the last weak inequality follows from the fact that a2 ¼ 0 when l2 ¼ 0 from Lemma 1. This contradicts (IA2). &
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Using the complementary slackness condition, Lemma 2 implies that (IA2) is binding. ¯ , which will be veri?ed later. From the Assume for a moment that (IA1) is slack for RoR complementary slackness condition, the assumption of (IA1) being slack implies that l1 ¼ 0. Then a1 ¼ 0 from Lemma 1. For a2 , from Corollary 1, Lemma 2 implies that a2 40. For the optimal capital allocations, from the ?rst-order conditions stated in (A.1)–(A.4), FB à FB using the facts that l1 ¼ 0 and that l2 40, one can easily see that kà n1 4k 1 ; kn2 ok2 , and à FB ka2 ¼ k2 . Finally, by substituting the solution into (IA1), it can be checked that the FB à ¯ p1 p2 aà assumption of (IA1) being slack is indeed satis?ed only if RoR 2 bk 2 , where a2 is ¯. & evaluated at R ¼ R
Proof of proposition 2. Using the binding (IA2) constraint and the fact that a1 ¼ 0 in equilibrium, kn1 ¼ ð1 À a2 Þkn2 þ R . p1 b (A.7)
Headquarters’ problem (PH) can then be written as R Max p1 V 1 ð1 À a2 Þkn2 þ þ p2 ½ð1 À a2 ÞV 2 ðkn2 Þ þ a2 V 2 ðka2 Þ À C ða2 Þ?. p1 b Then the solutions for kn1(R), kn2(R) and a2(R) satisfy the following ?rst-order conditions for this problem:
à 0 p1 V 01 ðkà n1 ðRÞÞ þ p2 V 2 ðkn2 ðRÞÞ ¼ 0, à FB à 0 à À p1 V 01 ðkà n1 ðRÞÞk n2 ðRÞ þ p2 ½V 2 ðk 2 Þ À V 2 ðk n2 ðRÞÞ À C ða2 ðRÞÞ? ¼ 0.
Taking the derivative of these with respect to R gives:
à 00 à p1 V 00 1 kn1 ðRÞ þ p2 V 2 kn2 ðRÞ ¼ 0, à à 00 à Àp1 V 00 1 kn1 ðRÞkn2 À p2 C a2 ðRÞ ¼ 0,
0 0 0 0
(A.8) (A.9)
respectively. Using (A.8), (A.9) becomes
à à 00 à V 00 2 kn2 kn2 ðRÞ À C a2 ðRÞ ¼ 0.
0 0
(A.10)
As the solutions satisfy (A.7), taking the derivative of (A.7) with respect to R gives:
à à à à kà n1 ðRÞ ¼ ð1 À a2 Þk n2 ðRÞ À kn2 a2 ðRÞ þ
0 0 0
1 . p1 b
(A.11)
Using (A.10) and (A.11), (A.8) becomes: Á Ã0 1 À V 00 Ã 2 00 00 Ã 00 00 00 00 1 ¼ 0. 00 p1 ð1 À a2 ÞV 1 C þ p2 V 2 C À p1 ðk n2 Þ V 1 V 2 k n2 ðRÞ þ C b
à 00 00 From this, kà n2 ðRÞo0 as C 40 and V i o0.0 Then, with this result that k n2 ðRÞo0, it is Ã0 à evident that kn1 ðRÞ40 from (A.8) and that a2 ðRÞ40 from (A.10).
0 0
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Proof of proposition 3. For the ?rst part of the proposition, it is clear that (IA1) holds ¯ from the proof of Proposition 1. This implies that (IA1) inequality (slack) only if RoR ¯ . Then the complementary slackness condition implies that l1 X0. holds equality if RXR ¯ . As they are not From Lemma 1 and Corollary 1, it must be the case that a1 X0 if RXR ¯. constrained by the size of R, Lemma 2 and Corollary 1 imply that a2 40 if RXR The following lemma is used for the rest of the proof. ¯. Lemma 3. l1 40 if R4R ¯ ð1 þ eÞ, where e40. As shown in Proof. Suppose to the contrary that l1 ¼ 0 and R ¼ R Lemma 2, l2 40. Then the complementary slackness condition implies that (IA2) is binding. Using the binding (IA2) constraint and R ¼ p1 p2 bkFB 2 ð1 þ eÞ, bkn1 þ R ¼ ð1 À a2 Þbkn2 þ bkFB 2 ð 1 þ eÞ . p2 (A.12)
Given that l1 ¼ 0, the ?rst-order condition stated in (A.4) implies that ka2 ¼ kFB 2 . With this, the following inequality holds: ð1 À a2 Þbkn2 þ a2 bka2 oð1 À a2 Þbkn2 þ bkFB 2 ð1 þ eÞ ¼ ð1 À a1 Þbk n1 þ R , p2
where the last equality follows from (A.12) and the fact that a1 ¼ 0 if l1 ¼ 0 from Lemma 1. This contradicts (IA1). & From Corollary 1, l1 being positive implies that aÃÃ 1 40. Again, Lemma 2 and Corollary 1 ¯ imply that aÃÃ 2 40. Finally, given that both l1 and l2 are positive for R4R, the ?rst-order conditions stated in (A.1)–(A.4) prove the characterization of the optimal capital allocations kÃÃ ni ÃÃ ÃÃ ÃÃ ÃÃ ÃÃ FB ÃÃ FB and kÃÃ & ai stated in the proposition: k n1 oka1 ; kn2 oka2 ; ka1 4k 1 , and k a2 4k 2 .
References
Aghion, P., Tirole, J., 1997. Formal and real authority in organizations. Journal of Political Economy 105, 1–29. Antle, R., Eppen, G., 1985. Capital rationing and organizational slack in capital budgeting. Management Science 31, 163–174. Antle, R., Fellingham, J., 1995. Information rents and preferences among information systems in a model of resource allocation. Journal of Accounting Research 33, S41–S63. Arya, A., Fellingham, J., Glover, J., Sivaramakrishnan, K., 2000. Capital budgeting, the hold-up problem, and information system design. Management Science 46, 205–216. Baldenius, T., 2003. Delegated investment decisions and private bene?ts of control. Accounting Review 78, 909–930. Baron, D., Besanko, D., 1984. Regulation, asymmetric information, and auditing. Rand Journal of Economics 15, 447–470. Berkovitch, E., Israel, R., 2004. Why the NPV criterion does not maximize NPV? Review of Financial Studies 17, 239–255. Bernardo, A., Cai, H., Luo, J., 2001. Capital budgeting and compensation with asymmetric information and moral hazard. Journal of Financial Economics 61, 311–344. Bernardo, A., Cai, H., Luo, J., 2004. Capital budgeting in multi-division ?rms: information, agency, and incentives. Review of Financial Studies 17, 739–767. ´ mer, J., Khalil, F., Rochet, J.-C., 1998. Contracts and productive information gathering. Games and Cre Economic Behavior 25, 174–194. Dewatripont, M., Tirole, J., 1999. Advocate. Journal of Political Economy 107, 1–39.
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270 D. Kim / Journal of Accounting and Economics 41 (2006) 257–270 Dutta, S., 2003. Capital budgeting and managerial compensation: incentive and retention effects. Accounting Review 78, 71–93. Dutta, S., Reichelstein, S., 2002. Controlling investment decisions: depreciation and capital charges. Review of Accounting Studies 7, 253–281. Fama, E., Jensen, M., 1983. Separation of ownership and control. Journal of Law and Economics 26, 301–325. Ferreira, E., Brooks, L., 1988. Capital budgeting: a key management challenge. Business 38, 22–29. Fremgen, J., 1973. Capital budgeting practices: a survey. Management Accounting 54, 19–25. Gitman, L., Forrester Jr., J., 1977. A survey of capital budgeting techniques used by major U.S. ?rms. Financial Management 6, 66–71. Harris, M., Raviv, A., 1996. The capital budgeting process: incentives and information. Journal of Finance 51, 1139–1174. Harris, M., Raviv, A., 1998. Capital budgeting and delegation. Journal of Financial Economics 50, 259–289. Harris, M., Kriebel, C., Raviv, A., 1982. Asymmetric information, incentives, and intra?rm resource allocation. Management Science 28, 604–620. Holmstro ¨ m, B., Ricart I Costa, J., 1986. Managerial incentives and capital management. Quarterly Journal of Economics 101, 835–860. Ittner, C., Lambert, R., Larcker, D., 2003. The structure and performance consequences of equity grants to employees of new economy ?rms. Journal of Accounting and Economics 34, 89–127. Kessler, A., 1998. The value of ignorance. Rand Journal of Economics 29, 339–354. Khalil, F., 1997. Auditing without commitment. Rand Journal of Economics 28, 629–640. Lambert, R., 1986. Executive effort and selection of risky projects. Rand Journal of Economics 17, 77–88. Lambert, R., 2001. Contracting theory and accounting. Journal of Accounting and Economics 32, 3–87. Laux, C., 2001. Delegated information acquisition and capital budgeting: on the separation of project evaluation and project management. Journal of Institutional and Theoretical Economics 157, 591–607. Lewis, T., Sappington, D., 1997. Information management in incentive problems. Journal of Political Economy 105, 796–821. Marino, A., Matsusaka, J., 2005. Decision processes, agency problems, and information: an economic analysis of capital budgeting procedures. Review of Financial Studies 18, 301–325. Murphy, K.J., 2003. Stock-based pay in new economy ?rms. Journal of Accounting and Economics 34, 129–147. Stein, J., 2002. Information production and capital allocation: decentralized versus hierarchical ?rms. Journal of Finance 57, 1891–1921. Taggart Jr., R., 1987. Allocating capital among a ?rm’s divisions: hurdle rates vs. budgets. Journal of Financial Research 10, 177–190. Zhang, G., 1997. Moral hazard in corporate investment and the disciplinary role of voluntary capital rationing. Management Science 43, 737–750.
doc_872696282.pdf
A budget is a quantitative expression of a plan for a defined period of time. It may include planned sales volumes and revenues, resource quantities
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Journal of Accounting and Economics 41 (2006) 257–270 www.elsevier.com/locate/jae
Capital budgeting for new projects: On the role of auditing in information acquisition$
Doyoung KimÃ
College of Business and Economics, University of Idaho, Moscow, ID 83844-3161, USA Received 24 February 2005; received in revised form 5 March 2006; accepted 15 March 2006 Available online 9 June 2006
Abstract This article studies capital budgeting for new projects in which information is acquired by managers. When information acquisition costs are small, optimal capital budgeting is not qualitatively different from that for routine projects where managers have pre-existing information. However, the need to provide incentives to acquire information results in more intensive auditing and further distortions in capital allocations. When information acquisition costs are large, optimal capital budgeting differs from that for routine projects. To provide strong incentives for information acquisition, auditing becomes more extensive, and more than the ?rst best amount of capital is allocated whenever auditing occurs. r 2006 Elsevier B.V. All rights reserved.
JEL classi?cation: D82; D83; G31; M42 Keywords: Capital allocation; Auditing; Information acquisition
1. Introduction Capital budgeting may be the most important decision made by corporations. While the net present value rule has long been a standard prescription, it does not guide the practical details of internal capital allocation (Harris and Raviv, 1996). In response to this, researchers have paid attention to asymmetric information in internal capital markets. In
$
ÃTel.: +1 208 885 7006.
I thank Rick Antle (the referee) and Ross Watts (the editor) for helpful comments.
E-mail address: [email protected]. 0165-4101/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jacceco.2006.03.001
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particular, they posit that division managers have better information than headquarters about the prospects of projects at the outset.1 Due to unaligned interests, information asymmetry creates agency problems. Capital budgeting is then optimally structured to provide managerial incentives for information revelation. While the assumption of endowed information is a reasonable approximation in some settings, managers may not be omniscient from the outset in many relevant settings (Lewis and Sappington, 1997). In particular, while managers could have information about the ongoing projects they have routinely managed, they might not have superior information about new or non-routine projects. For the successful implementation of such projects, it is important to motivate managers to acquire valuable information about the projects. However, there has been scant attention to this issue in the literature on capital budgeting. This article studies optimal capital budgeting from the standpoint of ef?ciently motivating managers to acquire and reveal information. The paper ?rst discovers an important, but unexplored, role of auditing. For routine projects where information is given to a manager at the outset, auditing is simply used to discipline the manager’s incentive to lie about his pre-existing private information. However, for new projects where information should be acquired by the manager, auditing is used to motivate him to acquire that information. More intensive and extensive auditing encourages the manager to acquire information by decreasing the opportunity cost of information acquisition. In addition to the disciplinary role the literature has presumed, this paper thus emphasizes the positive incentive role of auditing. Next, it ?nds that the optimal allocation of capital for new projects is qualitatively different from that for routine projects. The ef?cient provision of incentives to acquire information results in the distortion of resource allocations beyond that involved in the creation of incentives to disclose pre-existing private information. Formally, this study extends a standard capital budgeting model to consider a project where information is endogenously acquired by a manager. In particular, at the outset neither the manager nor headquarters has information about the productivity of capital, which will determine the net present value of the project. The manager, who has a preference for empire building, can acquire the information by incurring the personal costs of the information acquisition effort. Headquarters audits the manager’s report on the productivity of capital to verify the truthfulness of the report.2 Optimal capital budgeting can be characterized according to information acquisition costs. When these costs are small, headquarters audits (randomly) only if the manager reports that the productivity of capital is high. To discipline the manager’s incentive to overstate capital productivity, headquarters allocates less than the ?rst best amount of capital for the high-productivity state and more than the ?rst best amount of capital for the low-productivity state. Since these results hold unless information acquisition costs are large, they can also characterize the capital budgeting process for routine projects where information acquisition costs are zero, or equivalently where information is given to the manager at the outset. A contribution of the paper is then to show that headquarters
1 See for instance Harris et al. (1982), Antle and Eppen (1985), Holmstro ¨ m and Ricart I Costa (1986), Harris and Raviv (1996, 1998), Zhang (1997), Bernardo et al. (2001, 2004), Dutta and Reichelstein (2002), Dutta (2003), Baldenius (2003), Berkovitch and Israel (2004), and Marino and Matsusaka (2005). 2 The model follows Fama and Jensen (1983) in that the manager is responsible for project initiation (by information acquisition) and implementation, and headquarters is in charge of project rati?cation and auditing.
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distorts capital allocations and auditing intensities from their optimal levels for routine projects. In particular, more capital is allocated for the low-productivity state, less capital is allocated for the high-productivity state, and auditing occurs with higher probabilities. These distortions arise in order to motivate the manager to acquire information. Due to the manager’s preference for empire building, he has an incentive to request more capital by overstating the productivity of capital even without acquiring information. A more intensive audit of the manager’s capital request and a more stringent allocation of capital for the high-productivity state then discourage the manager from pretending to be informed and requesting more capital. That is, they reduce the opportunity cost of becoming informed. When information acquisition costs are large, optimal capital budgeting for new projects is qualitatively different from that for routine projects. With large information acquisition costs, the provision of incentives to acquire information becomes a serious issue. To provide strong incentives, headquarters broadens its audit such that it investigates the manager’s report even if the report says that the productivity of capital is low. A more extensive audit provides better incentives to acquire information by reducing the opportunity cost of becoming informed. In addition, for any productivity state the manager reports, headquarters allocates more than the ?rst best amount of capital when it audits the manager’s report, and more capital when it audits than when it does not. Linking more capital to the audit that veri?es the manager’s report increases the bene?t of becoming informed and, at the same time, decreases the opportunity cost of becoming informed. This article is closely related to the paper by Harris and Raviv (1996) as it extends their model.3 While they consider a project where information is given at the outset, this article focuses on a project where information must be acquired by a manager. With this difference, it shows that the capital budgeting process identi?ed in their paper may not always be optimal and that auditing is used as an incentive device to motivate a manager to acquire information. There are a few papers that model information acquisition in capital budgeting.4 Arya et al. (2000) study the effect of information systems (early vs. late information) on a manager’s incentive to exert research effort on the choice of projects.5 They show that information systems can serve as a commitment device to allow the manager to enjoy information rents that motivate him to exert research effort. Laux (2001) studies who (a division manager vs. a budget center) should be assigned the task of information acquisition. He shows that it depends on expected wage costs in inducing information acquisition. Stein (2002) studies which type of organization, a small decentralized ?rm or a large hierarchical ?rm, better motivates a manager to acquire information. He shows that it depends on the veri?ability of information. While the papers
The paper is also broadly related to the literature on capital budgeting for a single division under asymmetric information. The list includes Harris et al. (1982), Antle and Eppen (1985), Holmstro ¨ m and Ricart I Costa (1986), Zhang (1997), Bernardo et al. (2001), Dutta and Reichelstein (2002), Dutta (2003), and Baldenius (2003). In these papers, information is given at the outset. See also a review by Lambert (2001). 4 In a project choice model, Lambert (1986) pioneered the modeling of information acquisition. See also Lewis ´ mer et al. (1998), and Kessler (1998) for the modeling of information acquisition in and Sappington (1997), Cre procurement contracts, and Aghion and Tirole (1997) and Dewatripont and Tirole (1999) in the allocation of authority and organizational design. 5 Antle and Fellingham (1995) also study the choice of (public) information systems. But they focus on the effect of information systems on a manager’s activities that in?uence the ?rm’s choice of information systems.
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mentioned above take an organizational focus, this article examines the distortions in resource allocation and auditing involved in the provision of incentives to acquire information. The rest of the paper is organized as follows. Section 2 presents the model of capital budgeting with information acquisition. Section 3 characterizes optimal capital budgeting with regard to ef?ciently motivating a manager to acquire and reveal information. Section 4 concludes the paper. All proofs are relegated to the Appendix A. 2. Model The model consists of a risk-neutral headquarters and a risk-neutral division manager. Headquarters has access to capital k, which will be allocated to the manager for the implementation of a new or non-routine project.6 The state of nature, indexed by i 2 f1; 2g, determines the net present value of the project Vi(k), where Vi( Á ) is a strict concave function with V i ð0Þ ¼ 0 and V 0i ð0Þ ¼ 1, and has unique value-maximizing capital kFB i such FB that V 0i ðkFB Þ ¼ 0. k is then the ?rst best amount of capital that headquarters would i i allocate if it were informed about the state of the project. The productivity of capital is higher under i ¼ 2 than under i ¼ 1. That is, V 01 ðkÞoV 02 ðkÞ for all k, implying that FB kFB 1 ok2 . At the outset, neither headquarters nor the manager has information about the state of the project or, equivalently, the productivity of capital. Instead, they share the common belief that state i will be realized with probability pi 2 ð0; 1Þ, where p1 þ p2 ¼ 1. It is essential for headquarters to know the productivity of capital since capital can then be more ef?ciently allocated according to its productivity. Headquarters lets the manager acquire information about the productivity of capital, which can be justi?ed by the fact that corporate managers are responsible for the planning of projects in which information acquisition may be a main task. In other words, the manager’s experience in project planning (his human capital) makes it much less costly for headquarters to have the manager acquire information than to undertake its own information acquisition effort. The manager can acquire information about the productivity of capital by incurring a cost R, which can be seen as the manager’s personal, unobservable cost of research effort. This is the only chance to learn the productivity of capital, so capital allocation and project implementation will be done without information if the manager chooses not to acquire information. To make information acquisition relevant in the model, it is assumed that the information acquisition cost R is not too large, so that headquarters prefers to induce the manager to acquire information. As is standard in the literature, it is assumed that information acquisition is a private action by the manager and that the information acquired is private and unveri?able. Headquarters is then unaware of not only whether the manager acquired the information, but also which state he discovered.7 The manager has a preference for empire building.8 He derives higher utility from managing more capital and bigger projects. This may be due to the fact that a manager’s
It is thereby assumed that the manager has no resources of his own and that all resources must come from headquarters. 7 ´ mer et al. For this modeling of endogenous information acquisition, see Lewis and Sappington (1997) and Cre (1998). 8 For discussions about corporate managers’ preference for empire building and its modeling, see for instance Harris and Raviv (1996, 1998), Bernardo et al. (2001, 2004), and Baldenius (2003).
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reputation is enhanced by the size of the projects or divisions he has managed. It may also be the case that managers have more opportunities to enjoy perquisite consumption when they manage more capital. To model this, let the manager’s utility from managing capital be bk, where b is the manager’s marginal bene?t associated with additional capital. To simplify the analysis, monetary compensation is assumed away from the manager’s utility function, re?ecting the case where headquarters’ capital allocation decision is independent of its decision about managerial compensation. Technically, this simpli?cation implies that capital allocations and auditing are the only incentive devices available to headquarters. However, it must be noted that optimal capital budgeting as characterized in this simple model would still qualitatively survive even in a model that explicitly considers monetary incentives.9 The manager reports information about the productivity of capital to headquarters. Notice that the manager’s report does not have to be truthful since it is not veri?able. With the manager’s preference for empire building, this creates an agency problem, which in turn creates a need for monitoring. Headquarters therefore audits the manager’s report. Given that the manager reports state i, headquarters can choose auditing intensity ai 2 ½0; 1? by incurring a cost of C(ai), where C( Á ) is a strict convex function with C ð0Þ ¼ 0 and C 0 ð0Þ ¼ 0. In particular, with probability ai, headquarters audits the manager’s report and discovers whether the manager has reported truthfully or not. However, with probability (1–ai), headquarters does not audit and thereby receives no information about the veracity of the manager’s report. Based on the outcome of the audit, headquarters allocates capital to the manager for the implementation of the project. There are four possible capital allocations in equilibrium: kni and kai for all i 2 f1; 2g. kni is the amount of capital allocated when auditing does not occur for the manager’s report i, and kai is the amount of capital allocated when auditing reveals that the manager’s report i is true. Notice that there is no need for the introduction of capital allocations when auditing reveals that the manager’s report i is untruthful as these allocations never occur in equilibrium due to the direct revelation mechanism employed in solving the model. Out-of-equilibrium capital allocations in such a case are optimally zero. Intuitively, headquarters would not receive any gains from allowing the manager to lie. Since headquarters would not bear any cost of imposing a maximum penalty for the manager’s lie, it would allocate zero capital—the maximum penalty in the model—when it discovers the manager’s lie.10 The interaction between headquarters and the manager can be summarized as follows. Headquarters announces a capital budgeting scheme consisting of capital allocations and auditing intensities: {kni, kai, ai}. As is standard in the literature, it is assumed that
As will be shown later, headquarters distorts capital allocations to motivate the manager to acquire and reveal information. Headquarters can also provide such an incentive by giving the manager information rents in the form of monetary compensation. However, notice that the manager cannot take or use capital for his personal purposes in the model. Then, as is standard in hidden information models, monetary compensation in?icts a ?rstorder loss while a distortion in capital allocations in?icts a second-order loss to headquarters. Hence, in a model with monetary compensation, headquarters ?rst distorts capital allocations and then gives information rents as the distortion gets worse, though headquarters ?nally use both in equilibrium in order to minimize the loss. It implies that the optimal capital allocations are qualitatively the same regardless of the inclusion of monetary compensation in the model. 10 See Baron and Besanko (1984) for the principle of maximum penalties.
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headquarters can commit to this scheme.11 Given this scheme, the manager acquires information about the productivity of capital. The manager then reports this information. Headquarters audits the manager’s report i with auditing intensity ai as announced at the beginning. Headquarters allocates capital to the manager who then implements the project and renders the net present value of the project to headquarters. 3. Results 3.1. Headquarters’ problem Since information is essential for the successful implementation of the project, headquarters lets the manager acquire information. However, notice that information acquisition is not observable, which may imply that headquarters faces the manager’s moral hazard problem in the task of information acquisition. The manager in fact has an incentive not to acquire information in order to save his personal acquisition cost R, but to nonetheless pretend that he did acquire information. Headquarters then has to motivate the manager to acquire information. It can do so by making the manager’s expected payoff when he becomes informed at least as large as his expected payoff when he remains uninformed, as shown in the following information acquisition constraints: ðIAi Þ p1 ½ð1 À a1 Þbkn1 þ a1 bka1 ? þ p2 ½ð1 À a2 Þbkn2 þ a2 bka2 ? À RXð1 À ai Þbkni þ pi ai bkai 8i. The LHS is the manager’s expected payoff when he acquires information, and the RHS is his expected payoff when he is uninformed and claims that he discovered state i. Even without acquiring information, he can claim that he discovered state i since headquarters cannot distinguish between an informed and uninformed manager. If he reports state i, with probability (1–ai), no auditing occurs and he receives kni. With probability ai, auditing occurs and reveals whether the manager’s report is correct or not, and the manager receives kai if his report is correct (with probability pi) and no capital allocation otherwise. After inducing information acquisition, headquarters faces another agency problem. The information acquired is private to the manager. Given his preference for empire building, the manager lies about the true productivity of capital in a way to maximize his payoff. To discipline this incentive, headquarters’ decision on capital budgeting must be subject to the following incentive compatibility constraints: ðIC i Þ ð1 À ai Þbkni þ ai bkai Xð1 À aj Þbknj 8iaj 2 f1; 2g.
These constraints guarantee that the manager is better off by reporting truthfully.
11 Whether headquarters can commit to auditing may be an issue since the manager will be induced to report truthfully in equilibrium, which in turn makes auditing useless ex post. Without a commitment to auditing, a question is then whether there exists a mixed strategy equilibrium where the manager lies with some probabilities as in Khalil (1997) who studies auditing in a procurement model. A necessary condition for the existence of the mixed strategy equilibrium is that headquarters receives ex-post gains by letting the manager lie since it creates a trade-off between the manager’s truth-telling and lying from headquarters’ perspective. However, one can easily see that this condition does not hold in the model since headquarters does not receive any ex-post gains by letting the manager lie and then punishing him for his lie. Thus, the mixed strategy equilibrium would not arise in the model considered here.
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With the information acquisition and incentive compatibility constraints, headquarters’ optimal decision on {kni, kai, ai} is max
2 P i ¼1
pi ½ð1 À ai ÞV i ðkni Þ þ ai V i ðkai Þ À C ðai Þ? 8i:
ðPH Þ
s:t: ðIAi Þ; ðIC i Þ; 0pai p1; kni X0; kai X0
Notice that capital allocations are restricted to be non-negative, making the individual rationality conditions redundant by guaranteeing non-negative payoffs to the manager. 3.2. Capital budgeting when information acquisition costs are small The solution to headquarters’ problem (PH) above depends on information acquisition costs. The following proposition characterizes the optimal capital budgeting scheme, à à denoted by kà ni ; kai and ai , when information acquisition costs are small. ¯ , the optimal auditing intensities and capital allocations are Proposition 1. For RoR à FB à FB à FB FB à à ¯ characterized by a1 ¼ 0, aà 2 40, k n1 4k 1 , k n2 ok2 , and k a2 ¼ k 2 , where R p1 p2 a2 bk 2 . Headquarters chooses to audit only if the manager reports that the productivity of à capital is high ðaà 1 ¼ 0; a2 40Þ. For the manager’s report of high capital productivity, headquarters allocates the ?rst best amount of capital if it audits the manager’s report ex FB à FB post ðkà a2 ¼ k 2 Þ and less than the ?rst best amount otherwise ðk n2 ok2 Þ. When the manager reports that the productivity of capital is low, headquarters allocates more than FB à the ?rst best amount without auditing ðkà n1 4k 1 Þ. As a1 ¼ 0, ka1 is irrelevant. Since the manager with a preference for empire building would not understate the productivity of capital, there is no need for an audit when he reports that the productivity of capital is low ðaà 1 ¼ 0Þ. However, an audit is necessary when he reports that the productivity of capital is high since otherwise he would always overstate the productivity of capital regardless of whether or not he is informed about the productivity ðaà 2 4 0Þ . Headquarters distorts capital allocations to discipline the manager’s incentive to lie when there is no audit. In particular, headquarters provides the manager with an incentive not to overstate capital productivity by increasing the amount of capital allocated for the FB low-productivity state above the ?rst best level ðkà n1 4k 1 Þ, and discourages the manager from overstating capital productivity by decreasing the amount of capital allocated for the FB high-productivity state below the ?rst best level ðkà n2 ok2 Þ. Of course, there is no need for a distortion in capital allocations when an audit reveals that the manager’s report is FB truthful ðkà a2 ¼ k 2 Þ. To summarize, over-investment occurs when the productivity of FB capital is low ðkà n1 4k 1 Þ, whereas under-investment occurs when the productivity of à FB FB 12 capital is high ðkn2 ok2 ; kà a2 ¼ k 2 Þ.
12 Following Harris and Raviv (1996), one can argue that the optimal capital budgeting scheme described in Proposition 1 can be also implemented by a typical capital budgeting process consisting of an initial spending limit and additional capital allocations upon approval (see Fremgen (1973), Gitman and Forrester (1977), Taggart (1987), and Ferreira and Brooks (1988) for a survey of this capital budgeting process in corporations). Headquarters sets an initial spending limit equal to kà n1 . When the manager discovers that capital productivity is à high, he requests more capital. Then headquarters allocates additional capital equal to ðkà a2 À kn1 Þ when it audits à à and veri?es the manager’s request, and ðkn2 À kn1 Þ otherwise.
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Since these results hold unless information acquisition costs are large, they prevail even in routine projects where information is freely given at the outset, i.e., where information acquisition costs are zero in the model. A question then arises as to the role of endogenous information acquisition in optimal capital budgeting. To explore this issue, let the information acquisition cost R increase above zero. A comparative static analysis with respect to R, summarized in the following proposition, reveals the role of endogenous information acquisition.
à à ¯ , aà Proposition 2. For RoR 2 and kn1 increase with R, whereas k n2 decreases with R.
The probability of auditing ðaà 2 Þ increases with the cost of information acquisition. When there is no audit, the amount of capital allocated for the low-productivity state ðkà n1 Þ increases, while the amount of capital allocated for the high-productivity state ðkà n2 Þ decreases with the cost of information acquisition. If the cost of information acquisition is zero, headquarters deals with the manager only for information revelation. However, as the cost increases above zero, the manager’s moral hazard in information acquisition becomes an issue, and headquarters has to motivate the manager to acquire information in the ?rst place. To do so, as shown above, headquarters distorts capital allocations further from the optimal levels for routine projects. The intuition behind these distortions is straightforward. The manager acquires information if the bene?t of becoming informed, i.e., the expected payoff from becoming informed, is larger than the opportunity cost of becoming informed, i.e., the expected payoff when he remains uninformed and pretends to be informed. To motivate the manager to acquire information, headquarters has to increase the bene?t of becoming informed, while decreasing the opportunity cost of becoming informed. To increase the bene?t of becoming informed, headquarters may have to increase the amount of capital allocated to the à à manager, kà n1 or kn2 or both. However, an increase in k n2 increases the opportunity cost of becoming informed as well. With a preference for empire building, the uninformed manager would claim to have discovered the high-productivity state in order to receive à more capital, kà n2 , which constitutes the opportunity cost. Indeed, as k n2 increases, the opportunity cost increases more than the bene?t of becoming informed. Thus, while à headquarters increases kà n1 to improve the bene?t of becoming informed, it decreases k n2 to reduce the opportunity cost of becoming informed. An interesting result here is that, to motivate the manager to acquire information, headquarters increases the probability of auditing above the optimal level for routine projects. Again, since the uninformed manager would claim to have discovered the highproductivity state, a more intensive audit of the manager’s report of the high-productivity state, a2, discourages him from pretending to be informed and thereby decreases the opportunity cost of becoming informed.
3.3. Capital budgeting when information acquisition costs are large As shown below, optimal capital budgeting is qualitatively different when information acquisition costs are large as compared to when they are small. The following proposition characterizes the solution to headquarters’ capital budgeting problem (PH), denoted by ÃÃ ÃÃ kÃÃ ni ; kai and ai , when information acquisition is costly.
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ÃÃ Proposition 3. The optimal auditing intensities are characterized by aÃÃ 1 X0 and a2 40 for FB ¯ ¯ ÃÃ ÃÃ ¯ , and a1 40 for R4R, where R p1 p2 bk2 . Given that a1 40, the optimal capital RXR ÃÃ ÃÃ ÃÃ ÃÃ FB ÃÃ FB allocations are characterized by kÃÃ n1 oka1 , kn2 oka2 ; ka1 4k 1 , and ka2 4k 2 .
¯ Þ, headquarters may choose to When information acquisition costs are not small ðRXR audit the manager’s report of the low-productivity state ðaÃÃ 1 X0Þ. As information ¯ Þ, headquarters certainly chooses to audit the acquisition costs become larger ðR4R manager’s report of the low-productivity state ðaÃÃ 1 40Þ. In such a case, headquarters allocates a larger amount of capital when it audits, ex post, the manager’s report than ÃÃ ÃÃ ÃÃ when it does not ðkÃÃ n1 oka1 ; kn2 oka2 Þ, and more than the ?rst best amount of capital when FB ÃÃ FB it audits the manager’s report ðkÃÃ a1 4k 1 ; ka2 4k 2 Þ. Of course, headquarters always chooses to audit the manager’s report of the high-productivity state regardless of information acquisition costs ðaÃÃ 2 40Þ. If headquarters chooses, ex ante, not to audit the manager’s report of the low-productivity state ðaÃÃ 1 ¼ 0Þ, the optimal capital allocations are qualitatively the same as those when information acquisition costs are small. In contrast to the case where information acquisition costs are small, an interesting result here is that headquarters audits the manager’s report even if the report says that the productivity of capital is low ðaÃÃ 1 40Þ. As shown in Proposition 2, when information acquisition costs are small, the amount of capital allocated for the low-productivity state, kn1, increases as R increases in order to motivate the manager to acquire information. As R increases further so that the amount of capital allocated for the low-productivity state becomes larger, the uninformed manager now develops an incentive to understate the productivity of capital as well. This is because given that there is no audit when he understates (a1 ¼ 0), he can secure a non-trivial amount of capital with certainty. In other words, the distortion of capital allocations to motivate the manager to acquire information creates an incentive to understate the productivity of capital even without acquiring information. To counter this, headquarters also audits the manager’s report of low capital productivity to decrease his opportunity cost of becoming informed. Another contrasting result is that headquarters over-invests in capital whenever it audits FB the manager’s report ðkÃÃ ai 4ki Þ. As R increases, in order to motivate the manager to acquire information, headquarters has to increase the manager’s bene?t of becoming informed, while decreasing the opportunity cost of becoming informed. When information acquisition costs are small, this is accomplished by an increase in kn1 and a decrease in kn2 as shown in Proposition 2. However, when information acquisition costs are large, an increase in kn1 increases the opportunity cost of becoming informed since the uninformed manager now has an incentive to claim that he has learned that capital productivity is low. Headquarters then has to lower kn1, which decreases the opportunity cost, while setting ka1 above the ?rst best level, which increases the bene?t of becoming informed. Similarly, headquarters allows ka2 to be above the ?rst best level to increase the bene?t of becoming informed. In sum, when information acquisition costs are large, the creation of incentives for the manager to acquire information becomes a serious issue. To strengthen incentives, headquarters audits more extensively (such that auditing occurs in every state reported) and allocates more capital when it audits, ex post, the manager’s report than when it does not. By doing so, headquarters can increase the bene?t of becoming informed and, at the same time, decrease the opportunity cost of becoming informed. The analysis has shown that optimal capital budgeting varies with information acquisition costs. In practice, these costs depend on managerial, project, division, and ?rm
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characteristics.13 Including the case of pre-existing information, i.e., zero cost of information acquisition, certainly makes the capital budgeting scheme characterized in Proposition 1 plausible. A question then arises as to how large information acquisition costs must be for headquarters to choose the capital budgeting scheme characterized in ¯ becomes largest ð0:25bkFB Þ. Proposition 3. In the extreme case where p1 ¼ p2 ¼ 0:5, R 2 However, this is still only 25% of the manager’s personal bene?ts when he manages the ?rst best amount of capital for the high-productivity state. It is therefore likely that information acquisition costs can be larger than this amount in many cases, making the capital budgeting scheme characterized in Proposition 3 plausible. 4. Conclusion Corporate managers may not have superior information about the prospects of new or non-routine projects. Instead, they acquire information in connection with normal project initiation or planning tasks. It is then important for headquarters to motivate them to acquire and reveal that information. This paper studies the optimal capital budgeting process that provides such incentives. This article shows that the characteristics of optimal capital budgeting vary with information acquisition costs. For routine projects where information acquisitions costs are zero, headquarters audits the manager’s report only if the manager reports that the productivity of capital is high. When the manager reports a high-productivity state, headquarters allocates the ?rst best amount of capital if it audits the report, and less than the ?rst best amount if it does not audit. When the manager reports a low-productivity state, headquarters allocates more than the ?rst best amount without auditing. Unless information costs are large, this characterization still prevails for new, nonroutine projects where information acquisitions costs are positive. However, there are further distortions in capital allocations and auditing intensities from their optimal levels for routine projects. In particular, optimal capital budgeting consists of relatively larger amounts of capital for the low-productivity state, relatively smaller amounts of capital for the high-productivity state, and more intensive auditing. These distortions occur in order to motivate the manager to acquire information. When information acquisition costs are large, optimal capital budgeting is qualitatively different from that for routine projects. As information acquisitions costs increase, the provision of incentives to acquire information becomes a serious issue. To strengthen incentives, headquarters audits the manager’s report regardless of the productivity state reported therein and allocates more capital when it audits the manager’s report than when it does not. Furthermore, for any productivity state the manager reports, headquarters allocates more than the ?rst best amount of capital when it audits. To conclude, this article offers two important economic insights. First, it suggests that auditing is a robust incentive device for motivating managers to acquire information. Second, it shows that the ef?cient provision of incentives to acquire information involves the distortion of resource allocations beyond that required creating incentives to disclose pre-existing private information.
13 For instance, one can argue that information acquisition is more important and costly in R&D, marketing, and new product divisions. One may also identify new economy ?rms and ?rms in emerging markets as the ?rms where information acquisition is important (see Ittner et al. (2003) and Murphy (2003) for the de?nition of new economy ?rms).
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Appendix A Proof of proposition 1. Simplifying the information acquisition constraints gives ðIAi Þ ð1 À aj Þbknj þ aj bkaj Xð1 À ai Þbkni þ R pj 8iaj .
From these, it is clear that (IAi) implies (ICj) for all iaj , making both (IC) constraints redundant. Let li be the Lagrange multiplier for (IAi), then the ?rst-order conditions for interior solutions for kni, kai, and ai are given by p1 V 01 ðkn1 Þ ¼ l1 À l2 , p1 V 01 ðka1 Þ ¼ Àl2 , p2 V 02 ðkn2 Þ ¼ Àl1 þ l2 , p2 V 02 ðka2 Þ ¼ Àl1 , p1 C 0 ða1 Þ ¼ p1 ½V 1 ðka1 Þ À V 1 ðkn1 Þ? þ l2 ðka1 À kn1 Þ þ l1 kn1 , p2 C 0 ða2 Þ ¼ p2 ½V 2 ðka2 Þ À V 2 ðkn2 Þ? þ l1 ðka2 À kn2 Þ þ l2 kn2 . The following lemmas are used for the rest of the proof. Lemma 1. ai ¼ 0 if and only if li ¼ 0 for all i. Proof. To show ?rst that ai ¼ 0 if li ¼ 0, suppose to the contrary that ai 40 and li ¼ 0. From the ?rst-order conditions stated in (A.1)–(A.4), li being zero implies that kai ¼ kni . Then, since C 0 ð0Þ ¼ 0 and C 0 ðai Þ40 for ai 40, the ?rst-order conditions for ai stated in (A.5) and (A.6) imply that ai ¼ 0, which is a contradiction. Next, to show that li ¼ 0 if ai ¼ 0, suppose to the contrary that li 40 and ai ¼ 0. From the ?rst-order conditions stated in (A.1)–(A.4), li being positive implies that kai 4kni . Then, since C 0 ð0Þ ¼ 0 and C 0 ðai Þ40 for ai 40, the ?rst-order conditions for ai stated in (A.5) and (A.6) imply that ai 40, which is a contradiction. & Corollary 1. ai 40 if and only if li 40 for all i. (A.1) (A.2) (A.3) (A.4) (A.5) (A.6)
Proof. It immediately follows from Lemma 1. Lemma 2. l2 40.
&
Proof. Suppose to the contrary that l2 ¼ 0. The ?rst-order conditions stated in (A.1) and (A.2) imply that kn1 pka1 ¼ kFB 1 , and the ?rst-order conditions stated in (A.3) and (A.4) FB imply that kn2 ¼ ka2 XkFB . Given that kFB 2 1 ok2 , the following inequality holds: ð1 À a1 Þbkn1 þ a1 bka1 obkFB 2 pð1 À a2 Þbk n2 þ R , p1
where the last weak inequality follows from the fact that a2 ¼ 0 when l2 ¼ 0 from Lemma 1. This contradicts (IA2). &
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Using the complementary slackness condition, Lemma 2 implies that (IA2) is binding. ¯ , which will be veri?ed later. From the Assume for a moment that (IA1) is slack for RoR complementary slackness condition, the assumption of (IA1) being slack implies that l1 ¼ 0. Then a1 ¼ 0 from Lemma 1. For a2 , from Corollary 1, Lemma 2 implies that a2 40. For the optimal capital allocations, from the ?rst-order conditions stated in (A.1)–(A.4), FB à FB using the facts that l1 ¼ 0 and that l2 40, one can easily see that kà n1 4k 1 ; kn2 ok2 , and à FB ka2 ¼ k2 . Finally, by substituting the solution into (IA1), it can be checked that the FB à ¯ p1 p2 aà assumption of (IA1) being slack is indeed satis?ed only if RoR 2 bk 2 , where a2 is ¯. & evaluated at R ¼ R
Proof of proposition 2. Using the binding (IA2) constraint and the fact that a1 ¼ 0 in equilibrium, kn1 ¼ ð1 À a2 Þkn2 þ R . p1 b (A.7)
Headquarters’ problem (PH) can then be written as R Max p1 V 1 ð1 À a2 Þkn2 þ þ p2 ½ð1 À a2 ÞV 2 ðkn2 Þ þ a2 V 2 ðka2 Þ À C ða2 Þ?. p1 b Then the solutions for kn1(R), kn2(R) and a2(R) satisfy the following ?rst-order conditions for this problem:
à 0 p1 V 01 ðkà n1 ðRÞÞ þ p2 V 2 ðkn2 ðRÞÞ ¼ 0, à FB à 0 à À p1 V 01 ðkà n1 ðRÞÞk n2 ðRÞ þ p2 ½V 2 ðk 2 Þ À V 2 ðk n2 ðRÞÞ À C ða2 ðRÞÞ? ¼ 0.
Taking the derivative of these with respect to R gives:
à 00 à p1 V 00 1 kn1 ðRÞ þ p2 V 2 kn2 ðRÞ ¼ 0, à à 00 à Àp1 V 00 1 kn1 ðRÞkn2 À p2 C a2 ðRÞ ¼ 0,
0 0 0 0
(A.8) (A.9)
respectively. Using (A.8), (A.9) becomes
à à 00 à V 00 2 kn2 kn2 ðRÞ À C a2 ðRÞ ¼ 0.
0 0
(A.10)
As the solutions satisfy (A.7), taking the derivative of (A.7) with respect to R gives:
à à à à kà n1 ðRÞ ¼ ð1 À a2 Þk n2 ðRÞ À kn2 a2 ðRÞ þ
0 0 0
1 . p1 b
(A.11)
Using (A.10) and (A.11), (A.8) becomes: Á Ã0 1 À V 00 Ã 2 00 00 Ã 00 00 00 00 1 ¼ 0. 00 p1 ð1 À a2 ÞV 1 C þ p2 V 2 C À p1 ðk n2 Þ V 1 V 2 k n2 ðRÞ þ C b
à 00 00 From this, kà n2 ðRÞo0 as C 40 and V i o0.0 Then, with this result that k n2 ðRÞo0, it is Ã0 à evident that kn1 ðRÞ40 from (A.8) and that a2 ðRÞ40 from (A.10).
0 0
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Proof of proposition 3. For the ?rst part of the proposition, it is clear that (IA1) holds ¯ from the proof of Proposition 1. This implies that (IA1) inequality (slack) only if RoR ¯ . Then the complementary slackness condition implies that l1 X0. holds equality if RXR ¯ . As they are not From Lemma 1 and Corollary 1, it must be the case that a1 X0 if RXR ¯. constrained by the size of R, Lemma 2 and Corollary 1 imply that a2 40 if RXR The following lemma is used for the rest of the proof. ¯. Lemma 3. l1 40 if R4R ¯ ð1 þ eÞ, where e40. As shown in Proof. Suppose to the contrary that l1 ¼ 0 and R ¼ R Lemma 2, l2 40. Then the complementary slackness condition implies that (IA2) is binding. Using the binding (IA2) constraint and R ¼ p1 p2 bkFB 2 ð1 þ eÞ, bkn1 þ R ¼ ð1 À a2 Þbkn2 þ bkFB 2 ð 1 þ eÞ . p2 (A.12)
Given that l1 ¼ 0, the ?rst-order condition stated in (A.4) implies that ka2 ¼ kFB 2 . With this, the following inequality holds: ð1 À a2 Þbkn2 þ a2 bka2 oð1 À a2 Þbkn2 þ bkFB 2 ð1 þ eÞ ¼ ð1 À a1 Þbk n1 þ R , p2
where the last equality follows from (A.12) and the fact that a1 ¼ 0 if l1 ¼ 0 from Lemma 1. This contradicts (IA1). & From Corollary 1, l1 being positive implies that aÃÃ 1 40. Again, Lemma 2 and Corollary 1 ¯ imply that aÃÃ 2 40. Finally, given that both l1 and l2 are positive for R4R, the ?rst-order conditions stated in (A.1)–(A.4) prove the characterization of the optimal capital allocations kÃÃ ni ÃÃ ÃÃ ÃÃ ÃÃ ÃÃ FB ÃÃ FB and kÃÃ & ai stated in the proposition: k n1 oka1 ; kn2 oka2 ; ka1 4k 1 , and k a2 4k 2 .
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