Description
Cost allocation is a process of providing relief to shared service organization's cost centers that provide a product or service. In turn, the associated expense is assigned to internal clients' cost centers that consume the products and services.
Cost Allocation For Capital Budgeting Decisions
Tim Baldenius? Sunil Dutta† Stefan Reichelstein‡
September 2006
Graduate School of Business, Columbia University, [email protected] Haas School of Business, University of California at Berkeley, [email protected] ‡ Graduate School of Business, Stanford University, [email protected] We thank seminar participants at the following universities: Columbia, Washington, Ohio State, Boston, Duke, Michigan State and the Norwegian School of Business and Economics for their many comments on this paper. We are also grateful for detailed suggestions from two anonymous reviewers and the editor, Jack Hughes.
†
?
Cost Allocation For Capital Budgeting Decisions Abstract Investment decisions frequently require coordination across multiple divisions of a ?rm. This paper explores a class of capital budgeting mechanisms in which the divisions issue reports regarding the anticipated pro?tability of proposed projects. To hold the divisions accountable for their reports, the central o?ce ties the project acceptance decision to a system of cost allocations comprised of depreciation and capital charges. If the proposed project concerns a common asset that bene?ts multiple divisions, our analysis derives a sharing rule for dividing the asset among the users. Capital charges are based on a hurdle rate determined by the divisional reports. We ?nd that this hurdle rate deviates from the ?rm’s cost of capital in a manner that depends crucially on whether the coordination problem is one of implementing a common asset or choosing among multiple competing projects. We also ?nd that more severe divisional agency problems will increase the hurdle rate for common assets, yet this is generally not true for competing projects.
1
Introduction
Capital budgeting decisions frequently a?ect multiple entities, such as departments or divisions, within a ?rm. Decision externalities may arise because divisional investment opportunities are mutually exclusive due to limited investment budgets or due to a required intermediate good that can be procured from more than one division. Alternatively, ?rms may be in a position to acquire common assets that are of use to multiple divisions at the same time. The capital budgeting process must then determine whether the aggregated individual bene?ts justify the common investment expenditure.1 The management literature provides only scant evidence regarding ?rms’ actual capital budgeting practices; see for example Kaplan and Atkinson (1998) and Taggart (1988). Some authors have suggested that ?rms could operate market mechanisms, such as auctions, in order to solve their capital budgeting problems, e.g., Hodak (1997). One approach that seems prevalent in practice is that ?rms set hurdle rates which must be met by individual projects in order to receive funding. However, there does not seem to be a well accepted methodology either for setting hurdle rates or for including them in the system of divisional performance measurement. This paper explores a class of capital budgeting mechanisms for which the ?rm’s central o?ce commits to a decision rule in response to divisional reports about the projected pro?tability of particular projects. To hold the divisions accountable for their reports, the central o?ce speci?es cost charges to be imposed in subsequent periods. These cost charges comprise both depreciation and capital charges, with the latter based on a hurdle rate that is determined by the initial divisional reports. We focus on charging mechanisms that satisfy the usual accounting convention that the
A recent illustration of such “public goods” problems was provided to us by a California semiconductor manufacturer where several product lines (pro?t centers) were using a common fabrication facility. The product line managers regularly encouraged the ?rm’s central o?ce to upgrade the equipment in the manufacturing facility. However, the ?rm’s central o?ce was reluctant to undertake these upgrades as it worried about “creative optimism” by the divisions, partly because the ?rm did not have a methodology for allocating investment expenditures among the users of the facility.
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sum of all depreciation charges across time periods and across divisions be equal to the initial investment expenditure. The survey evidence of Poterba and Summers (1995) and others has shown that internal hurdle rates frequently exceed a ?rm’s actual cost of capital by a substantial margin. Consistent with this evidence, the capital budgeting literature has shown that with private managerial information second-best mechanisms entail capital rationing.2 This tendency emerges because the principal is better o? foregoing marginally pro?table projects in order to economize on the manager’s informational rents (the premium a manager must be paid for truthful reporting). One way to implement such capital rationing is to raise the hurdle rate above the ?rm’s cost of capital. Consequently, these models predict that higher agency costs will result in more severe capital rationing, and thus higher hurdle rates. In our multidivisional setting, hurdle rates can be interpreted as a “shadow price” of capital which re?ects two distinct though related factors: (i) the interdivisional coordination problem due to decision externalities, and (ii) agency costs at the divisional level. In contrast to single division settings, we ?nd that the hurdle rate will di?er from the ?rm’s cost of capital even in the absence of any agency costs. Whether the hurdle rate will be set above or below the ?rm’s cost of capital depends on the nature of the decision externality, i.e., whether the asset is shared or exclusively used by only one division, and the relative magnitude of the divisional agency costs. Suppose ?rst that the capital budgeting problem is such that each division has an investment opportunity but at most one of the n divisional projects can be funded. In such settings, we identify a Competitive Hurdle Rate (CHR) mechanism which e?ectively is a multiperiod version of the second-price auction. In particular, the winning division faces a hurdle rate which is una?ected by its own report. In the special case where the divisional projects are ex-ante identical, the competitive hurdle rate is simply the second highest reported internal rate of return. The CHR mechanism also speci?es a depreciation schedule for the initial investment. If this schedule conforms
2
See, for instance, Antle and Eppen (1985).
2
to the so-called relative bene?t rule, the project’s npv is re?ected in a time consistent manner in the divisional performance measure.3 These features make the CHR mechanism strongly incentive compatible in the sense that divisional managers have a dominant-strategy incentive to report their information truthfully regardless of the weights (determined by intertemporal preferences and compensation payments) they attach to outcomes in di?erent time periods. We establish that the CHR mechanism is the only satisfactory capital budgeting mechanism that achieves both strong incentive compatibility and e?cient project selection. This uniqueness ?nding relies in part on a well-known result by Green and La?ont (1979) in the earlier public choice literature: the discounted total charges to the divisions must conform to a Groves scheme (Groves 1973) in order to obtain dominant strategies. Since we postulate that a division will not be charged unless its project is funded, the class of feasible Groves mechanisms reduces to the so-called Pivot mechanism: a division is charged if and only if its report is pivotal in the sense that it changes the resource allocation decision. Finally, it has been shown that the Pivot mechanism amounts to a second-price auction in settings where an indivisible good is allocated among n participants.4 Our results are consistent with Poterba and Summers (1995) insofar as the competitive hurdle rate exceeds the ?rm’s cost of capital. Nonetheless, the CHR mechanism is balanced in the sense that the discounted sum of depreciation and capital charges is equal to the initial investment expenditure provided the future cost charges are discounted at the competitive hurdle rate. This is a direct consequence of the postulate of comprehensive income measurement requiring the sum of the depreciation charges to equal the initial investment expenditure incurred by the winning division.5
This feature has been established in the earlier work on goal congruent performance measures for investment decisions involving a single division, e.g., Rogerson (1997), Reichelstein (1997). 4 See Mas-Colell et al. (1995) and Krishna (2002). 5 In one-period settings, Pivot mechanisms are well-known to attain a budget surplus. Since the CHR mechanism is a multiperiod version of the Pivot mechanism surplus, one would expect the CHR mechanism to overcharge the winning division. This is true in the sense that the discounted sum of depreciation and capital charges exceeds the initial investment if the discount rate is given by the ?rm’s cost of capital which is below the competitive hurdle rate.
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We identify a class of environments for which the CHR mechanism is optimal among all revelation mechanisms. In contrast to single-agent settings, however, our analysis shows that the hurdle rate will be una?ected if the agency costs for each division go up by the same factor and the divisions face comparable projects, ex ante.6 At the same time, it will be bene?cial to use the hurdle rate as an instrument for “handicapping” divisions which (i) are ex-ante likely to have access to more pro?table projects and (ii) face relatively high agency costs. We demonstrate that optimal capital budgeting mechanisms have the feature that ceteris paribus those divisions face higher hurdle rates. When the capital budgeting problem concerns the acquisition of a common or shared asset to which all divisions have access, our ?ndings are in several respects “dual” to the ones obtained in connection with exclusive assets. Invoking again the criterion of satisfactory mechanisms, we identify the so-called Pay-the-MinimumNecessary (PMN) mechanism as the unique mechanism that results in e?cient project selection decisions in a (strongly) incentive compatible manner. In present value terms, the PMN mechanism charges every division its “critical value”, the present value that this division would have to obtain from the joint asset in order for the investment to just break even. To implement this rule in a time consistent fashion, each division is assigned a share of the joint investment expenditure as a divisional asset. The shares are set in proportion to the critical values and the capital charge rate under the PMN mechanism is given by the project’s internal rate of return evaluated at the critical values. In contrast to our ?ndings for exclusive assets, the capital charge rate under the PMN mechanism is below the ?rm’s cost of capital. As a consequence, the PMN mechanism e?ectively runs a de?cit in present value terms.7 While the sum of all
This ?nding is based on the same argument showing that under certain conditions the second price auction is a revenue maximizing mechanism; see, for example, Myerson (1981). In particular, this result obtains if the bidder with the highest intrinsic value for the object also has the highest virtual value. 7 Consistent with this observation, the PMN mechanism is not a multiperiod version of the Pivot mechanism.
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depreciation charges across time periods and divisions is equal to the acquisition cost of the joint asset, the discounted sum of all depreciation and capital charges is less than the initial acquisition cost, provided these charges are discounted at the ?rm’s cost capital. Intuitively, the need for such subsidization arises because of the public goods nature of the common asset. The divisions no longer compete but instead exert a positive externality upon one another. In order for the PMN mechanism to be a second-best contracting mechanism for shared assets, the capital charge rate must increase monotonically in the agency costs of each division. This re?ects that the total net pro?tability of the shared asset decreases with the sum of the individual agency costs (for exclusive assets, in contrast, it is the relative magnitude of the individual agency costs that matters). We conclude that for shared assets the PMN mechanism may result in an agency-adjusted capital charge rate that can either be above or below the ?rm’s cost of capital. The criterion for e?cient project selection tends to push the capital charge rate below the ?rm’s cost of capital, while agency costs tend to push in the opposite direction. In terms of prior literature, our work builds on a range of studies that have examined the role of cost allocation mechanisms in guiding intra?rm resource allocation; papers in this category include Zimmerman (1979), Baiman and Noel (1985), Rajan (1992) and Pfa? (1994), among others. These studies have focuses on cross-sectional cost allocations in a one-period setting. Similarly, most of the existing work on multiagent budgeting mechanisms has been con?ned to single-period studies, e.g., Harris, Kriebel and Raviv (1982), Kanodia (1994), Balakrishnan (1995), Arya et al. (1996), Harris and Raviv (1996), Chen (2003), and Bernardo, Cai and Luo (2004).8 Intertemporal cost allocations have been central to the prior work on accrual accounting for performance measurement, such as Rogerson (1997) and Reichelstein
Our ?nding regarding capital charge rates that di?er from the ?rm’s actual cost of capital is reminiscent of earlier work on “strategic” internal pricing, such as Hughes and Kao (1997), Alles and Datar (1998), Arya and Mittendorf (2004). The main theme in these models is that a ?rm’s central o?ce sets internal prices (e.g., cost allocations) so as to a?ect the competitive outcome between the ?rm’s divisions and external rivals. Goex and Schiller (2006) provide a survey of this literature.
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(1997), Wei (2004), and Bareket and Mohnen (2006). One conclusion in these models is that in the absence of moral hazard, the capital charge rate should be set equal to the ?rm’s cost of capital in order to obtain goal congruent performance measures. In contrast, Dutta and Reichelstein (2002) and Christensen, Feltham and Wu (2002) question the usual textbook recommendation of setting the capital charge rate equal to the ?rm’s weighted average cost of capital if risky projects are to be selected by a risk- and e?ort averse manager.9 All of these studies focus on a single representative agent without addressing coordination issues across multiple divisions.10 The remainder of the paper is organized as follows. Section 2 examines satisfactory capital budgeting mechanisms for both exclusive and shared assets. Agency costs and second-best contracting mechanisms are the subject of Section 3. We conclude in Section 4.
2
Satisfactory Capital Budgeting Mechanisms
We examine mechanisms for coordinating investment decisions in a ?rm comprised of n divisions and a central o?ce. At the initial date, the ?rm can acquire a capital asset with a useful life of T years. The ?rm’s central o?ce faces an incentive and coordination problem because the division managers have private information regarding the subsequent operating pro?tability of the asset. To study the most common forms of interdivisional coordination problems, we consider two scenarios: (i) shared and (ii) exclusive assets. In the case of shared assets, the ?rm has access to a common investment project that, if undertaken, generates future revenues for all n divisions. In that sense, the common investment project is a “public good” which bene?ts all divisions. For
Baldenius (2003) links the choice of the capital charge rate to empire bene?ts that managers derive from new investments. Baldenius and Ziv (2003) consider the e?ect of incomes taxes on capital charge rates. Dutta (2003) examines a ?rm’s choice of capital charge rate when its manager has the option of pursuing the investment project as an outside venture on his own. 10 An exception is Wei (2004) who studies a setting with symmetric information across managers and shows that ?xed cost allocations can alleviate divisional underinvestment problems.
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the scenario of exclusive assets, in contrast, each division is assumed to have one investment opportunity, yet for exogenous reasons the ?rm can undertake at most one of these n divisional projects. Each project, if undertaken, generates subsequent cash ?ows only for that particular division. In this setting, the projects are “private goods” and the divisions compete for scarce investment capital. Without loss of generality, each division’s status quo operating cash ?ow (in the absence of any new investment) is normalized to zero. When division i has access to the new investment (exclusive or shared), its periodic operating cash ?ows take the form: cit = xit · ?i . (1)
¯i ] denotes the pro?tability parameter of division i, with ? ? Here, ?i ? ?i = [?i , ? i 0. The vector of pro?tability parameters of all n divisions will be denoted by ? = (?1 , ..., ?n ) and ??i ? (?1 , ..., ?i?1 , ?i+1 , ..., ?n ) denotes the pro?tability pro?le of all divisions other than i. While ?i is assumed to be known only to manager i, the intertemporal distribution of future cash ?ows, as represented by the vector Xi =
11 (x1i , ..., xiT ) ? RT + , is commonly known.
The ?rm’s cost of capital is given by r ? 0 with ? = 1/(1 + r) representing the discount factor. The present value of division i’s cash ?ows is
T
P Vi (?i ) =
t=1
? t · xit · ?i
? ? · Xi · ?i , where ? ? (?, ..., ? T ). Initially, we ignore moral hazard problems and their associated agency costs and instead focus on the choice of goal congruent performance measures
The e?ective useful life of a project can be ? < T periods, in which case xit = 0 for all ? ? t ? T . The central assumption underlying (1) is that the designer knows not only the useful life of the asset but also the intertemporal pattern of cash ?ows. For instance, if the investment pertains to a new production facility, the parameters xit may re?ect the known production capacity available in di?erent periods, while ?i represents the (expected) contribution margin which is known only to the divisional manager. Similar assumptions are made in Rogerson (1997), Reichelstein (1997), Baldenius and Ziv (2003), Wei (2004), Bareket and Mohnen (2006).
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for divisional managers. As shown in Section 3 below, the mechanisms identi?ed in this section can be adapted to optimal second-best mechanisms so as to include issues of moral hazard and managerial compensation. Following the terminology in earlier literature, a performance measure is said to be goal congruent if it induces managers to make decisions that maximize the present value of ?rm-wide cash ?ows. In our search for goal congruent performance measures, we con?ne attention to accounting-based metrics of the form ?it = Incit ? r ˆ · Ai,t?1 , (2)
where Ait denotes book value of division i’s asset at the end of period t, and r ˆ is a capital charge rate applied to the beginning book value. The net asset value at the end of period t is given by Ait = Ai,t?1 ? dit · Ai0 , where dit denotes the period-t depreciation percentage for division i in period t and Ai0 represents the initial asset value assigned to division i. Given comprehensive income measurement, income in period t is calculated as: Incit = cit + Ait ? Ai,t?1 = cit ? dit · Ai0 . We note that the class of performance measures in (2) encompasses the most common accounting performance metrics such as income, residual income, and operating cash ?ow. To create reporting incentives, the central o?ce has two principal instruments: the capital charge rate r ˆ and the depreciation rules {dit }T t=1 . In our search for alternative capital budgeting mechanisms, we focus on balanced cost allocations satisfying the property that the sum of all depreciation charges across agents and across time periods is equal to the amount initially invested. Performance measures are required to have a “robustness” property such that the desired incentives hold even if a divisional manager attaches weights to future outcomes that di?er from those of the ?rm which is interested in the present value of 8
future cash ?ows. Let ui = (ui1 , ..., uiT ) denote non-negative weights that manager i attaches to the sequence of performance measures ?i = (?1i , ..., ?iT ). At the beginning of period 1, manager i’s objective function can thus be written as
T t=1
uit · E [?it ].
One can think of the weights ui as re?ecting a manager’s discount factor as well as the bonus coe?cients attached to the periodic performance measures. We require goal congruence for all ui in some open set in Vi ? RT + . For instance, Vi could be a neighborhood around (u · ?, u · ? 2 , ..., u · ? T ) for some constant bonus coe?cient u.12 ˜i , ? ˜?i |?i ) denote manager i’s period-t performance measure when his true Let ?it (? ˜i and the other n ? 1 managers report ? ˜?i . A performance type is ?i , but he reports ? measure satis?es strong incentive compatibility if:
T T
uit · ?it (?i , ??i |?i ) ?
t=1 t=1
˜i , ??i |?i ), uit · ?it (?
˜i , ??i , ui ? Vi . for all i, ?i , ?
(3)
Our requirement of strong incentive compatibility amounts to dominant-strategy incentives for truthful reporting in a setting where the designer is also uncertain about the weights that managers attach to their performance measures in di?erent periods. This form of strong incentive compatibility will prove useful below when moral hazard is added to the model and the intertemporal weights uit are determined by the managerial compensation schemes.
2.1
Shared Assets
We ?rst study the design of capital budgeting mechanisms for a setting in which the asset acquired is shared among the n divisions in the sense that all divisions can have simultaneous access to the asset and derive future cash bene?ts from it. Applicable examples include cost reducing investments in a manufacturing process that is used
In order to assess the robustness of a particular mechanism one would like Vi to be as large as possible, e.g, the entire RT + . On the other hand, any necessity result pointing to the uniqueness of a particular mechanism becomes more powerful if derived with reference to a smaller set Vi . For now, it is useful to view uit as a summary statistic for the manager’s discount factor and the bonus coe?cients in his compensation function. In Section 3, the coe?cients uit will emerge endogenously from the underlying agency problem.
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by all divisions, certain forms of IT services or “lumpy” investments in capacity which alleviate any subsequent capacity constraints. The “common good” requires an initial cash outlay of b and generates (incremental) operating cash ?ows in the amount of xit · ?i for divisions 1 ? i ? n in periods 1 ? t ? T . For each division, the pro?tability parameter ?i is private information. The corporate net present value (npv) of the project then equals
n
NP V (?) =
i=1
P Vi (?i ) ? b.
The corresponding internal rate of return, denoted by ro (?), is implicitly de?ned by
n T
(1 + ro (?))?t · xit · ?i = b.
i=1 t=1
We note that the internal rate of return ro (?) is unique because ?i > 0 and xit ? 0. The indicator variable I ? {0, 1} will be used to represent whether the common project is undertaken. The ?rst-best investment rule I ? (?) calls for the investment to be made whenever NP V (?) ? 0 or, equivalently, whenever ro (?) ? r. To obtain goal congruence, a capital budgeting mechanism for shared assets relies on four instruments: an investment rule, an asset allocation rule, a capital charge rate, and depreciation schedules. To compute the divisional performance measure in (2), the asset allocation rule assigns each division a share, ?i · b, of the joint asset’s initial investment cost b. This amount is capitalized on the divisional balance sheet and subsequently depreciated over the next T periods according to the speci?ed depreciation schedule di = (di1 , ..., dit ). The book value of division i’s asset at date t is therefore given by:
t
Ait = Ai,t?1 ? dit · ?i · b =
1?
? =1
di?
· ?i · b.
(4)
Formally, a capital budgeting mechanism for shared assets speci?es the following items: 10
• Investment rule I : ? ? {0, 1}; • Asset allocation rule ?i : ? ? [0, 1] satisfying the requirement • Capital charge rate r ˆ : ? ? (?1, ?); • Depreciation schedules di : ? ? RT + satisfying the conditions that 1 if I (?) = 1, and dit (?) ? 0 if I (?) = 0. We note that mechanisms are restricted to satisfy a “no-play-no-pay” condition: the divisions can be charged only if the joint asset is acquired, i.e., I (?) = 1. A capital budgeting mechanism is called satisfactory if (i) it is strongly incentive compatible as de?ned in (3), and (ii) the project is undertaken, if and only if NP V (?) ? 0. Given the asset valuation rule in (4), the divisional performance measure in (2) becomes: ?it = (xit · ?i ? zit · ?i · b) · I, where zit = dit + r ˆ· 1?
? =1 t?1 T t=1 n i=1
?i (?) = 1;
dit (?) =
(5)
di?
(6)
denotes the sum of depreciation and interest charge in period t. Following Rogerson (1997), we refer to {zit }T t=1 as an intertemporal cost allocation. Earlier studies on goal congruence have observed that for any given capital charge rate, r ˆ, there is a one-to-one mapping between depreciation and intertemporal cost allocation schemes. In particular, there exists a unique intertemporal cost allocation such that zit = xit
T ? =1 (1
+r ˆ)?? · xi?
.
(7)
The signi?cance of this relative bene?t cost allocation is that, for a given asset allocation rule {?i }n i=1 , division i’s residual income measure, relative to the capital charge rate r ˆ, will in each period be proportional to the divisional npv, P Vi (?i |r ˆ)??i ·b, which is obtained by discounting future cash ?ows at the rate r ˆ. Thus, for a capital 11
budgeting problem with a single agent, the principal can achieve goal congruence by setting r ˆ = r. The unique depreciation schedule that gives rise to the cost allocation charges in (7) is referred to as the relative bene?t depreciation rule.13
? Let ?i (??i ) denote the critical pro?tability parameter of division i, that is, the
value of ?i at which the project breaks even, given the valuations ??i of all other
? divisions. Thus, ?i (??i ) is de?ned implicitly by ? P Vi (?i (??i )) + j =i ? Assumption 1 Each division is essential in the sense that ?i (??i ) ? ?i for all i and
P Vj (?j ) = b.
(8)
??i . Assumption 1 stipulates that
j =i
P Vj (?j ) + P Vi (?i ) < b for all i and ??i . The
corresponding restriction on the range of divisional pro?tability parameters ?i is easier to satisfy in a setting with just a “small” number of participating divisions. As demonstrated at the end of this subsection, our main results can be extended to environments beyond those conforming to Assumption 1. The following capital budgeting mechanism will be referred to as the Pay-theMinimum-Necessary (PMN) mechanism: (i) I (?) = I ? (?);
? ? (ii) r ˆ = r? (?) ? ro (?1 (??1 ), ..., ?n (??n ));
(iii) ?i (?) =
? ?? (?) · Xi · ?i (??i ) ; n ? ? j =1 ? (? ) · Xj · ?j (??j )
(iv) Depreciation is calculated according to the relative bene?t rule based on the capital charge rate r? (?);
It is well known that this rule amounts to the annuity depreciation method in case the xit ’s are constant across time periods. On the other hand, if cash ?ows were to decline geometrically at a rate of ? over an in?nite horizon, relative bene?t depreciation would amount to a declining balance method with decline factor 1 ? ?.
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where ?? (?) ? (1 + r? (?))?1 , ..., (1 + r? (?))?T . The PMN mechanism sets the capital charge rate equal to the project’s internal rate of return at the critical pro?tability
? levels ?i (??i ). The asset sharing rule in (iii) allocates the initial investment cost in
proportion to the divisional present values generated by the common asset, evaluated at the critical pro?tability parameters. If the aggregate present value of all divisions other than i approaches b (note that it cannot exceed b by Assumption 1), then division i’s assigned ownership share ?i (?) will tend to be small; whereas it will tend be large when the aggregate valuation of the other divisions is small. These characterizations follow directly upon observing that, by the de?nition of the internal rate of return,
n j =1 ? ?? (?) · Xj · ?j (??j ) = b.
The relative bene?t depreciation rule in (iv) ensures that the cost charge to division i in period t is given by: zit (?) · ?i · b = Using again the fact that
n j =1
xit ? ? (?) ·
Xi
· ?i · b.
? ?? (?) · Xj · ?j (??j ) = b, it follows that the charge to
division i in period t is given by
? zit (?) · ?i (?) · b = xit · ?i (??i ).
(9)
Thus the performance measure under the PMN mechanism reduces to
? ?it (?) = xit · [?i ? ?i (??i )],
(10)
making it a dominant strategy for each manager to report truthfully. Equation (10) shows that this incentive holds not only in the aggregate, i.e., over the entire planning horizon, but on a period-by-period basis. We have demonstrated that the PMN mechanism is satisfactory. The following result shows that within the class of capital budgeting mechanisms we consider there is in fact no other satisfactory mechanism.14
14
All proofs are in Appendix A.
13
Proposition 1 Given Assumption 1, the Pay-the-Minimum-Necessary (PMN) mechanism is the unique satisfactory capital budgeting mechanism for shared assets. To provide intuition for the uniqueness of the PMN mechanism, it is useful ?rst to consider settings in which each manager attaches the same weights to future payo?s as the ?rm’s owners, that is, it is commonly known that uit = ? t · u for each i and t. In such a setting, intertemporal matching is of no importance and it su?ces to ensure dominant strategy incentive compatibility over the entire planning horizon, rather than on a period-by-period basis. As demonstrated in the public choice literature (e.g., Green and La?ont 1979), every dominant-strategy mechanism must be a Groves scheme if the organization is to achieve e?cient outcomes. Under a Groves scheme each participant’s payo? is the same as the social surplus up to a constant which does not depend on the report by that participant. Since manager i already internalizes his own surplus, P Vi (?i ), the transfer payment must be equal to the maximized surplus of the remaining n ? 1 divisions plus some amount which is a constant from i’s perspective. Given the performance measure in (5), the present value of the cost charges to division i must satisfy:
T
zit (?) · ?i (?) · b · ? t = ?
t=1 j =i
P Vj (?j ) ? b
· I ? (?i , ??i ) + hi (??i ),
where hi (·) is an arbitrary function of ??i . In conjunction with Assumption 1, the no-play-no-pay condition requires that this “remainder” term hi (·) must be equal to zero. By construction of the critical types, we ?nd that:
T ? (??i )) · I ? (?i , ??i ). zit (?) · ?i (?) · b · ? t = P Vi (?i t=1
(11)
When expressed in this form, the label “Pay-the-Minimum-Necessary” becomes transparent: given the valuations reported by the other divisions, the present value charge to each division is the minimum amount required in order for the project to break even. 14
Our notion of strong incentive compatibility requires truthful reporting to be a dominant strategy for each manager for an entire neighborhood of ui . As a consequence, the aggregate cost charge in (11) must be annuitized so that
? (??i ) · I ? (?i , ??i ). zit (?) · ?i (?) · b = xit · ?i
(12)
The remaining question is whether there exist other combinations of asset allocation rules, capital charge rates, and depreciation schedules that can yield the same periodic cost charges as those in (12). The answer to this question is provided by the following observation which we state as separate lemma because it will be used again later.15 Lemma 1 Let D = {(d1 , ..., dT ) |
T t=1
dt = 1} denote the set of all depreciation
schedules. The mapping ft : D × (?1, ?) ? RT , given by
t?1
ft (d, r ˆ) = dt + r ˆ· is one-to-one. Its image contains RT + and
1?
? =1 T t=1 (1
d?
(13)
+r ˆ)?t · ft (d, r ˆ) = 1 for all (d,r ˆ).
For a given asset sharing rule, Lemma 1 implies that the intertemporal cost allocation charge in (12) can only be generated by the relative bene?t depreciation rule and the capital charge rate r? (?). Given this observation, it follows directly that the asset shares ?i (?) must be the ones speci?ed under the PMN mechanism. The public choice literature has shown that it is generally impossible to attain e?cient outcomes in dominant strategies while keeping the budget balanced. In contrast, Proposition 1 demonstrates the possibility of an e?cient capital budgeting mechanism by means of conventional cost allocations that are nominally balanced. However, in present value terms investment in the shared assets must be “subsidized” by the central o?ce by way of a reduced capital charge rate. To see this, we note
15
This result generalizes Corollary 3 in Rogerson (1997).
15
that:
? r ? ro (??i , ?i (??i )) ? ? ? ro (?1 (??1 ), ..., ?n (??n ))
= r? (?), where the inequality holds because the internal rate of return ro (?1 , ..., ?n ) is monotone
? increasing in each of its arguments and ?i ? ?i (??i ) for any positive-npv project.16
Corollary 1 Under the PMN mechanism, the capital charge rate, r? (?), is less than the ?rm’s cost of capital, r. An alternative way to attain the same outcomes would be for the central o?ce to hold the capital charge rate at r while lowering the initial capital allocation for each division to ˆ i (??i ) ? ? However, because
n i=1
b?
j =i
P Vj (?j |r) . b
n i=1
ˆ i (??i ) = 1 [n·b?(n?1)· ? b
P Vi (?i )] < 1 whenever I (?) = 1,
this approach violates the basic accrual accounting requirement that the sum of all depreciation charges across time periods and participating divisions be equal to the initial investment outlay. It is instructive to examine how division i’s assigned share of the joint investment, ?i (?), depends on that division’s report. In the special case of identical Xj , the PMN mechanism divides the initial investment cost simply in proportion to the critical pro?tability levels, that is, ?i (?) =
16
? ?i (??i ) . n ? j =1 ?j (??j )
Because of the public goods nature of the shared asset, the capital charge rate is (weakly) o o decreasing in the number of participants. Speci?cally, suppose ?o ? (?1 , ..., ?n ) is the pro?tability ? o pro?le of the existing n divisions and let r (? ) be the capital charge rate for the corresponding PMN mechanism. It is readily veri?ed that if an additional division with pro?tability parameter ?n+1 participates in the asset acquisition decision, then r? (?o , ?n+1 ) < r? (?o ), regardless of the type ?n+1 of the new participant.
16
? Since the critical pro?tability level ?j (??j ) is decreasing in ?i for all i = j , the share
of initial investment cost allocated to division i is increasing in its own pro?tability parameter ?i . This comparative statics continues to hold more generally when the Xj di?er across divisions. Corollary 2 Under the PMN mechanism, the share of investment cost allocated to division i, ?i (?), is increasing in the pro?tability parameter ?i . A necessary feature of any dominant strategy mechanisms for a binary decision is that, contingent on the project being undertaken, the cost charge to any participant must be independent of its own report. Corollary (2) and the capital charge rate
? ? r? (?) = ro (?1 (??1 ), ..., ?n (??n )) may suggest that the PMN mechanism violates this
property. However, while both r? (?) and ?i (?) vary with manager i’s report ?i , the
? fact that zit (?) · ?i (?) · b = xit · ?i (??i ) · I ? (?i , ??i ) shows that these partial e?ects
exactly o?set each other, i.e., the resulting sum of depreciation and capital charges, zit (?) · ?i (?) · b, is indeed independent of ?i . It is instructive to note that the PMN mechanism is not a multiperiod version of the so-called Pivot mechanism. In the public choice literature, it is usually assumed that the cost of the public investment is divided in some arbitrary, say equal, fashion among all participants (e.g., Kreps 1990, Mas-Colell et al. 1995) and in addition the participants make (receive) transfer payments. One of the key features of the Pivot mechanism is that an agent receives a non-zero additional transfer payment only if he is “pivotal” in the sense that his report alters the social decision. While the PMN mechanism also satis?es this feature (in fact, our Assumption 1 implies that all agents are pivotal), the Pivot mechanism does not satisfy our no-play-no-pay condition since agents can be charged even if the project is not undertaken. One of the attractive features of Pivot mechanisms for public choice problems is that they always run a “budget surplus”. In contrast, the PMN mechanism runs a de?cit in real terms in the sense that the sum of all divisional charges, discounted at the ?rm’s cost of capital, r, is less than the initial investment expenditure. 17
To conclude this section, we demonstrate that the PMN mechanism can be extended to settings in which Assumption 1 does not hold. The immediate issue then is
? that the critical pro?tability cuto?s ?i (??i ) may be less than ?i . We extend the PMN
mechanism to environments satisfying the following weaker form of Assumption 1.
? Assumption 2 For all ?, there exists some division i such that ?i (??i ) ? ?i .
Thus, for all type pro?les ?, there must be at least one essential division. Assumption 2 is likely to be satis?ed in settings where there are either one or a few larger divisions (in terms of the magnitude of their P Vi ’s) whose valuation is critical for the project’s overall pro?tability. Given Assumption 2, the PMN mechanism can be
? ? modi?ed by replacing ?i (??i ) with max{?i , ?i (??i )}. As a consequence, non-essential ? agents, for whom ?i (??i ) < ?i , will be assigned an ownership share ?i (?) = 0. While
the modi?ed PMN mechanism is again satisfactory, it is no longer the unique solution among the class of capital budgeting mechanisms we consider. In particular, it is not necessary that non-essential divisions be charged zero. However, uniqueness of this modi?ed PMN mechanism can be restored if one imposes an additional participation constraint akin to Moulin’s (1986) “no-free-ride” condition. Speci?cally, this ex-post participation condition requires that no division be worse o? by participating in the mechanism. As a consequence, non-essential divisions must then be charged zero.
2.2
Exclusive Assets
We now examine the design of capital budgeting mechanisms for a setting in which the divisions face a limited capital budget. For simplicity, we suppose that the capital budget is su?cient to fund at most one of the divisional projects.17 One interpretation of this speci?cation is that, while the ?rm’s capital cost for ?nancing a single project is r, this cost would increase su?ciently fast with additional investments such
It is readily seen that the mechanisms we identify in this subsection can be extended to settings where the ?rm can fund k out of n possible projects. However, such an extension would add signi?cant clutter to the notation without adding any substantive insight.
17
18
that the additional cash returns could not possibly cover the higher cost of capital. An alternative interpretation is that the ?rm has already decided to undertake a particular project, e.g., the ?rm has entered into a procurement contract, yet the capital budgeting process is to determine which of the n divisions could implement this project most pro?tably. Division i’s investment opportunity requires an initial cash outlay of bi . The npv of division i’s investment project is then given by NP Vi (?i ) ? P Vi (?i ) ? bi . As before, the pro?tability parameters ?i are divisional private information. Division
o i’s internal rate of return is denoted by ri (?i ), that is, T o (1 + ri (?i ))?t · xit · ?i = bi . t=1
(14)
The ?rst-best investment decision rule calls for selecting the highest npv project provided that npv is positive; i.e., the corresponding internal rate of return exceeds the ?rm’s cost of capital r. We use the indicator variable Ii ? {0, 1} to represent whether division i’s investment project is undertaken. Given our speci?cation that the ?rm can fund at most one project, a feasible investment policy must satisfy budgeting mechanism for exclusive assets speci?es: • An investment decision rule, Ii : ? ? {0, 1} such that i’s beginning balance equals Ai0 = bi · Ii (?); • A capital charge rate, r ˆ : ? ? (?1, ?); • A depreciation schedule, di : ? ? RT , satisfying and dit (?) ? 0 if Ii (?) = 0.
T t=1 n i=1 Ii (? ) n i=1 Ii
? 1. A capital
? 1. Division
dit (?) = 1 if Ii (?) = 1,
19
Note that this class of mechanisms again imposes a no-play-no-pay condition, since a division can be charged only if its project receives funding.18 In direct analogy to our earlier terminology, a capital budgeting mechanism is said to be satisfactory if (i) it is strongly incentive compatible for each manager, and (ii) it selects the highest positive npv project. It will be notationally convenient to denote the highest positive npv project by NP V 1 (?) ? max {NP Vi (?i ), 0}.
i
E?cient project selection requires that Ii? (?) = 1 only if NP V 1 (?) = NP Vi (?i ). We also de?ne NP V 1 (??i ) = max {NP Vj (?j ), 0}.
j =i ? For a given ??i , let ?i (??i ) denote the lowest value of division i’s pro?tability
parameter for which its project is at least as pro?table as any of the other n ? 1 projects. That is,
? NP Vi (?i (??i )) = NP V 1 (??i ). ? Put di?erently, at ?i = ?i (??i ), division i’s npv would just tie with the highest npv
of the remaining divisions, provided that value is positive.19 Note that this de?nition
? ? of ?i (??i ) implies that Ii? (?) = 1 only if ?i ? ?i (??i ).20
The following capital budgeting mechanism will be referred to as the Competitive Hurdle Rate (CHR) mechanism: (i) Ii (?) = Ii? (?);
o ? (?i (??i )) if Ii? (?) = 1; (ii) r ˆ = r? (?) ? ri
Because division i will be burdened with depreciation and capital charges only if Ii = 1, it is without loss of generality to set a uniform, ?rmwide capital charge rate r ˆ(·). 19 ? While ?i (??i ) depends on all distributional cash ?ow parameters (X1 , ..., Xn ) and on the cash outlay amounts (b1 , ..., bn ), we suppress this dependence for notational ease. 20 ? To rule out uninteresting corner solutions and to ensure the critical pro?tability type ?i (??i ) is ¯i ) = H always well de?ned for all ??i , we assume throughout that, for all i, NP Vi (?i ) = L and NP Vi (? for some H > L.
18
20
(iii) {dit (r? (?))}T t=1 is the relative bene?t depreciation schedule based on the competitive hurdle rate r? (?). The competitive hurdle rate, r? (?), is the internal rate of return of the winning
? division evaluated at the critical pro?tability type, ?i (??i ). Division i’s report does ? not a?ect the competitive hurdle rate provided ?i > ?i (??i ). The CHR mechanism
simpli?es considerably when all divisions are ex-ante identical with regard to Xi and bi . The rank order of the npv’s then is identical to the rank order of the internal rates of return and therefore the competitive hurdle rate for the winning division
? o simply equals the second-highest internal rate of return: ri = maxj =i {rj (?j ), r}. In
this context, it is also readily seen that the CHR mechanism can be viewed as a delegation mechanism: divisions report their internal rates of return and decide on their own whether to proceed with their divisional projects with the capital charge rate set at the second-highest reported internal rate of return. The relative bene?t depreciation schedule in (iii) ensures that the cost charge to division i is given by zit (?) · bi = xit · [
T ? =1 o ? [1 + ri (?i (??i ))]?? · xi? ]?1 · bi . It then
follows from equation (14) that the charge to division i is given by
? zit (?) · bi = xit · ?i (??i )
(15)
? and hence ?it (?) ? ?it (?i , ??i |?i ) = xit · [?i ? ?i (??i )] · Ii (?). Therefore, The CHR
mechanism provides strong incentives for each manager to report his information truthfully. Proposition 2 The Competitive Hurdle Rate (CHR) mechanism is the unique satisfactory capital budgeting mechanism for exclusive assets. To see the uniqueness of the CHR mechanism, consider again the case where divisional managers receive a constant share of residual income in each period as compensation and they discount future payo?s at the ?rm’s cost of capital r. Since any satisfactory dominant-strategy mechanism must then be a Groves mechanism, 21
we have:
T
? t · zit (?) · bi = ?
t=1 j =i
? NP Vj (?j ) · Ij (?) + hi (??i ),
(16)
where hi (·) is an arbitrary function of ??i . The total surplus of the remaining n ? 1 divisions, given by the ?rst term on the right hand side of (16), is equal to zero when Ii (?i ) = 1, and equal to NP V 1 (??i ) when Ii? (?) = 0. Equation (16) can therefore be written as
T
? t · zit (?) · bi = ?NP V 1 (??i ) · [1 ? Ii? (?)] + hi (??i ).
t=1
The no-play-no-pay condition immediately implies that hi (??i ) = NP V 1 (??i ) and therefore:
T T
? · ?it = [NP Vi (?i ) ? NP V
t=1
t
1
(??i )] · Ii? (?i , ??i )
=
t=1
? ? t · xit · [?i ? ?i (??i )] · Ii? (?i , ??i ).
(17)
Equation (17) shows that when all divisions discount future payo?s at the principal’s cost of capital, r, a satisfactory capital budgeting mechanism for exclusive assets must take the form of a second-price auction in which the present value of all charges to the winning division is the second highest npv. Strong incentive compatibility requires that the managerial performance measure re?ects the divisional npv not only over the entire planning horizon but also on a period-by-period basis. As a consequence, the aggregate cost charge must be annuitized as in equation (15). Uniqueness of the CHR mechanism now follows from the fact that the competitive hurdle rate combined with the relative bene?t depreciation implement the cost charges in (15) and, by Lemma 1, there can be only one such solution. The CHR mechanism is a multiperiod version of the Pivot mechanism. It is well known from the public choice literature that Pivot mechanisms always attain a budget surplus in the sense that the sum of all monetary transfers to the agents, in addition to cost sharing payments, is negative. Accordingly, we obtain the following result. Corollary 3 The hurdle rate under the CHR mechanism, r? (?), exceeds the ?rm’s cost of capital, r. 22
We note that Propositions 1 and 2 di?er markedly in their prescriptions regarding the capital charge rates. In order to obtain strong incentives within the class of accounting based mechanisms we consider, the capital charge rate for shared assets must be lower than the owner’s cost of capital (Corollary 1), while the reverse conclusion is obtained for exclusive assets.21 While the usual comprehensive income measurement conditions
T t=1
dit = 1 and Ai0 = bi · Ii hold, the investment expenditure is less than
the present value of the cost charges to the winning division, discounted at the ?rm’s cost of capital, r:
T T
(1 + r)?t · zit (?) · bi >
t=1 t=1
(1 + r? )?t · zit (?) · bi = bi .
If one were to impose the requirement that the capital charge rate be equal to the ?rm’s cost of capital, i.e., r ˆ(?) ? r, incentive compatibility would require a departure from comprehensive income measurement. Speci?cally, the same incentives could
? be generated by capitalizing the initial amount Ai0 = P Vi (?i (??i )) for the winning
division, yet this amount would exceed the actual investment expenditure bi .
3
3.1
Hidden Actions and Incentive Contracting
Second-Best Mechanisms
Our analysis has so far taken as given that divisional managers seek to maximize their performance measures and that the objective of the central o?ce is to maximize the ?rmwide npv by choice of goal congruent performance measures. It is natural to ask which properties of goal congruent mechanisms carry over to second-best mechanisms once we include an explicit agency problem and the desired incentives are derived from
Corollary 3 seems at odds with the results in Bareket and Mohnen (2006) who also consider the problem of picking one among several mutually exclusive projects, yet in their model the hurdle rate can be kept at r. The essential di?erence between the models is that we consider multiple divisions competing for funds, whereas in Bareket and Mohnen (2006) a single division can have only one of several projects approved. Dougart (2005) studies decision problems in which projects impose externalities on other divisions and argues that in such settings goal congruent performance measures require depreciation rules that di?er from the relative bene?t rule.
21
23
a uni?ed optimization program. Speci?cally, suppose divisional operating cash ?ow in period t is now given by cit = ait + xit · ?i · Ii , where ait ? [0, a ¯it ] denotes productive e?ort chosen privately by manager i in period t and Ii ? {0, 1} indicates generically (for shared and exclusive assets) whether division i has access to the asset. Manager i observes ?i before contracting. The central o?ce and all other managers share the same beliefs about ?i given by the cumulative distribution, Fi (?i ), with strictly positive density, fi (?i ), over the entire support ?i .22 We assume the usual monotone inverse hazard rate condition; i.e., Hi (?i ) ? [1 ? Fi (?i )]/fi (?i ) is decreasing in ?i for all i. While the central o?ce can observe the divisional operating cash ?ows in each period, it is unable to disentangle the investment-related from the e?ort-related components. Manager i’s date-0 utility payo? is given by
T
Ui =
t=1
? t · [sit ? vit (ait )],
where sit denotes his compensation in period t and vit (·) is his disutility from exerting e?ort ait in period t. The function vit is increasing and convex with vit (0) = 0, for all i and t. Given this structure, it is only the present value of compensation payments that matters to each manager, provided all parties can commit to a T -period contract. ˜) ? In our setting, a revelation mechanism speci?es an investment decision rule Ii (? ˜) ? (ci1 (? ˜), . . . , ciT (? ˜)) to be delivered by each division, {0, 1}, “target cash ?ows” ci (? ˜) ? (si1 (? ˜), . . . , siT (? ˜)), contingent on and managerial compensation payments si (? ˜. For any such mechanism, let Ui (? ˜i , ? ˜?i |?i ) denote manager i’s utility the reports ? ˜?i submitted by the contingent on his own true pro?tability parameter ?i , reports ? ˜i . Given truthful reporting on the part of the other managers, and his own report ?
Alternatively, the managers may learn their ?i -parameters after entering into the contract, but they cannot be prevented from quitting the job if their participation constraints are not satis?ed.
22
24
other managers, this yields
T
˜i , ??i | ?i ) ? Ui (?
t=1
˜i , ??i ) ? vit (ait (? ˜i , ??i | ?i ))], ? t · [sit (?
where ˜i , ??i | ?i ) ? min ait ait + xit · ?i · Ii (? ˜i , ??i ) ? cit (? ˜i , ??i ) ait (? is the minimum e?ort that manager i has to exert so as to achieve the required ˜i , ??i ) in each period. periodic cash ?ow target cit (? By the Revelation Principle we may restrict attention to mechanisms which induce managers to reveal their information truthfully. The central o?ce’s optimization problem can then be stated as follows:
n T
P:
(ci (? ),si (? ),Ii (? ))n i=1
max
E?
i=1 t=1
? t · [cit (?) ? sit (?)] ? B (?)
subject to: (ie ) for exclusive assets:
n i=1 Ii (? )
? 1 and B (?) =
n i=1 bi
· Ii (?),
(is ) for shared assets: Ii (?) = Ij (?) = I (?), for all i, j , and B (?) = b · I (?), ˜i , ??i | ?i )], for all ?i , ? ˜i and i, (ii) E??i [Ui (?i , ??i | ?i )] ? E??i [Ui (? (iii) E??i [Ui (?i , ??i | ?i )] ? 0, for all ?i and i. Constraints (ie ) and (is ) ensure feasibility of the investment rule for exclusive and shared assets, respectively, and specify the resulting initial investment amounts. The incentive compatibility constraints (ii) require that truthful reporting constitute a Bayesian-Nash equilibrium. The participation constraints (iii) are required to hold on an interim basis, i.e., each manager must break even in expectation over
25
the other managers’ possible types. We denote the solution to this program by
? ? n (c? i (? ), si (? ), Ii (? ))i=1 and refer to it as the second-best solution. ˜i < ?i and at the same time reduce his e?ort whenever A manager can underreport ?
˜i , ??i ) = 1. Therefore managers will earn informational rents on account of their Ii (? private information. The basic tradeo? for the central o?ce is that manager i’s information rents will be increasing both in the induced e?ort levels, (ait , ..., aiT ), as well as in the set of states in which division i has access to the asset. Applying standard arguments from the adverse selection literature based on “local” incentive constraints (e.g., La?ont and Tirole 1993), it can be shown that manager i’s interim informational rent for any ?i (i.e., in expectation over other managers’ types) equals
?i T
E??i [U (?i , ??i | ?i )] = E??i
?i
? t · vit (ait (qi , ??i | qi )) · xit · Ii (qi , ??i ) dqi .
t=1
(18)
The expression in (18) illustrates the above tradeo?: manager i’s informational rent can be reduced either by creating lower powered e?ort incentives (there will be no rent if ait = 0) or by curtailing the set of states ? in which Ii (?) = 1. To simplify the exposition and sharpen our predictions regarding the second-best investment decision rule, we focus on a setting in which the central o?ce always ?nds it worthwhile to induce maximum e?ort in each period, i.e., a? it (?i , ??i | ?i ) ? a ¯it . Inducing the maximum level of e?ort will indeed be optimal provided vit (¯ ait ) is su?ciently small relative to the other parameters of the model. For brevity, we denote vit ? vit (¯ ait ). As shown in Appendix B, the following condition ensures that the central o?ce will indeed seek to induce the maximum level of e?ort in each period: ait ) · xit · Hi (?i ) ? 0, 1 ? vit ? vit (¯ (19)
for each i and t. Condition (19) says that even if division i has access to the asset for all ?i ? ?i , the marginal return from e?ort, which has been normalized to one, is su?ciently large for the central o?ce to prefer high e?ort despite the corresponding increase in expected informational rents. We will discuss below how our results would be a?ected if (19) were relaxed so that interior levels of e?ort would be optimal. 26
We use the expression in (18), evaluated at ait = a ¯it , to solve for each manager’s compensation payments sit (?) in P . The central o?ce’s optimization problem then simpli?es to the following program (see Appendix B for a detailed derivation):
n T
P
:
(Ii (? ))n i=1
max E?
i=1 t=1
? t · [¯ ait ? vit (¯ ait )] + [P Vi (?i ) ? ?i · Hi (?i )] · Ii (?) ? B (?) ,
subject to (ie ), (is ), where ?i ?
T t=1
? t · vit · xit . The reduced objective function in P re?ects that the
expected value of manager i’s interim informational rents (i.e., E? [U (?i , ??i | ?i )]) is equal to the expected value of ?i · Hi (?i ) · Ii (?). To characterize the second-best investment decision rule, it will be useful to de?ne ?i (?i ) ? P Vi (?i ) ? ?i · Hi (?i ) (20)
as the present value of division i’s virtual cash ?ows (i.e., the present value of cash ?ows net of the manager’s informational rents). The second-best investment rule for exclusive assets then is given by Ii (?) = 1 if and only if VNP Vi (?i ) ? max{VNP Vj (?j ), 0},
j
(21)
n i=1
for VNP Vi (?i ) ? ?i (?i )?bi as the virtual divisional npv. Let VNP V (?) ? assets requires VNP V (?) ? 0.
?i (?i )?
b denote the virtual corporate npv, so that the second-best investment rule for shared (22)
We now demonstrate that the satisfactory mechanisms identi?ed in Section 2 can be adapted to generate optimal incentives in the presence of agency problems. A capital budgeting mechanism is said to be optimal if and only if there exist linear compensation schemes ˜) = ?it (? ˜) + ?it (? ˜) · ?it (·)}, {sit (?it | ? (23)
for all 1 ? t ? T and 1 ? i ? n, that achieve the same payo? for the principal as the second-best mechanism identi?ed above. 27
3.2
Optimal Mechanisms: Exclusive Assets
The second-best investment rule calls for funding a project if and only if its virtual npv exceeds both zero and the virtual npvs of all other projects. That is, Ii (?) = 1, if and only if VNP Vi (?i ) = VNP V 1 (?), where VNP V 1 (?) ? max1?j ?n {VNP Vj (?j ), 0}. Accordingly, we now denote the agency-adjusted pro?tability cuto? for division i by
?? ?i (??i ) which is implicitly de?ned by23 ?? VNP Vi (?i (??i )) = VNP V 1 (??i ). ?? The corresponding agency-adjusted competitive hurdle rate of division i, ri (?), is
de?ned to be the internal rate of return of its agency-adjusted critical project:
?? o ?? ri (?) ? ri (?i (??i )).
(24)
Suppose now that each manager is o?ered a linear compensation scheme of the form in (23) with bonus coe?cients ?it = vit . If the performance measure is based on the competitive hurdle rate mechanism (i.e., the capital charge rate is equal to the agency-adjusted competitive hurdle rate and the asset valuation is based on the relative bene?t depreciation rule), then ˜i , ??i | ?i ) = v · Ii (? ˜i , ??i ) · xit · [?i ? ??? (??i )]. ?it (? it i (25)
Each manager has dominant-strategy incentives to report his information truthfully because a project makes a positive contribution to his performance measure if
?? and only if ?i > ?i (??i ). Our next result shows that this mechanism is indeed op-
timal. Furthermore, even though the optimization program in P is stated in terms of Bayesian-Nash incentive compatibility and interim participation constraints, the central o?ce obtains dominant strategy incentives and ex-post satisfaction of the participation constraints for “free.” This characterization applies to a broader class of
To ensure that well-de?ned pro?tability cuto?s exist, we again assume that, for all i, ¯i ) = H for some H > L. VNP Vi (?i ) = L and VNP Vi (?
23
28
mechanisms (Mookherjee and Reichelstein 1992).24 Proposition 3 The Competitive Hurdle Rate (CHR) mechanism based on the agency?? adjusted capital charge rate r ˆ = ri (?) is an optimal mechanism.
Equation (25) shows that the present value of the intertemporal cost charges for the winning division is equal to the second highest virtual npv. The CHR mechanism can therefore again be interpreted as a multiperiod version of the second-price auction mechanism. If managers are ex-ante identical with regard to Fi (·) ? F (·), ?i = ?, bi = b and Xi = X , it will su?ce to ask each manager to report his internal rate of return ri . Denoting by rc the internal rate of return of a project whose virtual npv is zero, it is then optimal to set the hurdle rate for the winning division equal to
?? ri (r1 , · · · , rn ) = maxj =i {rj , rc }. In general, however, the rank order of the divisional
internal rates of return need not agree with the rank order of the virtual npvs. The case of ex-ante identical managers also illustrates the impact of an exogenous change in the number of divisions competing for the scarce capital. The principal will bene?t from increased competition among divisions in two ways. First, the informational rents of the winning manager will be reduced. For instance, with exante identical agents, the winner’s informational rent is given by Ui (?i , ??i | ?i ) =
?? ? · [?i ? ?i (??i )] = ? · [?i ? maxj =i {?j , ?c }], where ?c denotes the agency adjusted
break-even pro?tability in a single-agent setting (i.e., rc ? ro (?c )). Clearly, the winning agent’s informational rents are decreasing in n, in expectation. Second, the underinvestment problem will be mitigated. To see this, note that the ex-ante probability of underinvestment is equal to [F (?c ) ? F (?o )]n , where ?o denotes the break-even pro?tability (i.e., NP V (?o ) = 0).
Unlike Proposition 2, Proposition 3 only speaks to the su?ciency, and not necessity, of the CHR mechanism. Given that the parties can commit to the contract and they have the same intertemporal preferences, there is considerable indeterminacy in distributing the divisional cost charges across di?erent periods. Additional robustness criteria, like those proposed in Dutta and Reichelstein (2002) or Arya, Demski, Glover and Liang (2005), may narrow the class of “feasible” mechanisms.
24
29
A natural question is how the agency-adjusted hurdle rate under the CHR mechanism, r?? (?), compares with the hurdle rate identi?ed in Proposition 2 in the absence of an explicit agency problem. In single-period capital budgeting problems, such as the one posed in Antle and Eppen (1985), the hurdle rate always increases as the agency problem becomes more severe: the virtual npv of the project decreases while the opportunity cost of investing is given by the ?rm’s cost of capital r, and therefore una?ected. We ?nd that this prediction does not generalize straightforwardly to our multi-divisional setting. To illustrate, let n = 2.25 The parameters (?1 , ?2 ) measure the relative severity of the two agency problems. Denoting by ?i (?i | ?i ) the present value of virtual cash ?ows as a function of ?i , division 1’s critical type is given by:
?? ?1 (?1 (?2 | ?1 , ?2 ) | ?1 ) ? b1 ? ?2 (?2 | ?2 ) ? b2 ,
(26)
?? o ?? and the resulting hurdle rate equals r1 (? | ?1 , ?2 ) = r1 (?1 (?2 | ?1 , ?2 )). We note
from (20) that ?i (?i | ?i ) is increasing in ?i and decreasing in ?i . In the case of ex-ante identical agents, i.e., Fi (·) ? F (·), ?i = ?, bi = b, Xi = X, for i = 1, 2, the condition characterizing division 1’s cuto? pro?tability type in (26)
?? reduces to ?1 (?2 | ?, ?) ? ?2 . To understand the irrelevance of the agency parameter
? for computing hurdle rates with ex-ante identical agents, note that an increase in ? reduces the virtual npv of division 1’s project, but at the same time it also reduces the opportunity cost (i.e., the virtual npv of division 2’s project) by the same amount. As a consequence, hurdle rates are una?ected. Generalizing this insight, the natural question is whether the hurdle rate can be viewed as an instrument for “handicapping” certain divisions.26 Consider an exogenous increase in ?2 , i.e., manager 2’s agency problem becomes more severe, ceteris paribus. As a result, the right-hand side of (26) decreases. To restore the identity,
?? (?2 | ?1 , ?2 ) must go down and so does the competitive manager 1’s cuto? type ?1 ?? (·). The reverse prediction emerges for an increase in ?1 since now hurdle rate r1
The following discussion can be extended in a straightforward fashion to arbitrary n. The auctions literature has also explored the di?erential treatment of agents based on di?erences in their ex-ante characteristics; see Krishna (2002).
26
25
30
division 1’s virtual npv decreases, which results in a higher pro?tability cuto?. We summarize these observations in the following corollary.
?? Corollary 4 For n = 2, the agency-adjusted competitive hurdle rate ri (? | ?1 , ?2 )
under the CHR mechanism is increasing in ?i , but decreasing in ?j , j = i. The principal thus calibrates the CHR mechanism so as to handicap a division whose agency problem is more severe. In particular, suppose the two projects are ex-ante identical but ?1 > ?2 so that the ?rm faces higher agency costs with division 1. Corollary 4 then implies that division 1 will face a higher hurdle rate than division 2. This ?nding is related to the corporate ?nance literature on investments in diversi?ed ?rms (e.g., Shin and Stulz 1998).27 That literature has pointed to “corporate socialism” insofar as stronger divisions get allocated relatively less capital than weaker divisions, as a result of information asymmetry,. This ?nding has been explained by managers exerting in?uence activities (Rajan et al. 2000). Our results generate similar empirical predictions, though in our model handicapping “stronger” divisions actually improves overall e?ciency. Consider two divisions that are identical in terms of ?i = ?, bi = b, and Xi = X, for i = 1, 2, but their pro?tability types ?i are drawn from di?erent distributions Fi over identical supports ?i = ?. In particular, suppose that division 1’s type distribution is more favorable in the sense of (inverse) hazard rate dominance: H1 (y ) > H2 (y ) for all y . Applying similar arguments as in Corollary 4, it can be shown that, all else equal, division 1 will be charged more for its allocated capital. Since division 1’s project is more pro?table ex ante than that of division 2, rent extraction weighs more heavily for division 1’s manager with the consequence of a higher hurdle rate. To conclude this section, we demonstrate that the CHR mechanism remains optimal for settings in which the second-best solution entails interior e?ort choices. When condition (19) does not hold, the optimal e?ort choice a? it (? | ?i ) will satisfy
27
We thank an anonymous referee for pointing out this connection.
31
the following ?rst-order condition:
? 1 ? vit (a? it (? | ?i )) ? vit (ait (? | ?i )) · xi · Hi (?i ) · Ii (? ) ? 0,
(27)
such that, by complementary slackness, this inequality will hold as an equality when a? ¯it .28 Condition (27) shows that the central o?ce may ?nd it optimal it (? | ?i ) < a to create lowered-powered e?ort incentives in order to economize on the winning division’s informational rents. Furthermore, the winning division’s e?ort incentives will increase in its reported productivity type. The reason is that higher-powered incentives for unproductive types result in higher information rents for more favorable type realizations.29 We note, however, that the second-best investment rule is again given by (21) with VN P Vi (?i ) ? P Vi (?i )?bi ?
T t=1
? t ·vit (a? it (? | ?i ))·xit ·Hi (?i )·Ii (? ). If
the winning division’s periodic bonus coe?cients are chosen as ?it (?) = vit (a? it (? | ?i )) and the performance measure is based on the agency-adjusted competitive hurdle rate, a linear compensation scheme of the form in (23) can achieve the same payo?s for the principal as the direct revelation mechanism. Thus, the CHR mechanism remains optimal even when we allow for interior e?ort choices.30
3.3
Optimal Mechanisms: Shared Assets
For shared assets, we recall from (22) that the second-best investment decision calls for the joint asset to be acquired if and only if the ?rmwide virtual npv is positive. That is, I (?) = 1, if and only if
n
VNP V (?) ?
i=1
28
?i (?i ) ? b ? 0.
See Appendix B in Dutta and Reichelstein (2002) for a complete derivation in a single agent setting. 29 A qualitatively similar outcome would obtain if e?ort and investment productivity were complements, such that cit = ait · xit · ?i · Ii . As demonstrated in Baldenius and Reichelstein (2005), the second-best bonus coe?cients for the winning manager will be an increasing function of his reported type ?i , provided the periodic cash ?ows are multiplicatively separable in ?i and ait . 30 ˜, whereas the ?xed When condition (19) holds, only the performance measures ?i (·) depend on ? ˜) ? ?i and ?i (? ˜) ? ?i . salaries and bonus coe?cients are independent of the reported types: ?i (? This no longer holds once we allow for interior e?ort choices because condition (19) is violated.
32
? In analogy with our de?nition of ?i (??i ), we now de?ne the agency-adjusted critical ?? values by ?i (??i ) by ?? ?i (?i (??i )) ? b ? j =i ?? ? By construction, ?i (??i ) > ?i (??i ). In the presence of agency costs, the capital
?j (?j ).
(28)
charge rate under the PMN mechanism is given by
?? ?? r ˆ(?) = r?? (?) ? ro (?1 (??1 ), ..., ?n (??n )),
(29)
and the divisional asset shares are calculated according to: ?i (?) =
?? (??i ) ??? (?) · Xi · ?i , n ?? ?? i=j ? (? ) · Xj · ?j (??j )
(30)
with ??? (?) ? ((1 + r?? (?))?1 , ..., (1 + r?? (?))?T ) If divisional managers are compensated on the basis of linear schemes of the form in (23) with bonus coe?cients ?it = vit , they have incentives to provide the targeted amount of e?ort in each period. Furthermore, the PMN mechanism results in the following contribution to manager i’s compensation in period t: ˜i , ??i | ?i ) = ?it · Ii (? ˜i , ??i ) · xit · [?i ? ??? (??i )]. ?it · ?it (? i (31)
As a consequence, the joint project leaves manager i better o? if and only if ?i >
?? ?i (??i ), i.e., VNP V (?) ? 0. Truthful reporting therefore again is a dominant-strategy
equilibrium and we obtain the following:31 Proposition 4 Given Assumption 1, the Pay-the-Minimum-Necessary (PMN) mechanism, based on the agency-adjusted capital charge rate of r?? (?) in (29) and the asset sharing rule in (30), is optimal.
Assumption 1 was introduced in Section 2.1 to ensure that the pro?tability cuto?s are in the ?? interior of the managers’ type supports. Since ?i (??i ) is higher for any positive ?i than for ?i = 0 (i.e., absent a moral hazard problem), Assumption 1 ensures interior cuto?s also in the presence of agency problems.
31
33
In contrast to the exclusive asset scenario, the cuto? condition in (28) implies an unambiguous comparative statics result on how the relative severity of the agency problem a?ects the capital charge rate.32 To this end, we again express the hurdle rate r?? (? | ?1 , ..., ?n ) as a function of the agency cost parameters ?i . Corollary 5 The agency-adjusted hurdle rate r?? (? | ?1 , ..., ?n ) under the PMN mechanism is increasing in ?i for all i. As the agency problem for any division becomes more severe, its contribution ?i (·) to the total virtual npv declines. This in turn has a negative externality on the other divisions re?ected in a higher capital charge rate. It is straightforward to show that for su?ciently high agency costs (high ?i ’s), the resulting agency-adjusted hurdle rate, r?? (?), will exceeds the ?rm’s cost of capital r. Conversely, by continuity our result in Corollary 1 shows that the hurdle rate r?? (?) will be below r for su?ciently low levels of agency costs.
4
Conclusion
Interdependencies between divisions are ubiquitous when ?rms make capital budgeting decisions. The framework developed in this paper allows for positive externalities (shared assets) or negative externalities (exclusive assets) among the divisions. Our analysis has illustrated commonalities between these two scenarios as well as distinct di?erences, in particular with regard to the emerging hurdle rates. We found that capital charge rates without managerial moral hazard tend to exceed the cost of capital r for exclusive assets, while the reverse holds for shared assets. As incentive problems become more severe, the hurdle rate will go up unambiguously for shared assets, but not necessarily so for exclusive assets. Our analysis generates a rich set of predictions regarding the cross-sectional variation in hurdle rates, which may prove useful in future empirical research in this area.
Based on the similar arguments as in the exclusive asset case, it can be easily shown that the PMN mechanism remains optimal even when we allow for interior e?ort choices.
32
34
For the exclusive asset setting, our analysis has taken it as given that the ?rm faces a ?xed investment budget which forces capital rationing. While this approach appears descriptive of actual practice in many ?rms, it would be instructive to examine a more continuous problem in which a ?rm faces the tradeo? that more capital spending results in a higher cost of capital for all funded projects. Such an approach seems particularly appealing in settings where the decisions regarding divisional projects are not binary, but instead the scale of the project is also a choice variable. Most long-term investment projects involve multiple rounds of funding. In such settings, it is likely that the participating divisions will receive new information at intermediate dates. Recent research has begun to analyze the dynamics of both “abandonment” and “growth” options for single-agent problems.33 The introduction of multiple project milestones leads naturally to additional cost allocation issues, such as full cost versus successful e?orts accounting. Sequential investment problems also raise the question of how hurdle rates evolve over time. It would be desirable to develop a theory that predicts how abandonment or growth options should be managed for common assets that bene?t multiple divisions within a ?rm. Another promising avenue for future research is to relax our assumption of risk neutrality. As mentioned in the Introduction, earlier work for single-agent models has investigated how capital charge rates should be set for risk averse managers who face projects; see Christensen et al. (2002) and Dutta and Reichelstein (2002). In multi-divisional ?rms aggregate ?rm risk also becomes a coordination issue. To that end, Stoughton and Zechner (2004) explore negative externalities across divisions as higher aggregate risk may necessitate a higher proportion of relatively expensive equity (rather than debt) capital. In contrast, Homburg and Scherpereel (2004) emphasize positive externalities across divisions due to possible diversi?cation e?ects. Either way, incremental risk must be charged for as part of the capital budgeting process.
33
See, for instance, Arya and Glover (2001), Friedl (2003) and Pfei?er and Schneider (2006)
35
Appendix A: Proofs
Proof of Proposition 1 We ?rst prove that the PMN mechanism is a satisfactory mechanism. Under the PMN mechanism, division i’s performance measure in period t is given by: ˜i , ??i | ?i ) = [xit · ?i ? zit (? ˜i , ??i ) · ?i (? ˜i , ??i ) · b] · I ? (? ˜i , ??i ) ?it (? b ˜i , ??i ) · I ? (? ˜i , ??i ) = xit · ?i ? · ?i (? ? ˜ ? (?i , ??i ) · Xi ˜i , ??i ) = Since ?i (? takes the form: ˜i , ??i | ?i ) = xit · [?i ? ?? (??i )] · I ? (? ˜i , ??i ) ?it (? i (32)
˜i ,??i )·Xi ·?? (??i ) ?? (? i Pn ? ? ˜ j =1 ? (?i ,??i )·Xj ·?j (??j )
and the denominator is just equal to b, by the
de?nition of the internal rate of return, manager i’s performance measure in period t
From (32), we note that in each period, manager i’s payo? depends on his own report ˜i only through its impact on the decision rule I ? (·). Furthermore, (32) shows that ?
? each manager has strong incentives to induce I ? = 1 when ?i ? ?i (??i ), and I ? = 0 ? when ?i < ?i (??i ). Consequently, the PMN mechanism satis?es our requirement of
strong incentive compatibility ˜i , ??i | ?i ), ?it (?i , ??i | ?i ) ? ?it (? ˜i , and ??i . for all i, t, ?i , ? We now prove the necessity part, i.e., the PMN mechanism is the unique satisfactory mechanism in the class described in (5). We ?rst show that for, a given asset allocation rule {?i }n i=1 , any strongly incentive compatible mechanism must have the
? (??i ) property that for all ?i > ?i ? (??i ) xit · ?i (33) ?i · b If ui can vary in some open neighborhood, strong incentive compatibility implies ? ˜i > ?? (??i ): (??i ) and ? directly that for any ?i > ?i i
zit (?i , ??i ) =
˜i , ??i ) = zit (?i , ??i ). zit (? 36
Suppose now that for some period t,
? xit · ?i (??i ) + ?t (??i ). ?i · b ˜i = ?? (??i ) + i , incentive compatibility If division i’s pro?tability parameter is ? i ? zit (?i (??i ), ??i ) =
requires that:
T
uit · [ i · xit ? ?t (??i ) · ?i · b] ? 0.
t=1
˜i = ?? (??i ) ? Conversely, for ? i
T
i
incentive compatibility requires:
uit · [? i · xit ? ?t (??i ) · ?i · b] ? 0.
t=1
Since both of the above inequalities have to hold for any
i
> 0 and for all ui in some
open neighborhood, (33) must hold. Finally, by Lemma 1 there exist a unique capital charge rate r ˆ and a unique depreciation schedule, d, implementing the intertemporal cost allocation
? ?i ?? (??i ) (??i ) , ..., xiT · i . ?i (?) · b ?i (?) · b It remains to demonstrate the uniqueness of the asset sharing rule, (?i (?), ..., ?n (?)).
zi (?) =
xi1 ·
Relative bene?t depreciation ensures that
? xit · ?i (??i ) = zit (?) · ?i (?) · b xit · ?i (?) · b. = ? ? (?) · Xi ? ?? (?) · Xi · ?i (??i ) ?? ?i (?) = b ? ? ? (?) · Xi · ?i (??i ) = . n ? ? j =1 ? (? ) · Xj · ?j (??j )
Proof of Lemma 1 For any given z ? (z1 , · · · , zT ) ? RT + , the mapping
t?1
ft (d, r ˆ) ? dt + r ˆ· 37
1?
? =1
d?
= zt
(34)
de?nes a set of T non-linear equations in (d, r ˆ). We show that the above system of equations has a unique solution in D × (?1, ?) for each z ? RT + . Solving the ?rst T ? 1 equations in (34) recursively for (d1 , · · · , dT ?1 ), we get dt = zt + r ˆ · zt?1 + r ˆ(1 + r ˆ)zt?2 + ... + r ˆ(1 + r ˆ)t?2 z1 ? r ˆ(1 + r ˆ)t?1 , (35)
for t = 1, ..., T ? 1. Substituting this solution into the last component of equation (34) gives zT = (1+ˆ r)· (1 + r ˆ)T ?1 ? (1 + r ˆ)T ?2 · z1 ? (1 + r ˆ)T ?3 · z2 ? ... ? (1 + r ˆ) · zT ?2 ? zT ?1 . Multiplying both sides by ? ˆ T ? (1 + r ˆ)?T and simplifying, this last equation reduces to:
T
? ˆ t · zt = 1 .
t=1
(36)
T t=1
For any z ? (z1 , · · · , zT ) ? RT + , the polynomial
? ˆ t · zt is increasing in ? ˆ and
therefore (36) will be satis?ed for a unique r ˆ. In conjunction with (35) this shows that there is a unique solution to the system of T equations.
Proof of Corollary 2 Since
n j =1 ? ?? (?) · Xj · ?j (??j ) = b, the asset sharing rule in the PMN mechanism
? (? ) ?? (? )·Xi ·?i ?i . b
simpli?es to ?i (?) =
To prove the result, therefore, it su?ces to show
that r? (?) is decreasing in ?i . This follows because
? ? r? (?) = ro (?1 (??1 ), · · · , ?n (??n )), ? (??j ) is decreasing in ?i for all ro (·) is increasing in each of its arguments, and ?j
i = j.
38
Proof of Proposition 2. We again begin with the su?ciency part. Under the competitive hurdle rate mechanism, period t performance measure is given by: ˜i , ??i | ?i ) = ?it (? xit · ?i ? xit
T ? =1 [1
+
o ? ri (?i (??i ))]??
· xi?
· bi
˜i , ??i ) · Ii? (? (37)
? ˜i , ??i ), = xit · [?i ? ?i (??i )] · Ii? (?
It follows from (37) that each manager has strong incentives to induce Ii? = 1 when
? ? ?i ? ?i (??i ), and Ii? = 0 when ?i < ?i (??i ). This proves the su?ciency part of
Proposition (2). To prove the necessity part, we note that manager i’s performance measure in period t is given by ?( ?i , ??i ) = [xit · ?i ? zit (?i , ??i ) · bi ]. · Ii (?i , ??i ) Using arguments parallel to the ones in the proof of Proposition 1, we can show that zit (?i , ??i ) =
xit bi ? ? · ?i (??i ) for any ?i > ?i (??i ). By Lemma 1, there exists a
unique depreciation schedule and a unique capital charge rate that implements the intertemporal cost allocation scheme: zi (?) = xiT ? xi1 ? · ?i (??i ), ..., · ?i (??i ) . bi bi
As argued in the text, one solution is provided by the capital charge rate r ˆ = r? (?) combined with the relative bene?t depreciation rule and this must therefore be the unique solution.
Proof of Proposition 3 ˜i , ??i | ?i ) denote type For a given pro?le of types other than division i, ??i , let Uit (? ˜i . Therefore, the manager’s ?i manager’s utility payo? in period t when he reports ?
39
˜i , ??i | ?i ) ? total utility payo?s are given by Ui (? competitive hurdle rate mechanism,
T t=1
˜i , ??i | ?i ). Under the ? t · Uit (?
˜i , ??i | ?i ) = ?(? ˜i , ??i ) + ? (? ˜i , ??i ) · [ait + I ? (? ˜i , ??i ) · (xit · ?i ? zit (? ˜i , ??i ))] ? vit (ait ), Uit (? i where Ii? (·, ·) denotes the optimal investment decision rule and zit (·, ·) denotes the sum of depreciation and interest charges, as de?ned in (6). For each division, the central o?ce chooses the compensation parameters {?it , ?it }T t=1 ˜ ˜ ˜ such that ?it (?) = v and ?it (?) = vit (¯ ait ) ? v · a ¯it for any report pro?le ?. This
it it
choice of bonus coe?cients ensures that ait = a ¯it for each i and t. Furthermore, the relative bene?t depreciation schedule corresponding to the agency-adjusted compet˜ ??i ) = xit · ??? (??i ). As a consequence, itive hurdle rate r?? (??i ) implies that zit (?,
i i
manager i’s total utility payo? becomes: ˜i , ??i | ?i ) = ?i · I ? (? ˜i , ??i ) · [?i ? ??? (??i )] Ui (? i i where ?i ?
T t=1
(38)
? t · vit · xit .
It is clear from (38) that manager i’s participation constraint holds for each ??i , and each manager has a strong dominant strategy incentive to report his information truthfully. Furthermore, a comparison with (40) reveals that manager i’s interim utility payo?s E??i [Ui (?i , ??i | ?i )] from (38) coincide with those of the optimal revelation mechanism.
Proof of Corollary 4
?? (??i |?1 , ?2 ) is de?ned For given ?1 and ?2 , division i’s cuto? pro?tability parameter ?i
by the equation:
?? VNP Vi (?i (??i | ?1 , ?2 )) = max{VNP Vj (?j | ?j ), 0}, j = i.
Since the virtual npv is given by VNP Vi (?i | ?i ) = NP Vi (?i ) ? ?i · Hi (?i ), it follows that VNP Vi (?i | ?i ) is uniformly decreasing in ?i . This immediately implies that 40
?? ?i (??i | ·) is uniformly increasing in ?i and uniformly decreasing in ?j , j = i. The ?? result follows since the agency-adjusted competitive hurdle rate ri (? | ?1 , ?2 ) is equal o ?? o to ri (?i (??i | ?1 , ?2 )), and ri (·) is an increasing function.
Proof of Proposition 4 Under the PMN mechanism, manager i’s utility payo?s in period t are given by: ˜i , ??i | ?i ) = ?(? ˜i , ??i ) + ? (? ˜i , ??i ) · [ait + I ? (? ˜i , ??i ) · (xit · ?i ? zit (? ˜i , ??i ))] ? vit (ait ). Uit (? As in the proof of Proposition 3, high managerial e?orts can be induced by ˜) = v for each report ? ˜. Combined with the PMN asset allocation setting ?it (? it ˜i , ?i )}N , relative bene?t depreciation corresponding to the hurdle rate rule {?i (?
i=1
˜i , ??i ) ensures that zit (?, ˜ ??i ) = xit · ??? (??i ). When the ?xed payments are r (? i ˜ chosen such that ?it (?) = vit (¯ ait ) ? vit · a ¯it , manager i’s total utility payo?s become:
??
˜i , ??i | ?i ) = ?i · I ? (? ˜i , ?i ) · [?i ? ??? (??i )]. Ui (? i
(39)
Clearly, the above PMN mechanism ensures that manager i’s participation constraint holds for each ??i , and each manager has a strong dominant strategy incentive to report his information truthfully. Furthermore, each manager’s interim utility payo?s E??i [Ui (?i , ??i | ?i )] from (39) coincide with those of the optimal revelation mechanism as given in (40). Proof of Corollary 5 Since ?i (?i ) = Vi (?i ) ? ?i · Hi (?i ), equation (28) implies that the cuto?-pro?tability
?? (??i | ?1 , ..., ?n ) is uniformly increasing in ?j for all 1 ? j ? n. The parameter ?i
result follows because:
?? ?? r?? (? | ?1 , ..., ?n ) = ro (?1 (??1 | ?1 , ..., ?n ), · · · , ?n (??n | ?1 , ..., ?n )) ,
and ro (·) increases in each of its arguments. 41
Appendix B: Derivation of Relaxed Program P
The participation constraint E??i [Ui (?i , ??i | ?i )] ? 0 in the principal’s original program P will hold with equality for the lowest type ?i . This boundary condition combined with the fact that ait (?i , ??i | ?i ) = cit (?i , ??i ) ? xit · ?i · Ii (?i , ??i ) and the “local” incentive compatibility condition implies that manager i will earn the following informational rents:
?i T
E??i [Ui (?i , ??i | ?i )] = E??i
?i
? t · vit (ait (qi , ??i | qi )) · xit · Ii (qi , ??i ) dqi .
t=1
(40)
Integrating by parts yields that manager i’s expected informational rents are given by:
¯i ? T
E? [Ui (?i , ??i | ?i )] = E??i
?i
? t · [vit (ait (?)) · xit · Ii (?) · Hi (?i )] fi (?i )d?i ,
t=1
(41)
where ait (?) ? ait (?i , ??i | ?i ). Since Ui (?i , ??i | ?i ) ?
T t=1
? t · [sit (?i , ??i | ?i ) ? vit (ait (?))], substituting (41) into
n
the objective function in P yields: max E?
i=1
(ai (? ),Ii (?))n i=1
?i (?) ? B (?)
subject to(ie ), (is ), where
T
?i (?) ?
t=1
? t · {ait (?) ? vit (ait (?)) + xit · [?i ? vit (ait (?)) · Hi (?i )] · Ii (?)} .
For any given investment rule Ii (?), the central o?ce will choose ait to maximize: ait ? vit (ait ) ? vit (ait ) · xit · Hi (?i ) · Ii (?) 42
Condition (19) implies that 1 ? vit (¯ ait ) ? vit (¯ ait ) · xit · Hi (?i ) · I (?) > 0 for all ?i and each Ii (?) ? {0, 1}. It is therefore optimal to induce the highest level of e?ort a ¯it for all ?i . Consequently, the optimization program in P simpli?es to the program in P , and the optimal investment rules for the exclusive and shared asset settings are as given by (21) and (22), respectively. To complete the proof, we need to show that the resulting scheme is globally incentive compatible. As shown in Mirrlees (1971), a mechanism is incentive compatible ˜ ˜i . provided it is locally incentive compatible, and ?Ui (?i ,??i |?i ) is weakly increasing in ?
??i
For the above mechanism: ˜i , ??i | ?i ) ?Ui (? = ??i
T
˜i , ??i | ?i )) · xit · Ii (? ˜i , ??i ), ? t · vit (ait (?
t=1
˜i since v (ait (·, ??i | ?i )) is increasing in ? ˜i and the optimal which is increasing in ? it Ii (·, ??i ) is an upper-tail investment policy in both settings.
43
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Christensen, P., G. Feltham and M. Wu (2002), “ ‘Cost of Capital’ in Residual Income Measurement Under Moral Hazard,” The Accounting Review 77(1), 1-23. Daugart, J., (2005), “Divisional Performance Measurement and Investment Incentives: Residual Income, Multiple Divisions, and Externalities,” mimeo, University of Hanover. Dutta, S. (2003), “Capital Budgeting and Managerial Compensation: Incentive and Retention E?ects,” The Accounting Review 78(1), 71-93. Dutta, S., and S. Reichelstein (2002), “Controlling Investment Decisions: Depreciation and Capital Charges,” Review of Accounting Studies 7, 253-281. Dutta, S., and S. Reichelstein (2005), “Accrual Accounting for Performance Evaluation,” Review of Accounting Studies, 10(4),527-552. Friedl, G. (2003), “Growth Options, Organizational Slack, and Managerial Investment Incentives,” mimeo, University of Munich. Goex, R., and U. Schiller (2006), “An Economic Perspective on Transfer Pricing”, forthcoming in Management Accounting Research, C. Chapman, A. Hopwood and M. Shields (eds., Oxford University Press, Oxford. Green, J., and J.-J. La?ont (1979), Incentives in Public Decision-Making, NorthHolland, Amsterdam. Groves, T. (1973), “Incentives in Teams,” Econometrica 41, 617-631. Harris, M., and A. Raviv (1996), “The Capital Budgeting Process: Incentives and Information,” Journal of Finance 51(4), 1139-1174. Hodak, M. (1997), “The End of Cost Allocations as We Know Them,” Journal of Applied Corporate Finance 10(3), 117-124. Homburg, C., and P. Scherpereel (2004), “How Should the Cost of Joint Risk Capital be Allocated for Performance Measurement?”, mimeo, University of Cologne. Hughes, J., and J. Kao (1997), “Cross Subsidization, Cost Allocation and Tacit Collusion,” Review of Accounting Studies 2(3), 265-293. Kanodia, C. (1993), “Participative Budgets as Coordination and Motivational Devices,” Journal of Accounting Research 31(2), 172-189. 45
Kaplan, R., and A. Atkinson (1998), Advanced Management Accounting, Third Ed., Prentice-Hall, Englewood Cli?s. Kreps, D. (1990), A Course in Microeconomic Theory, Princeton University Press, Princeton, NJ. Krishna, V. (2002), Auction Theory, Academic Press, San Diego, CA. La?ont, J.-J., and J. Tirole (1993), A Theory of Incentives in Procurement and Regulation, MIT Press, Cambridge, MA. Mas-Colell, A., M Whinston and J. Green (1995), Microeconomic Theory, Oxford University Press, Oxford. Mirrlees, J. (1971), “An Exploration in the Theory of Optimal Income Taxation,” Review of Economic Studies 38, 175-208. Mookherjee, D., and S. Reichelstein (1992), “Dominant Strategy Implementation of Bayesian Incentive Compatible Allocation Rules,” Journal of Economic Theory 55(2), 378-399. Moulin, H. (1986), “Characterizations of the Pivotal Mechanism,” Journal of Public Economics 31, 53-78. Myerson, R. (1981), “Optimal Auction Design,” Mathematics of Operations Research 6(1), 58-73. Pfa?, D. (1994), “On the Allocation of Overhead Costs,” European Accounting Review 3, 49-70. Poterba, J., and L. Summers (1995), “A CEO Survey of U.S. Companies’ Time Horizons and Hurdle Rates,” Sloan Management Review 37(1), 43-53. Pfei?er, T., and G. Schneider (2006), “Residual Income Based Compensation Schemes for Controlling Investment Decisions under Sequential Private Information” forthcoming in Management Science. Rajan, M. (1992), “Management Control Systems and the Implementation of Strategies,” Journal of Accounting Research 30(2), 227-248. Rajan, R., H. Servaes, L. Zingales (2000),“The Cost of Diversity: The Diversi?cation Discount and Ine?cient Investment,” Journal of Finance 55(1), 35-80. 46
Reichelstein, S. (1997), “Investment Decisions and Managerial Performance Evaluation,” Review of Accounting Studies 2, 157-180. Rogerson, W. (1997), “Inter-Temporal Cost Allocation and Managerial Investment Incentives: A Theory Explaining the Use of Economic Value Added as a Performance Measure,” Journal of Political Economy 105, 770-795. Shin, H., and R. Stulz (1998), “Are Internal Capital Markets E?cient?,” Quarterly Journal of Economics 113(2), 531-52. Stoughton, N., and J. Zechner (2004), “Capital Allocation using RAROC and EVA”, Working paper No. 4169, Centre for Economic Policy Research, www.cepr.org Taggart, R. (1987), “Allocating Capital Among a Firm’s Divisions: Hurdle Rates vs. Budgets,” The Journal of Financial Research 10(3), 177-189. Wei, D. (2004), “Interdepartmental Cost Allocation and Investment Incentives,” Review of Accounting Studies Vol. 9, 97-116. Zimmerman, J. (1979), “The Costs and Bene?ts of Cost Allocations,” The Accounting Review 54, 504-521.
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doc_338213756.pdf
Cost allocation is a process of providing relief to shared service organization's cost centers that provide a product or service. In turn, the associated expense is assigned to internal clients' cost centers that consume the products and services.
Cost Allocation For Capital Budgeting Decisions
Tim Baldenius? Sunil Dutta† Stefan Reichelstein‡
September 2006
Graduate School of Business, Columbia University, [email protected] Haas School of Business, University of California at Berkeley, [email protected] ‡ Graduate School of Business, Stanford University, [email protected] We thank seminar participants at the following universities: Columbia, Washington, Ohio State, Boston, Duke, Michigan State and the Norwegian School of Business and Economics for their many comments on this paper. We are also grateful for detailed suggestions from two anonymous reviewers and the editor, Jack Hughes.
†
?
Cost Allocation For Capital Budgeting Decisions Abstract Investment decisions frequently require coordination across multiple divisions of a ?rm. This paper explores a class of capital budgeting mechanisms in which the divisions issue reports regarding the anticipated pro?tability of proposed projects. To hold the divisions accountable for their reports, the central o?ce ties the project acceptance decision to a system of cost allocations comprised of depreciation and capital charges. If the proposed project concerns a common asset that bene?ts multiple divisions, our analysis derives a sharing rule for dividing the asset among the users. Capital charges are based on a hurdle rate determined by the divisional reports. We ?nd that this hurdle rate deviates from the ?rm’s cost of capital in a manner that depends crucially on whether the coordination problem is one of implementing a common asset or choosing among multiple competing projects. We also ?nd that more severe divisional agency problems will increase the hurdle rate for common assets, yet this is generally not true for competing projects.
1
Introduction
Capital budgeting decisions frequently a?ect multiple entities, such as departments or divisions, within a ?rm. Decision externalities may arise because divisional investment opportunities are mutually exclusive due to limited investment budgets or due to a required intermediate good that can be procured from more than one division. Alternatively, ?rms may be in a position to acquire common assets that are of use to multiple divisions at the same time. The capital budgeting process must then determine whether the aggregated individual bene?ts justify the common investment expenditure.1 The management literature provides only scant evidence regarding ?rms’ actual capital budgeting practices; see for example Kaplan and Atkinson (1998) and Taggart (1988). Some authors have suggested that ?rms could operate market mechanisms, such as auctions, in order to solve their capital budgeting problems, e.g., Hodak (1997). One approach that seems prevalent in practice is that ?rms set hurdle rates which must be met by individual projects in order to receive funding. However, there does not seem to be a well accepted methodology either for setting hurdle rates or for including them in the system of divisional performance measurement. This paper explores a class of capital budgeting mechanisms for which the ?rm’s central o?ce commits to a decision rule in response to divisional reports about the projected pro?tability of particular projects. To hold the divisions accountable for their reports, the central o?ce speci?es cost charges to be imposed in subsequent periods. These cost charges comprise both depreciation and capital charges, with the latter based on a hurdle rate that is determined by the initial divisional reports. We focus on charging mechanisms that satisfy the usual accounting convention that the
A recent illustration of such “public goods” problems was provided to us by a California semiconductor manufacturer where several product lines (pro?t centers) were using a common fabrication facility. The product line managers regularly encouraged the ?rm’s central o?ce to upgrade the equipment in the manufacturing facility. However, the ?rm’s central o?ce was reluctant to undertake these upgrades as it worried about “creative optimism” by the divisions, partly because the ?rm did not have a methodology for allocating investment expenditures among the users of the facility.
1
1
sum of all depreciation charges across time periods and across divisions be equal to the initial investment expenditure. The survey evidence of Poterba and Summers (1995) and others has shown that internal hurdle rates frequently exceed a ?rm’s actual cost of capital by a substantial margin. Consistent with this evidence, the capital budgeting literature has shown that with private managerial information second-best mechanisms entail capital rationing.2 This tendency emerges because the principal is better o? foregoing marginally pro?table projects in order to economize on the manager’s informational rents (the premium a manager must be paid for truthful reporting). One way to implement such capital rationing is to raise the hurdle rate above the ?rm’s cost of capital. Consequently, these models predict that higher agency costs will result in more severe capital rationing, and thus higher hurdle rates. In our multidivisional setting, hurdle rates can be interpreted as a “shadow price” of capital which re?ects two distinct though related factors: (i) the interdivisional coordination problem due to decision externalities, and (ii) agency costs at the divisional level. In contrast to single division settings, we ?nd that the hurdle rate will di?er from the ?rm’s cost of capital even in the absence of any agency costs. Whether the hurdle rate will be set above or below the ?rm’s cost of capital depends on the nature of the decision externality, i.e., whether the asset is shared or exclusively used by only one division, and the relative magnitude of the divisional agency costs. Suppose ?rst that the capital budgeting problem is such that each division has an investment opportunity but at most one of the n divisional projects can be funded. In such settings, we identify a Competitive Hurdle Rate (CHR) mechanism which e?ectively is a multiperiod version of the second-price auction. In particular, the winning division faces a hurdle rate which is una?ected by its own report. In the special case where the divisional projects are ex-ante identical, the competitive hurdle rate is simply the second highest reported internal rate of return. The CHR mechanism also speci?es a depreciation schedule for the initial investment. If this schedule conforms
2
See, for instance, Antle and Eppen (1985).
2
to the so-called relative bene?t rule, the project’s npv is re?ected in a time consistent manner in the divisional performance measure.3 These features make the CHR mechanism strongly incentive compatible in the sense that divisional managers have a dominant-strategy incentive to report their information truthfully regardless of the weights (determined by intertemporal preferences and compensation payments) they attach to outcomes in di?erent time periods. We establish that the CHR mechanism is the only satisfactory capital budgeting mechanism that achieves both strong incentive compatibility and e?cient project selection. This uniqueness ?nding relies in part on a well-known result by Green and La?ont (1979) in the earlier public choice literature: the discounted total charges to the divisions must conform to a Groves scheme (Groves 1973) in order to obtain dominant strategies. Since we postulate that a division will not be charged unless its project is funded, the class of feasible Groves mechanisms reduces to the so-called Pivot mechanism: a division is charged if and only if its report is pivotal in the sense that it changes the resource allocation decision. Finally, it has been shown that the Pivot mechanism amounts to a second-price auction in settings where an indivisible good is allocated among n participants.4 Our results are consistent with Poterba and Summers (1995) insofar as the competitive hurdle rate exceeds the ?rm’s cost of capital. Nonetheless, the CHR mechanism is balanced in the sense that the discounted sum of depreciation and capital charges is equal to the initial investment expenditure provided the future cost charges are discounted at the competitive hurdle rate. This is a direct consequence of the postulate of comprehensive income measurement requiring the sum of the depreciation charges to equal the initial investment expenditure incurred by the winning division.5
This feature has been established in the earlier work on goal congruent performance measures for investment decisions involving a single division, e.g., Rogerson (1997), Reichelstein (1997). 4 See Mas-Colell et al. (1995) and Krishna (2002). 5 In one-period settings, Pivot mechanisms are well-known to attain a budget surplus. Since the CHR mechanism is a multiperiod version of the Pivot mechanism surplus, one would expect the CHR mechanism to overcharge the winning division. This is true in the sense that the discounted sum of depreciation and capital charges exceeds the initial investment if the discount rate is given by the ?rm’s cost of capital which is below the competitive hurdle rate.
3
3
We identify a class of environments for which the CHR mechanism is optimal among all revelation mechanisms. In contrast to single-agent settings, however, our analysis shows that the hurdle rate will be una?ected if the agency costs for each division go up by the same factor and the divisions face comparable projects, ex ante.6 At the same time, it will be bene?cial to use the hurdle rate as an instrument for “handicapping” divisions which (i) are ex-ante likely to have access to more pro?table projects and (ii) face relatively high agency costs. We demonstrate that optimal capital budgeting mechanisms have the feature that ceteris paribus those divisions face higher hurdle rates. When the capital budgeting problem concerns the acquisition of a common or shared asset to which all divisions have access, our ?ndings are in several respects “dual” to the ones obtained in connection with exclusive assets. Invoking again the criterion of satisfactory mechanisms, we identify the so-called Pay-the-MinimumNecessary (PMN) mechanism as the unique mechanism that results in e?cient project selection decisions in a (strongly) incentive compatible manner. In present value terms, the PMN mechanism charges every division its “critical value”, the present value that this division would have to obtain from the joint asset in order for the investment to just break even. To implement this rule in a time consistent fashion, each division is assigned a share of the joint investment expenditure as a divisional asset. The shares are set in proportion to the critical values and the capital charge rate under the PMN mechanism is given by the project’s internal rate of return evaluated at the critical values. In contrast to our ?ndings for exclusive assets, the capital charge rate under the PMN mechanism is below the ?rm’s cost of capital. As a consequence, the PMN mechanism e?ectively runs a de?cit in present value terms.7 While the sum of all
This ?nding is based on the same argument showing that under certain conditions the second price auction is a revenue maximizing mechanism; see, for example, Myerson (1981). In particular, this result obtains if the bidder with the highest intrinsic value for the object also has the highest virtual value. 7 Consistent with this observation, the PMN mechanism is not a multiperiod version of the Pivot mechanism.
6
4
depreciation charges across time periods and divisions is equal to the acquisition cost of the joint asset, the discounted sum of all depreciation and capital charges is less than the initial acquisition cost, provided these charges are discounted at the ?rm’s cost capital. Intuitively, the need for such subsidization arises because of the public goods nature of the common asset. The divisions no longer compete but instead exert a positive externality upon one another. In order for the PMN mechanism to be a second-best contracting mechanism for shared assets, the capital charge rate must increase monotonically in the agency costs of each division. This re?ects that the total net pro?tability of the shared asset decreases with the sum of the individual agency costs (for exclusive assets, in contrast, it is the relative magnitude of the individual agency costs that matters). We conclude that for shared assets the PMN mechanism may result in an agency-adjusted capital charge rate that can either be above or below the ?rm’s cost of capital. The criterion for e?cient project selection tends to push the capital charge rate below the ?rm’s cost of capital, while agency costs tend to push in the opposite direction. In terms of prior literature, our work builds on a range of studies that have examined the role of cost allocation mechanisms in guiding intra?rm resource allocation; papers in this category include Zimmerman (1979), Baiman and Noel (1985), Rajan (1992) and Pfa? (1994), among others. These studies have focuses on cross-sectional cost allocations in a one-period setting. Similarly, most of the existing work on multiagent budgeting mechanisms has been con?ned to single-period studies, e.g., Harris, Kriebel and Raviv (1982), Kanodia (1994), Balakrishnan (1995), Arya et al. (1996), Harris and Raviv (1996), Chen (2003), and Bernardo, Cai and Luo (2004).8 Intertemporal cost allocations have been central to the prior work on accrual accounting for performance measurement, such as Rogerson (1997) and Reichelstein
Our ?nding regarding capital charge rates that di?er from the ?rm’s actual cost of capital is reminiscent of earlier work on “strategic” internal pricing, such as Hughes and Kao (1997), Alles and Datar (1998), Arya and Mittendorf (2004). The main theme in these models is that a ?rm’s central o?ce sets internal prices (e.g., cost allocations) so as to a?ect the competitive outcome between the ?rm’s divisions and external rivals. Goex and Schiller (2006) provide a survey of this literature.
8
5
(1997), Wei (2004), and Bareket and Mohnen (2006). One conclusion in these models is that in the absence of moral hazard, the capital charge rate should be set equal to the ?rm’s cost of capital in order to obtain goal congruent performance measures. In contrast, Dutta and Reichelstein (2002) and Christensen, Feltham and Wu (2002) question the usual textbook recommendation of setting the capital charge rate equal to the ?rm’s weighted average cost of capital if risky projects are to be selected by a risk- and e?ort averse manager.9 All of these studies focus on a single representative agent without addressing coordination issues across multiple divisions.10 The remainder of the paper is organized as follows. Section 2 examines satisfactory capital budgeting mechanisms for both exclusive and shared assets. Agency costs and second-best contracting mechanisms are the subject of Section 3. We conclude in Section 4.
2
Satisfactory Capital Budgeting Mechanisms
We examine mechanisms for coordinating investment decisions in a ?rm comprised of n divisions and a central o?ce. At the initial date, the ?rm can acquire a capital asset with a useful life of T years. The ?rm’s central o?ce faces an incentive and coordination problem because the division managers have private information regarding the subsequent operating pro?tability of the asset. To study the most common forms of interdivisional coordination problems, we consider two scenarios: (i) shared and (ii) exclusive assets. In the case of shared assets, the ?rm has access to a common investment project that, if undertaken, generates future revenues for all n divisions. In that sense, the common investment project is a “public good” which bene?ts all divisions. For
Baldenius (2003) links the choice of the capital charge rate to empire bene?ts that managers derive from new investments. Baldenius and Ziv (2003) consider the e?ect of incomes taxes on capital charge rates. Dutta (2003) examines a ?rm’s choice of capital charge rate when its manager has the option of pursuing the investment project as an outside venture on his own. 10 An exception is Wei (2004) who studies a setting with symmetric information across managers and shows that ?xed cost allocations can alleviate divisional underinvestment problems.
9
6
the scenario of exclusive assets, in contrast, each division is assumed to have one investment opportunity, yet for exogenous reasons the ?rm can undertake at most one of these n divisional projects. Each project, if undertaken, generates subsequent cash ?ows only for that particular division. In this setting, the projects are “private goods” and the divisions compete for scarce investment capital. Without loss of generality, each division’s status quo operating cash ?ow (in the absence of any new investment) is normalized to zero. When division i has access to the new investment (exclusive or shared), its periodic operating cash ?ows take the form: cit = xit · ?i . (1)
¯i ] denotes the pro?tability parameter of division i, with ? ? Here, ?i ? ?i = [?i , ? i 0. The vector of pro?tability parameters of all n divisions will be denoted by ? = (?1 , ..., ?n ) and ??i ? (?1 , ..., ?i?1 , ?i+1 , ..., ?n ) denotes the pro?tability pro?le of all divisions other than i. While ?i is assumed to be known only to manager i, the intertemporal distribution of future cash ?ows, as represented by the vector Xi =
11 (x1i , ..., xiT ) ? RT + , is commonly known.
The ?rm’s cost of capital is given by r ? 0 with ? = 1/(1 + r) representing the discount factor. The present value of division i’s cash ?ows is
T
P Vi (?i ) =
t=1
? t · xit · ?i
? ? · Xi · ?i , where ? ? (?, ..., ? T ). Initially, we ignore moral hazard problems and their associated agency costs and instead focus on the choice of goal congruent performance measures
The e?ective useful life of a project can be ? < T periods, in which case xit = 0 for all ? ? t ? T . The central assumption underlying (1) is that the designer knows not only the useful life of the asset but also the intertemporal pattern of cash ?ows. For instance, if the investment pertains to a new production facility, the parameters xit may re?ect the known production capacity available in di?erent periods, while ?i represents the (expected) contribution margin which is known only to the divisional manager. Similar assumptions are made in Rogerson (1997), Reichelstein (1997), Baldenius and Ziv (2003), Wei (2004), Bareket and Mohnen (2006).
11
7
for divisional managers. As shown in Section 3 below, the mechanisms identi?ed in this section can be adapted to optimal second-best mechanisms so as to include issues of moral hazard and managerial compensation. Following the terminology in earlier literature, a performance measure is said to be goal congruent if it induces managers to make decisions that maximize the present value of ?rm-wide cash ?ows. In our search for goal congruent performance measures, we con?ne attention to accounting-based metrics of the form ?it = Incit ? r ˆ · Ai,t?1 , (2)
where Ait denotes book value of division i’s asset at the end of period t, and r ˆ is a capital charge rate applied to the beginning book value. The net asset value at the end of period t is given by Ait = Ai,t?1 ? dit · Ai0 , where dit denotes the period-t depreciation percentage for division i in period t and Ai0 represents the initial asset value assigned to division i. Given comprehensive income measurement, income in period t is calculated as: Incit = cit + Ait ? Ai,t?1 = cit ? dit · Ai0 . We note that the class of performance measures in (2) encompasses the most common accounting performance metrics such as income, residual income, and operating cash ?ow. To create reporting incentives, the central o?ce has two principal instruments: the capital charge rate r ˆ and the depreciation rules {dit }T t=1 . In our search for alternative capital budgeting mechanisms, we focus on balanced cost allocations satisfying the property that the sum of all depreciation charges across agents and across time periods is equal to the amount initially invested. Performance measures are required to have a “robustness” property such that the desired incentives hold even if a divisional manager attaches weights to future outcomes that di?er from those of the ?rm which is interested in the present value of 8
future cash ?ows. Let ui = (ui1 , ..., uiT ) denote non-negative weights that manager i attaches to the sequence of performance measures ?i = (?1i , ..., ?iT ). At the beginning of period 1, manager i’s objective function can thus be written as
T t=1
uit · E [?it ].
One can think of the weights ui as re?ecting a manager’s discount factor as well as the bonus coe?cients attached to the periodic performance measures. We require goal congruence for all ui in some open set in Vi ? RT + . For instance, Vi could be a neighborhood around (u · ?, u · ? 2 , ..., u · ? T ) for some constant bonus coe?cient u.12 ˜i , ? ˜?i |?i ) denote manager i’s period-t performance measure when his true Let ?it (? ˜i and the other n ? 1 managers report ? ˜?i . A performance type is ?i , but he reports ? measure satis?es strong incentive compatibility if:
T T
uit · ?it (?i , ??i |?i ) ?
t=1 t=1
˜i , ??i |?i ), uit · ?it (?
˜i , ??i , ui ? Vi . for all i, ?i , ?
(3)
Our requirement of strong incentive compatibility amounts to dominant-strategy incentives for truthful reporting in a setting where the designer is also uncertain about the weights that managers attach to their performance measures in di?erent periods. This form of strong incentive compatibility will prove useful below when moral hazard is added to the model and the intertemporal weights uit are determined by the managerial compensation schemes.
2.1
Shared Assets
We ?rst study the design of capital budgeting mechanisms for a setting in which the asset acquired is shared among the n divisions in the sense that all divisions can have simultaneous access to the asset and derive future cash bene?ts from it. Applicable examples include cost reducing investments in a manufacturing process that is used
In order to assess the robustness of a particular mechanism one would like Vi to be as large as possible, e.g, the entire RT + . On the other hand, any necessity result pointing to the uniqueness of a particular mechanism becomes more powerful if derived with reference to a smaller set Vi . For now, it is useful to view uit as a summary statistic for the manager’s discount factor and the bonus coe?cients in his compensation function. In Section 3, the coe?cients uit will emerge endogenously from the underlying agency problem.
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by all divisions, certain forms of IT services or “lumpy” investments in capacity which alleviate any subsequent capacity constraints. The “common good” requires an initial cash outlay of b and generates (incremental) operating cash ?ows in the amount of xit · ?i for divisions 1 ? i ? n in periods 1 ? t ? T . For each division, the pro?tability parameter ?i is private information. The corporate net present value (npv) of the project then equals
n
NP V (?) =
i=1
P Vi (?i ) ? b.
The corresponding internal rate of return, denoted by ro (?), is implicitly de?ned by
n T
(1 + ro (?))?t · xit · ?i = b.
i=1 t=1
We note that the internal rate of return ro (?) is unique because ?i > 0 and xit ? 0. The indicator variable I ? {0, 1} will be used to represent whether the common project is undertaken. The ?rst-best investment rule I ? (?) calls for the investment to be made whenever NP V (?) ? 0 or, equivalently, whenever ro (?) ? r. To obtain goal congruence, a capital budgeting mechanism for shared assets relies on four instruments: an investment rule, an asset allocation rule, a capital charge rate, and depreciation schedules. To compute the divisional performance measure in (2), the asset allocation rule assigns each division a share, ?i · b, of the joint asset’s initial investment cost b. This amount is capitalized on the divisional balance sheet and subsequently depreciated over the next T periods according to the speci?ed depreciation schedule di = (di1 , ..., dit ). The book value of division i’s asset at date t is therefore given by:
t
Ait = Ai,t?1 ? dit · ?i · b =
1?
? =1
di?
· ?i · b.
(4)
Formally, a capital budgeting mechanism for shared assets speci?es the following items: 10
• Investment rule I : ? ? {0, 1}; • Asset allocation rule ?i : ? ? [0, 1] satisfying the requirement • Capital charge rate r ˆ : ? ? (?1, ?); • Depreciation schedules di : ? ? RT + satisfying the conditions that 1 if I (?) = 1, and dit (?) ? 0 if I (?) = 0. We note that mechanisms are restricted to satisfy a “no-play-no-pay” condition: the divisions can be charged only if the joint asset is acquired, i.e., I (?) = 1. A capital budgeting mechanism is called satisfactory if (i) it is strongly incentive compatible as de?ned in (3), and (ii) the project is undertaken, if and only if NP V (?) ? 0. Given the asset valuation rule in (4), the divisional performance measure in (2) becomes: ?it = (xit · ?i ? zit · ?i · b) · I, where zit = dit + r ˆ· 1?
? =1 t?1 T t=1 n i=1
?i (?) = 1;
dit (?) =
(5)
di?
(6)
denotes the sum of depreciation and interest charge in period t. Following Rogerson (1997), we refer to {zit }T t=1 as an intertemporal cost allocation. Earlier studies on goal congruence have observed that for any given capital charge rate, r ˆ, there is a one-to-one mapping between depreciation and intertemporal cost allocation schemes. In particular, there exists a unique intertemporal cost allocation such that zit = xit
T ? =1 (1
+r ˆ)?? · xi?
.
(7)
The signi?cance of this relative bene?t cost allocation is that, for a given asset allocation rule {?i }n i=1 , division i’s residual income measure, relative to the capital charge rate r ˆ, will in each period be proportional to the divisional npv, P Vi (?i |r ˆ)??i ·b, which is obtained by discounting future cash ?ows at the rate r ˆ. Thus, for a capital 11
budgeting problem with a single agent, the principal can achieve goal congruence by setting r ˆ = r. The unique depreciation schedule that gives rise to the cost allocation charges in (7) is referred to as the relative bene?t depreciation rule.13
? Let ?i (??i ) denote the critical pro?tability parameter of division i, that is, the
value of ?i at which the project breaks even, given the valuations ??i of all other
? divisions. Thus, ?i (??i ) is de?ned implicitly by ? P Vi (?i (??i )) + j =i ? Assumption 1 Each division is essential in the sense that ?i (??i ) ? ?i for all i and
P Vj (?j ) = b.
(8)
??i . Assumption 1 stipulates that
j =i
P Vj (?j ) + P Vi (?i ) < b for all i and ??i . The
corresponding restriction on the range of divisional pro?tability parameters ?i is easier to satisfy in a setting with just a “small” number of participating divisions. As demonstrated at the end of this subsection, our main results can be extended to environments beyond those conforming to Assumption 1. The following capital budgeting mechanism will be referred to as the Pay-theMinimum-Necessary (PMN) mechanism: (i) I (?) = I ? (?);
? ? (ii) r ˆ = r? (?) ? ro (?1 (??1 ), ..., ?n (??n ));
(iii) ?i (?) =
? ?? (?) · Xi · ?i (??i ) ; n ? ? j =1 ? (? ) · Xj · ?j (??j )
(iv) Depreciation is calculated according to the relative bene?t rule based on the capital charge rate r? (?);
It is well known that this rule amounts to the annuity depreciation method in case the xit ’s are constant across time periods. On the other hand, if cash ?ows were to decline geometrically at a rate of ? over an in?nite horizon, relative bene?t depreciation would amount to a declining balance method with decline factor 1 ? ?.
13
12
where ?? (?) ? (1 + r? (?))?1 , ..., (1 + r? (?))?T . The PMN mechanism sets the capital charge rate equal to the project’s internal rate of return at the critical pro?tability
? levels ?i (??i ). The asset sharing rule in (iii) allocates the initial investment cost in
proportion to the divisional present values generated by the common asset, evaluated at the critical pro?tability parameters. If the aggregate present value of all divisions other than i approaches b (note that it cannot exceed b by Assumption 1), then division i’s assigned ownership share ?i (?) will tend to be small; whereas it will tend be large when the aggregate valuation of the other divisions is small. These characterizations follow directly upon observing that, by the de?nition of the internal rate of return,
n j =1 ? ?? (?) · Xj · ?j (??j ) = b.
The relative bene?t depreciation rule in (iv) ensures that the cost charge to division i in period t is given by: zit (?) · ?i · b = Using again the fact that
n j =1
xit ? ? (?) ·
Xi
· ?i · b.
? ?? (?) · Xj · ?j (??j ) = b, it follows that the charge to
division i in period t is given by
? zit (?) · ?i (?) · b = xit · ?i (??i ).
(9)
Thus the performance measure under the PMN mechanism reduces to
? ?it (?) = xit · [?i ? ?i (??i )],
(10)
making it a dominant strategy for each manager to report truthfully. Equation (10) shows that this incentive holds not only in the aggregate, i.e., over the entire planning horizon, but on a period-by-period basis. We have demonstrated that the PMN mechanism is satisfactory. The following result shows that within the class of capital budgeting mechanisms we consider there is in fact no other satisfactory mechanism.14
14
All proofs are in Appendix A.
13
Proposition 1 Given Assumption 1, the Pay-the-Minimum-Necessary (PMN) mechanism is the unique satisfactory capital budgeting mechanism for shared assets. To provide intuition for the uniqueness of the PMN mechanism, it is useful ?rst to consider settings in which each manager attaches the same weights to future payo?s as the ?rm’s owners, that is, it is commonly known that uit = ? t · u for each i and t. In such a setting, intertemporal matching is of no importance and it su?ces to ensure dominant strategy incentive compatibility over the entire planning horizon, rather than on a period-by-period basis. As demonstrated in the public choice literature (e.g., Green and La?ont 1979), every dominant-strategy mechanism must be a Groves scheme if the organization is to achieve e?cient outcomes. Under a Groves scheme each participant’s payo? is the same as the social surplus up to a constant which does not depend on the report by that participant. Since manager i already internalizes his own surplus, P Vi (?i ), the transfer payment must be equal to the maximized surplus of the remaining n ? 1 divisions plus some amount which is a constant from i’s perspective. Given the performance measure in (5), the present value of the cost charges to division i must satisfy:
T
zit (?) · ?i (?) · b · ? t = ?
t=1 j =i
P Vj (?j ) ? b
· I ? (?i , ??i ) + hi (??i ),
where hi (·) is an arbitrary function of ??i . In conjunction with Assumption 1, the no-play-no-pay condition requires that this “remainder” term hi (·) must be equal to zero. By construction of the critical types, we ?nd that:
T ? (??i )) · I ? (?i , ??i ). zit (?) · ?i (?) · b · ? t = P Vi (?i t=1
(11)
When expressed in this form, the label “Pay-the-Minimum-Necessary” becomes transparent: given the valuations reported by the other divisions, the present value charge to each division is the minimum amount required in order for the project to break even. 14
Our notion of strong incentive compatibility requires truthful reporting to be a dominant strategy for each manager for an entire neighborhood of ui . As a consequence, the aggregate cost charge in (11) must be annuitized so that
? (??i ) · I ? (?i , ??i ). zit (?) · ?i (?) · b = xit · ?i
(12)
The remaining question is whether there exist other combinations of asset allocation rules, capital charge rates, and depreciation schedules that can yield the same periodic cost charges as those in (12). The answer to this question is provided by the following observation which we state as separate lemma because it will be used again later.15 Lemma 1 Let D = {(d1 , ..., dT ) |
T t=1
dt = 1} denote the set of all depreciation
schedules. The mapping ft : D × (?1, ?) ? RT , given by
t?1
ft (d, r ˆ) = dt + r ˆ· is one-to-one. Its image contains RT + and
1?
? =1 T t=1 (1
d?
(13)
+r ˆ)?t · ft (d, r ˆ) = 1 for all (d,r ˆ).
For a given asset sharing rule, Lemma 1 implies that the intertemporal cost allocation charge in (12) can only be generated by the relative bene?t depreciation rule and the capital charge rate r? (?). Given this observation, it follows directly that the asset shares ?i (?) must be the ones speci?ed under the PMN mechanism. The public choice literature has shown that it is generally impossible to attain e?cient outcomes in dominant strategies while keeping the budget balanced. In contrast, Proposition 1 demonstrates the possibility of an e?cient capital budgeting mechanism by means of conventional cost allocations that are nominally balanced. However, in present value terms investment in the shared assets must be “subsidized” by the central o?ce by way of a reduced capital charge rate. To see this, we note
15
This result generalizes Corollary 3 in Rogerson (1997).
15
that:
? r ? ro (??i , ?i (??i )) ? ? ? ro (?1 (??1 ), ..., ?n (??n ))
= r? (?), where the inequality holds because the internal rate of return ro (?1 , ..., ?n ) is monotone
? increasing in each of its arguments and ?i ? ?i (??i ) for any positive-npv project.16
Corollary 1 Under the PMN mechanism, the capital charge rate, r? (?), is less than the ?rm’s cost of capital, r. An alternative way to attain the same outcomes would be for the central o?ce to hold the capital charge rate at r while lowering the initial capital allocation for each division to ˆ i (??i ) ? ? However, because
n i=1
b?
j =i
P Vj (?j |r) . b
n i=1
ˆ i (??i ) = 1 [n·b?(n?1)· ? b
P Vi (?i )] < 1 whenever I (?) = 1,
this approach violates the basic accrual accounting requirement that the sum of all depreciation charges across time periods and participating divisions be equal to the initial investment outlay. It is instructive to examine how division i’s assigned share of the joint investment, ?i (?), depends on that division’s report. In the special case of identical Xj , the PMN mechanism divides the initial investment cost simply in proportion to the critical pro?tability levels, that is, ?i (?) =
16
? ?i (??i ) . n ? j =1 ?j (??j )
Because of the public goods nature of the shared asset, the capital charge rate is (weakly) o o decreasing in the number of participants. Speci?cally, suppose ?o ? (?1 , ..., ?n ) is the pro?tability ? o pro?le of the existing n divisions and let r (? ) be the capital charge rate for the corresponding PMN mechanism. It is readily veri?ed that if an additional division with pro?tability parameter ?n+1 participates in the asset acquisition decision, then r? (?o , ?n+1 ) < r? (?o ), regardless of the type ?n+1 of the new participant.
16
? Since the critical pro?tability level ?j (??j ) is decreasing in ?i for all i = j , the share
of initial investment cost allocated to division i is increasing in its own pro?tability parameter ?i . This comparative statics continues to hold more generally when the Xj di?er across divisions. Corollary 2 Under the PMN mechanism, the share of investment cost allocated to division i, ?i (?), is increasing in the pro?tability parameter ?i . A necessary feature of any dominant strategy mechanisms for a binary decision is that, contingent on the project being undertaken, the cost charge to any participant must be independent of its own report. Corollary (2) and the capital charge rate
? ? r? (?) = ro (?1 (??1 ), ..., ?n (??n )) may suggest that the PMN mechanism violates this
property. However, while both r? (?) and ?i (?) vary with manager i’s report ?i , the
? fact that zit (?) · ?i (?) · b = xit · ?i (??i ) · I ? (?i , ??i ) shows that these partial e?ects
exactly o?set each other, i.e., the resulting sum of depreciation and capital charges, zit (?) · ?i (?) · b, is indeed independent of ?i . It is instructive to note that the PMN mechanism is not a multiperiod version of the so-called Pivot mechanism. In the public choice literature, it is usually assumed that the cost of the public investment is divided in some arbitrary, say equal, fashion among all participants (e.g., Kreps 1990, Mas-Colell et al. 1995) and in addition the participants make (receive) transfer payments. One of the key features of the Pivot mechanism is that an agent receives a non-zero additional transfer payment only if he is “pivotal” in the sense that his report alters the social decision. While the PMN mechanism also satis?es this feature (in fact, our Assumption 1 implies that all agents are pivotal), the Pivot mechanism does not satisfy our no-play-no-pay condition since agents can be charged even if the project is not undertaken. One of the attractive features of Pivot mechanisms for public choice problems is that they always run a “budget surplus”. In contrast, the PMN mechanism runs a de?cit in real terms in the sense that the sum of all divisional charges, discounted at the ?rm’s cost of capital, r, is less than the initial investment expenditure. 17
To conclude this section, we demonstrate that the PMN mechanism can be extended to settings in which Assumption 1 does not hold. The immediate issue then is
? that the critical pro?tability cuto?s ?i (??i ) may be less than ?i . We extend the PMN
mechanism to environments satisfying the following weaker form of Assumption 1.
? Assumption 2 For all ?, there exists some division i such that ?i (??i ) ? ?i .
Thus, for all type pro?les ?, there must be at least one essential division. Assumption 2 is likely to be satis?ed in settings where there are either one or a few larger divisions (in terms of the magnitude of their P Vi ’s) whose valuation is critical for the project’s overall pro?tability. Given Assumption 2, the PMN mechanism can be
? ? modi?ed by replacing ?i (??i ) with max{?i , ?i (??i )}. As a consequence, non-essential ? agents, for whom ?i (??i ) < ?i , will be assigned an ownership share ?i (?) = 0. While
the modi?ed PMN mechanism is again satisfactory, it is no longer the unique solution among the class of capital budgeting mechanisms we consider. In particular, it is not necessary that non-essential divisions be charged zero. However, uniqueness of this modi?ed PMN mechanism can be restored if one imposes an additional participation constraint akin to Moulin’s (1986) “no-free-ride” condition. Speci?cally, this ex-post participation condition requires that no division be worse o? by participating in the mechanism. As a consequence, non-essential divisions must then be charged zero.
2.2
Exclusive Assets
We now examine the design of capital budgeting mechanisms for a setting in which the divisions face a limited capital budget. For simplicity, we suppose that the capital budget is su?cient to fund at most one of the divisional projects.17 One interpretation of this speci?cation is that, while the ?rm’s capital cost for ?nancing a single project is r, this cost would increase su?ciently fast with additional investments such
It is readily seen that the mechanisms we identify in this subsection can be extended to settings where the ?rm can fund k out of n possible projects. However, such an extension would add signi?cant clutter to the notation without adding any substantive insight.
17
18
that the additional cash returns could not possibly cover the higher cost of capital. An alternative interpretation is that the ?rm has already decided to undertake a particular project, e.g., the ?rm has entered into a procurement contract, yet the capital budgeting process is to determine which of the n divisions could implement this project most pro?tably. Division i’s investment opportunity requires an initial cash outlay of bi . The npv of division i’s investment project is then given by NP Vi (?i ) ? P Vi (?i ) ? bi . As before, the pro?tability parameters ?i are divisional private information. Division
o i’s internal rate of return is denoted by ri (?i ), that is, T o (1 + ri (?i ))?t · xit · ?i = bi . t=1
(14)
The ?rst-best investment decision rule calls for selecting the highest npv project provided that npv is positive; i.e., the corresponding internal rate of return exceeds the ?rm’s cost of capital r. We use the indicator variable Ii ? {0, 1} to represent whether division i’s investment project is undertaken. Given our speci?cation that the ?rm can fund at most one project, a feasible investment policy must satisfy budgeting mechanism for exclusive assets speci?es: • An investment decision rule, Ii : ? ? {0, 1} such that i’s beginning balance equals Ai0 = bi · Ii (?); • A capital charge rate, r ˆ : ? ? (?1, ?); • A depreciation schedule, di : ? ? RT , satisfying and dit (?) ? 0 if Ii (?) = 0.
T t=1 n i=1 Ii (? ) n i=1 Ii
? 1. A capital
? 1. Division
dit (?) = 1 if Ii (?) = 1,
19
Note that this class of mechanisms again imposes a no-play-no-pay condition, since a division can be charged only if its project receives funding.18 In direct analogy to our earlier terminology, a capital budgeting mechanism is said to be satisfactory if (i) it is strongly incentive compatible for each manager, and (ii) it selects the highest positive npv project. It will be notationally convenient to denote the highest positive npv project by NP V 1 (?) ? max {NP Vi (?i ), 0}.
i
E?cient project selection requires that Ii? (?) = 1 only if NP V 1 (?) = NP Vi (?i ). We also de?ne NP V 1 (??i ) = max {NP Vj (?j ), 0}.
j =i ? For a given ??i , let ?i (??i ) denote the lowest value of division i’s pro?tability
parameter for which its project is at least as pro?table as any of the other n ? 1 projects. That is,
? NP Vi (?i (??i )) = NP V 1 (??i ). ? Put di?erently, at ?i = ?i (??i ), division i’s npv would just tie with the highest npv
of the remaining divisions, provided that value is positive.19 Note that this de?nition
? ? of ?i (??i ) implies that Ii? (?) = 1 only if ?i ? ?i (??i ).20
The following capital budgeting mechanism will be referred to as the Competitive Hurdle Rate (CHR) mechanism: (i) Ii (?) = Ii? (?);
o ? (?i (??i )) if Ii? (?) = 1; (ii) r ˆ = r? (?) ? ri
Because division i will be burdened with depreciation and capital charges only if Ii = 1, it is without loss of generality to set a uniform, ?rmwide capital charge rate r ˆ(·). 19 ? While ?i (??i ) depends on all distributional cash ?ow parameters (X1 , ..., Xn ) and on the cash outlay amounts (b1 , ..., bn ), we suppress this dependence for notational ease. 20 ? To rule out uninteresting corner solutions and to ensure the critical pro?tability type ?i (??i ) is ¯i ) = H always well de?ned for all ??i , we assume throughout that, for all i, NP Vi (?i ) = L and NP Vi (? for some H > L.
18
20
(iii) {dit (r? (?))}T t=1 is the relative bene?t depreciation schedule based on the competitive hurdle rate r? (?). The competitive hurdle rate, r? (?), is the internal rate of return of the winning
? division evaluated at the critical pro?tability type, ?i (??i ). Division i’s report does ? not a?ect the competitive hurdle rate provided ?i > ?i (??i ). The CHR mechanism
simpli?es considerably when all divisions are ex-ante identical with regard to Xi and bi . The rank order of the npv’s then is identical to the rank order of the internal rates of return and therefore the competitive hurdle rate for the winning division
? o simply equals the second-highest internal rate of return: ri = maxj =i {rj (?j ), r}. In
this context, it is also readily seen that the CHR mechanism can be viewed as a delegation mechanism: divisions report their internal rates of return and decide on their own whether to proceed with their divisional projects with the capital charge rate set at the second-highest reported internal rate of return. The relative bene?t depreciation schedule in (iii) ensures that the cost charge to division i is given by zit (?) · bi = xit · [
T ? =1 o ? [1 + ri (?i (??i ))]?? · xi? ]?1 · bi . It then
follows from equation (14) that the charge to division i is given by
? zit (?) · bi = xit · ?i (??i )
(15)
? and hence ?it (?) ? ?it (?i , ??i |?i ) = xit · [?i ? ?i (??i )] · Ii (?). Therefore, The CHR
mechanism provides strong incentives for each manager to report his information truthfully. Proposition 2 The Competitive Hurdle Rate (CHR) mechanism is the unique satisfactory capital budgeting mechanism for exclusive assets. To see the uniqueness of the CHR mechanism, consider again the case where divisional managers receive a constant share of residual income in each period as compensation and they discount future payo?s at the ?rm’s cost of capital r. Since any satisfactory dominant-strategy mechanism must then be a Groves mechanism, 21
we have:
T
? t · zit (?) · bi = ?
t=1 j =i
? NP Vj (?j ) · Ij (?) + hi (??i ),
(16)
where hi (·) is an arbitrary function of ??i . The total surplus of the remaining n ? 1 divisions, given by the ?rst term on the right hand side of (16), is equal to zero when Ii (?i ) = 1, and equal to NP V 1 (??i ) when Ii? (?) = 0. Equation (16) can therefore be written as
T
? t · zit (?) · bi = ?NP V 1 (??i ) · [1 ? Ii? (?)] + hi (??i ).
t=1
The no-play-no-pay condition immediately implies that hi (??i ) = NP V 1 (??i ) and therefore:
T T
? · ?it = [NP Vi (?i ) ? NP V
t=1
t
1
(??i )] · Ii? (?i , ??i )
=
t=1
? ? t · xit · [?i ? ?i (??i )] · Ii? (?i , ??i ).
(17)
Equation (17) shows that when all divisions discount future payo?s at the principal’s cost of capital, r, a satisfactory capital budgeting mechanism for exclusive assets must take the form of a second-price auction in which the present value of all charges to the winning division is the second highest npv. Strong incentive compatibility requires that the managerial performance measure re?ects the divisional npv not only over the entire planning horizon but also on a period-by-period basis. As a consequence, the aggregate cost charge must be annuitized as in equation (15). Uniqueness of the CHR mechanism now follows from the fact that the competitive hurdle rate combined with the relative bene?t depreciation implement the cost charges in (15) and, by Lemma 1, there can be only one such solution. The CHR mechanism is a multiperiod version of the Pivot mechanism. It is well known from the public choice literature that Pivot mechanisms always attain a budget surplus in the sense that the sum of all monetary transfers to the agents, in addition to cost sharing payments, is negative. Accordingly, we obtain the following result. Corollary 3 The hurdle rate under the CHR mechanism, r? (?), exceeds the ?rm’s cost of capital, r. 22
We note that Propositions 1 and 2 di?er markedly in their prescriptions regarding the capital charge rates. In order to obtain strong incentives within the class of accounting based mechanisms we consider, the capital charge rate for shared assets must be lower than the owner’s cost of capital (Corollary 1), while the reverse conclusion is obtained for exclusive assets.21 While the usual comprehensive income measurement conditions
T t=1
dit = 1 and Ai0 = bi · Ii hold, the investment expenditure is less than
the present value of the cost charges to the winning division, discounted at the ?rm’s cost of capital, r:
T T
(1 + r)?t · zit (?) · bi >
t=1 t=1
(1 + r? )?t · zit (?) · bi = bi .
If one were to impose the requirement that the capital charge rate be equal to the ?rm’s cost of capital, i.e., r ˆ(?) ? r, incentive compatibility would require a departure from comprehensive income measurement. Speci?cally, the same incentives could
? be generated by capitalizing the initial amount Ai0 = P Vi (?i (??i )) for the winning
division, yet this amount would exceed the actual investment expenditure bi .
3
3.1
Hidden Actions and Incentive Contracting
Second-Best Mechanisms
Our analysis has so far taken as given that divisional managers seek to maximize their performance measures and that the objective of the central o?ce is to maximize the ?rmwide npv by choice of goal congruent performance measures. It is natural to ask which properties of goal congruent mechanisms carry over to second-best mechanisms once we include an explicit agency problem and the desired incentives are derived from
Corollary 3 seems at odds with the results in Bareket and Mohnen (2006) who also consider the problem of picking one among several mutually exclusive projects, yet in their model the hurdle rate can be kept at r. The essential di?erence between the models is that we consider multiple divisions competing for funds, whereas in Bareket and Mohnen (2006) a single division can have only one of several projects approved. Dougart (2005) studies decision problems in which projects impose externalities on other divisions and argues that in such settings goal congruent performance measures require depreciation rules that di?er from the relative bene?t rule.
21
23
a uni?ed optimization program. Speci?cally, suppose divisional operating cash ?ow in period t is now given by cit = ait + xit · ?i · Ii , where ait ? [0, a ¯it ] denotes productive e?ort chosen privately by manager i in period t and Ii ? {0, 1} indicates generically (for shared and exclusive assets) whether division i has access to the asset. Manager i observes ?i before contracting. The central o?ce and all other managers share the same beliefs about ?i given by the cumulative distribution, Fi (?i ), with strictly positive density, fi (?i ), over the entire support ?i .22 We assume the usual monotone inverse hazard rate condition; i.e., Hi (?i ) ? [1 ? Fi (?i )]/fi (?i ) is decreasing in ?i for all i. While the central o?ce can observe the divisional operating cash ?ows in each period, it is unable to disentangle the investment-related from the e?ort-related components. Manager i’s date-0 utility payo? is given by
T
Ui =
t=1
? t · [sit ? vit (ait )],
where sit denotes his compensation in period t and vit (·) is his disutility from exerting e?ort ait in period t. The function vit is increasing and convex with vit (0) = 0, for all i and t. Given this structure, it is only the present value of compensation payments that matters to each manager, provided all parties can commit to a T -period contract. ˜) ? In our setting, a revelation mechanism speci?es an investment decision rule Ii (? ˜) ? (ci1 (? ˜), . . . , ciT (? ˜)) to be delivered by each division, {0, 1}, “target cash ?ows” ci (? ˜) ? (si1 (? ˜), . . . , siT (? ˜)), contingent on and managerial compensation payments si (? ˜. For any such mechanism, let Ui (? ˜i , ? ˜?i |?i ) denote manager i’s utility the reports ? ˜?i submitted by the contingent on his own true pro?tability parameter ?i , reports ? ˜i . Given truthful reporting on the part of the other managers, and his own report ?
Alternatively, the managers may learn their ?i -parameters after entering into the contract, but they cannot be prevented from quitting the job if their participation constraints are not satis?ed.
22
24
other managers, this yields
T
˜i , ??i | ?i ) ? Ui (?
t=1
˜i , ??i ) ? vit (ait (? ˜i , ??i | ?i ))], ? t · [sit (?
where ˜i , ??i | ?i ) ? min ait ait + xit · ?i · Ii (? ˜i , ??i ) ? cit (? ˜i , ??i ) ait (? is the minimum e?ort that manager i has to exert so as to achieve the required ˜i , ??i ) in each period. periodic cash ?ow target cit (? By the Revelation Principle we may restrict attention to mechanisms which induce managers to reveal their information truthfully. The central o?ce’s optimization problem can then be stated as follows:
n T
P:
(ci (? ),si (? ),Ii (? ))n i=1
max
E?
i=1 t=1
? t · [cit (?) ? sit (?)] ? B (?)
subject to: (ie ) for exclusive assets:
n i=1 Ii (? )
? 1 and B (?) =
n i=1 bi
· Ii (?),
(is ) for shared assets: Ii (?) = Ij (?) = I (?), for all i, j , and B (?) = b · I (?), ˜i , ??i | ?i )], for all ?i , ? ˜i and i, (ii) E??i [Ui (?i , ??i | ?i )] ? E??i [Ui (? (iii) E??i [Ui (?i , ??i | ?i )] ? 0, for all ?i and i. Constraints (ie ) and (is ) ensure feasibility of the investment rule for exclusive and shared assets, respectively, and specify the resulting initial investment amounts. The incentive compatibility constraints (ii) require that truthful reporting constitute a Bayesian-Nash equilibrium. The participation constraints (iii) are required to hold on an interim basis, i.e., each manager must break even in expectation over
25
the other managers’ possible types. We denote the solution to this program by
? ? n (c? i (? ), si (? ), Ii (? ))i=1 and refer to it as the second-best solution. ˜i < ?i and at the same time reduce his e?ort whenever A manager can underreport ?
˜i , ??i ) = 1. Therefore managers will earn informational rents on account of their Ii (? private information. The basic tradeo? for the central o?ce is that manager i’s information rents will be increasing both in the induced e?ort levels, (ait , ..., aiT ), as well as in the set of states in which division i has access to the asset. Applying standard arguments from the adverse selection literature based on “local” incentive constraints (e.g., La?ont and Tirole 1993), it can be shown that manager i’s interim informational rent for any ?i (i.e., in expectation over other managers’ types) equals
?i T
E??i [U (?i , ??i | ?i )] = E??i
?i
? t · vit (ait (qi , ??i | qi )) · xit · Ii (qi , ??i ) dqi .
t=1
(18)
The expression in (18) illustrates the above tradeo?: manager i’s informational rent can be reduced either by creating lower powered e?ort incentives (there will be no rent if ait = 0) or by curtailing the set of states ? in which Ii (?) = 1. To simplify the exposition and sharpen our predictions regarding the second-best investment decision rule, we focus on a setting in which the central o?ce always ?nds it worthwhile to induce maximum e?ort in each period, i.e., a? it (?i , ??i | ?i ) ? a ¯it . Inducing the maximum level of e?ort will indeed be optimal provided vit (¯ ait ) is su?ciently small relative to the other parameters of the model. For brevity, we denote vit ? vit (¯ ait ). As shown in Appendix B, the following condition ensures that the central o?ce will indeed seek to induce the maximum level of e?ort in each period: ait ) · xit · Hi (?i ) ? 0, 1 ? vit ? vit (¯ (19)
for each i and t. Condition (19) says that even if division i has access to the asset for all ?i ? ?i , the marginal return from e?ort, which has been normalized to one, is su?ciently large for the central o?ce to prefer high e?ort despite the corresponding increase in expected informational rents. We will discuss below how our results would be a?ected if (19) were relaxed so that interior levels of e?ort would be optimal. 26
We use the expression in (18), evaluated at ait = a ¯it , to solve for each manager’s compensation payments sit (?) in P . The central o?ce’s optimization problem then simpli?es to the following program (see Appendix B for a detailed derivation):
n T
P
:
(Ii (? ))n i=1
max E?
i=1 t=1
? t · [¯ ait ? vit (¯ ait )] + [P Vi (?i ) ? ?i · Hi (?i )] · Ii (?) ? B (?) ,
subject to (ie ), (is ), where ?i ?
T t=1
? t · vit · xit . The reduced objective function in P re?ects that the
expected value of manager i’s interim informational rents (i.e., E? [U (?i , ??i | ?i )]) is equal to the expected value of ?i · Hi (?i ) · Ii (?). To characterize the second-best investment decision rule, it will be useful to de?ne ?i (?i ) ? P Vi (?i ) ? ?i · Hi (?i ) (20)
as the present value of division i’s virtual cash ?ows (i.e., the present value of cash ?ows net of the manager’s informational rents). The second-best investment rule for exclusive assets then is given by Ii (?) = 1 if and only if VNP Vi (?i ) ? max{VNP Vj (?j ), 0},
j
(21)
n i=1
for VNP Vi (?i ) ? ?i (?i )?bi as the virtual divisional npv. Let VNP V (?) ? assets requires VNP V (?) ? 0.
?i (?i )?
b denote the virtual corporate npv, so that the second-best investment rule for shared (22)
We now demonstrate that the satisfactory mechanisms identi?ed in Section 2 can be adapted to generate optimal incentives in the presence of agency problems. A capital budgeting mechanism is said to be optimal if and only if there exist linear compensation schemes ˜) = ?it (? ˜) + ?it (? ˜) · ?it (·)}, {sit (?it | ? (23)
for all 1 ? t ? T and 1 ? i ? n, that achieve the same payo? for the principal as the second-best mechanism identi?ed above. 27
3.2
Optimal Mechanisms: Exclusive Assets
The second-best investment rule calls for funding a project if and only if its virtual npv exceeds both zero and the virtual npvs of all other projects. That is, Ii (?) = 1, if and only if VNP Vi (?i ) = VNP V 1 (?), where VNP V 1 (?) ? max1?j ?n {VNP Vj (?j ), 0}. Accordingly, we now denote the agency-adjusted pro?tability cuto? for division i by
?? ?i (??i ) which is implicitly de?ned by23 ?? VNP Vi (?i (??i )) = VNP V 1 (??i ). ?? The corresponding agency-adjusted competitive hurdle rate of division i, ri (?), is
de?ned to be the internal rate of return of its agency-adjusted critical project:
?? o ?? ri (?) ? ri (?i (??i )).
(24)
Suppose now that each manager is o?ered a linear compensation scheme of the form in (23) with bonus coe?cients ?it = vit . If the performance measure is based on the competitive hurdle rate mechanism (i.e., the capital charge rate is equal to the agency-adjusted competitive hurdle rate and the asset valuation is based on the relative bene?t depreciation rule), then ˜i , ??i | ?i ) = v · Ii (? ˜i , ??i ) · xit · [?i ? ??? (??i )]. ?it (? it i (25)
Each manager has dominant-strategy incentives to report his information truthfully because a project makes a positive contribution to his performance measure if
?? and only if ?i > ?i (??i ). Our next result shows that this mechanism is indeed op-
timal. Furthermore, even though the optimization program in P is stated in terms of Bayesian-Nash incentive compatibility and interim participation constraints, the central o?ce obtains dominant strategy incentives and ex-post satisfaction of the participation constraints for “free.” This characterization applies to a broader class of
To ensure that well-de?ned pro?tability cuto?s exist, we again assume that, for all i, ¯i ) = H for some H > L. VNP Vi (?i ) = L and VNP Vi (?
23
28
mechanisms (Mookherjee and Reichelstein 1992).24 Proposition 3 The Competitive Hurdle Rate (CHR) mechanism based on the agency?? adjusted capital charge rate r ˆ = ri (?) is an optimal mechanism.
Equation (25) shows that the present value of the intertemporal cost charges for the winning division is equal to the second highest virtual npv. The CHR mechanism can therefore again be interpreted as a multiperiod version of the second-price auction mechanism. If managers are ex-ante identical with regard to Fi (·) ? F (·), ?i = ?, bi = b and Xi = X , it will su?ce to ask each manager to report his internal rate of return ri . Denoting by rc the internal rate of return of a project whose virtual npv is zero, it is then optimal to set the hurdle rate for the winning division equal to
?? ri (r1 , · · · , rn ) = maxj =i {rj , rc }. In general, however, the rank order of the divisional
internal rates of return need not agree with the rank order of the virtual npvs. The case of ex-ante identical managers also illustrates the impact of an exogenous change in the number of divisions competing for the scarce capital. The principal will bene?t from increased competition among divisions in two ways. First, the informational rents of the winning manager will be reduced. For instance, with exante identical agents, the winner’s informational rent is given by Ui (?i , ??i | ?i ) =
?? ? · [?i ? ?i (??i )] = ? · [?i ? maxj =i {?j , ?c }], where ?c denotes the agency adjusted
break-even pro?tability in a single-agent setting (i.e., rc ? ro (?c )). Clearly, the winning agent’s informational rents are decreasing in n, in expectation. Second, the underinvestment problem will be mitigated. To see this, note that the ex-ante probability of underinvestment is equal to [F (?c ) ? F (?o )]n , where ?o denotes the break-even pro?tability (i.e., NP V (?o ) = 0).
Unlike Proposition 2, Proposition 3 only speaks to the su?ciency, and not necessity, of the CHR mechanism. Given that the parties can commit to the contract and they have the same intertemporal preferences, there is considerable indeterminacy in distributing the divisional cost charges across di?erent periods. Additional robustness criteria, like those proposed in Dutta and Reichelstein (2002) or Arya, Demski, Glover and Liang (2005), may narrow the class of “feasible” mechanisms.
24
29
A natural question is how the agency-adjusted hurdle rate under the CHR mechanism, r?? (?), compares with the hurdle rate identi?ed in Proposition 2 in the absence of an explicit agency problem. In single-period capital budgeting problems, such as the one posed in Antle and Eppen (1985), the hurdle rate always increases as the agency problem becomes more severe: the virtual npv of the project decreases while the opportunity cost of investing is given by the ?rm’s cost of capital r, and therefore una?ected. We ?nd that this prediction does not generalize straightforwardly to our multi-divisional setting. To illustrate, let n = 2.25 The parameters (?1 , ?2 ) measure the relative severity of the two agency problems. Denoting by ?i (?i | ?i ) the present value of virtual cash ?ows as a function of ?i , division 1’s critical type is given by:
?? ?1 (?1 (?2 | ?1 , ?2 ) | ?1 ) ? b1 ? ?2 (?2 | ?2 ) ? b2 ,
(26)
?? o ?? and the resulting hurdle rate equals r1 (? | ?1 , ?2 ) = r1 (?1 (?2 | ?1 , ?2 )). We note
from (20) that ?i (?i | ?i ) is increasing in ?i and decreasing in ?i . In the case of ex-ante identical agents, i.e., Fi (·) ? F (·), ?i = ?, bi = b, Xi = X, for i = 1, 2, the condition characterizing division 1’s cuto? pro?tability type in (26)
?? reduces to ?1 (?2 | ?, ?) ? ?2 . To understand the irrelevance of the agency parameter
? for computing hurdle rates with ex-ante identical agents, note that an increase in ? reduces the virtual npv of division 1’s project, but at the same time it also reduces the opportunity cost (i.e., the virtual npv of division 2’s project) by the same amount. As a consequence, hurdle rates are una?ected. Generalizing this insight, the natural question is whether the hurdle rate can be viewed as an instrument for “handicapping” certain divisions.26 Consider an exogenous increase in ?2 , i.e., manager 2’s agency problem becomes more severe, ceteris paribus. As a result, the right-hand side of (26) decreases. To restore the identity,
?? (?2 | ?1 , ?2 ) must go down and so does the competitive manager 1’s cuto? type ?1 ?? (·). The reverse prediction emerges for an increase in ?1 since now hurdle rate r1
The following discussion can be extended in a straightforward fashion to arbitrary n. The auctions literature has also explored the di?erential treatment of agents based on di?erences in their ex-ante characteristics; see Krishna (2002).
26
25
30
division 1’s virtual npv decreases, which results in a higher pro?tability cuto?. We summarize these observations in the following corollary.
?? Corollary 4 For n = 2, the agency-adjusted competitive hurdle rate ri (? | ?1 , ?2 )
under the CHR mechanism is increasing in ?i , but decreasing in ?j , j = i. The principal thus calibrates the CHR mechanism so as to handicap a division whose agency problem is more severe. In particular, suppose the two projects are ex-ante identical but ?1 > ?2 so that the ?rm faces higher agency costs with division 1. Corollary 4 then implies that division 1 will face a higher hurdle rate than division 2. This ?nding is related to the corporate ?nance literature on investments in diversi?ed ?rms (e.g., Shin and Stulz 1998).27 That literature has pointed to “corporate socialism” insofar as stronger divisions get allocated relatively less capital than weaker divisions, as a result of information asymmetry,. This ?nding has been explained by managers exerting in?uence activities (Rajan et al. 2000). Our results generate similar empirical predictions, though in our model handicapping “stronger” divisions actually improves overall e?ciency. Consider two divisions that are identical in terms of ?i = ?, bi = b, and Xi = X, for i = 1, 2, but their pro?tability types ?i are drawn from di?erent distributions Fi over identical supports ?i = ?. In particular, suppose that division 1’s type distribution is more favorable in the sense of (inverse) hazard rate dominance: H1 (y ) > H2 (y ) for all y . Applying similar arguments as in Corollary 4, it can be shown that, all else equal, division 1 will be charged more for its allocated capital. Since division 1’s project is more pro?table ex ante than that of division 2, rent extraction weighs more heavily for division 1’s manager with the consequence of a higher hurdle rate. To conclude this section, we demonstrate that the CHR mechanism remains optimal for settings in which the second-best solution entails interior e?ort choices. When condition (19) does not hold, the optimal e?ort choice a? it (? | ?i ) will satisfy
27
We thank an anonymous referee for pointing out this connection.
31
the following ?rst-order condition:
? 1 ? vit (a? it (? | ?i )) ? vit (ait (? | ?i )) · xi · Hi (?i ) · Ii (? ) ? 0,
(27)
such that, by complementary slackness, this inequality will hold as an equality when a? ¯it .28 Condition (27) shows that the central o?ce may ?nd it optimal it (? | ?i ) < a to create lowered-powered e?ort incentives in order to economize on the winning division’s informational rents. Furthermore, the winning division’s e?ort incentives will increase in its reported productivity type. The reason is that higher-powered incentives for unproductive types result in higher information rents for more favorable type realizations.29 We note, however, that the second-best investment rule is again given by (21) with VN P Vi (?i ) ? P Vi (?i )?bi ?
T t=1
? t ·vit (a? it (? | ?i ))·xit ·Hi (?i )·Ii (? ). If
the winning division’s periodic bonus coe?cients are chosen as ?it (?) = vit (a? it (? | ?i )) and the performance measure is based on the agency-adjusted competitive hurdle rate, a linear compensation scheme of the form in (23) can achieve the same payo?s for the principal as the direct revelation mechanism. Thus, the CHR mechanism remains optimal even when we allow for interior e?ort choices.30
3.3
Optimal Mechanisms: Shared Assets
For shared assets, we recall from (22) that the second-best investment decision calls for the joint asset to be acquired if and only if the ?rmwide virtual npv is positive. That is, I (?) = 1, if and only if
n
VNP V (?) ?
i=1
28
?i (?i ) ? b ? 0.
See Appendix B in Dutta and Reichelstein (2002) for a complete derivation in a single agent setting. 29 A qualitatively similar outcome would obtain if e?ort and investment productivity were complements, such that cit = ait · xit · ?i · Ii . As demonstrated in Baldenius and Reichelstein (2005), the second-best bonus coe?cients for the winning manager will be an increasing function of his reported type ?i , provided the periodic cash ?ows are multiplicatively separable in ?i and ait . 30 ˜, whereas the ?xed When condition (19) holds, only the performance measures ?i (·) depend on ? ˜) ? ?i and ?i (? ˜) ? ?i . salaries and bonus coe?cients are independent of the reported types: ?i (? This no longer holds once we allow for interior e?ort choices because condition (19) is violated.
32
? In analogy with our de?nition of ?i (??i ), we now de?ne the agency-adjusted critical ?? values by ?i (??i ) by ?? ?i (?i (??i )) ? b ? j =i ?? ? By construction, ?i (??i ) > ?i (??i ). In the presence of agency costs, the capital
?j (?j ).
(28)
charge rate under the PMN mechanism is given by
?? ?? r ˆ(?) = r?? (?) ? ro (?1 (??1 ), ..., ?n (??n )),
(29)
and the divisional asset shares are calculated according to: ?i (?) =
?? (??i ) ??? (?) · Xi · ?i , n ?? ?? i=j ? (? ) · Xj · ?j (??j )
(30)
with ??? (?) ? ((1 + r?? (?))?1 , ..., (1 + r?? (?))?T ) If divisional managers are compensated on the basis of linear schemes of the form in (23) with bonus coe?cients ?it = vit , they have incentives to provide the targeted amount of e?ort in each period. Furthermore, the PMN mechanism results in the following contribution to manager i’s compensation in period t: ˜i , ??i | ?i ) = ?it · Ii (? ˜i , ??i ) · xit · [?i ? ??? (??i )]. ?it · ?it (? i (31)
As a consequence, the joint project leaves manager i better o? if and only if ?i >
?? ?i (??i ), i.e., VNP V (?) ? 0. Truthful reporting therefore again is a dominant-strategy
equilibrium and we obtain the following:31 Proposition 4 Given Assumption 1, the Pay-the-Minimum-Necessary (PMN) mechanism, based on the agency-adjusted capital charge rate of r?? (?) in (29) and the asset sharing rule in (30), is optimal.
Assumption 1 was introduced in Section 2.1 to ensure that the pro?tability cuto?s are in the ?? interior of the managers’ type supports. Since ?i (??i ) is higher for any positive ?i than for ?i = 0 (i.e., absent a moral hazard problem), Assumption 1 ensures interior cuto?s also in the presence of agency problems.
31
33
In contrast to the exclusive asset scenario, the cuto? condition in (28) implies an unambiguous comparative statics result on how the relative severity of the agency problem a?ects the capital charge rate.32 To this end, we again express the hurdle rate r?? (? | ?1 , ..., ?n ) as a function of the agency cost parameters ?i . Corollary 5 The agency-adjusted hurdle rate r?? (? | ?1 , ..., ?n ) under the PMN mechanism is increasing in ?i for all i. As the agency problem for any division becomes more severe, its contribution ?i (·) to the total virtual npv declines. This in turn has a negative externality on the other divisions re?ected in a higher capital charge rate. It is straightforward to show that for su?ciently high agency costs (high ?i ’s), the resulting agency-adjusted hurdle rate, r?? (?), will exceeds the ?rm’s cost of capital r. Conversely, by continuity our result in Corollary 1 shows that the hurdle rate r?? (?) will be below r for su?ciently low levels of agency costs.
4
Conclusion
Interdependencies between divisions are ubiquitous when ?rms make capital budgeting decisions. The framework developed in this paper allows for positive externalities (shared assets) or negative externalities (exclusive assets) among the divisions. Our analysis has illustrated commonalities between these two scenarios as well as distinct di?erences, in particular with regard to the emerging hurdle rates. We found that capital charge rates without managerial moral hazard tend to exceed the cost of capital r for exclusive assets, while the reverse holds for shared assets. As incentive problems become more severe, the hurdle rate will go up unambiguously for shared assets, but not necessarily so for exclusive assets. Our analysis generates a rich set of predictions regarding the cross-sectional variation in hurdle rates, which may prove useful in future empirical research in this area.
Based on the similar arguments as in the exclusive asset case, it can be easily shown that the PMN mechanism remains optimal even when we allow for interior e?ort choices.
32
34
For the exclusive asset setting, our analysis has taken it as given that the ?rm faces a ?xed investment budget which forces capital rationing. While this approach appears descriptive of actual practice in many ?rms, it would be instructive to examine a more continuous problem in which a ?rm faces the tradeo? that more capital spending results in a higher cost of capital for all funded projects. Such an approach seems particularly appealing in settings where the decisions regarding divisional projects are not binary, but instead the scale of the project is also a choice variable. Most long-term investment projects involve multiple rounds of funding. In such settings, it is likely that the participating divisions will receive new information at intermediate dates. Recent research has begun to analyze the dynamics of both “abandonment” and “growth” options for single-agent problems.33 The introduction of multiple project milestones leads naturally to additional cost allocation issues, such as full cost versus successful e?orts accounting. Sequential investment problems also raise the question of how hurdle rates evolve over time. It would be desirable to develop a theory that predicts how abandonment or growth options should be managed for common assets that bene?t multiple divisions within a ?rm. Another promising avenue for future research is to relax our assumption of risk neutrality. As mentioned in the Introduction, earlier work for single-agent models has investigated how capital charge rates should be set for risk averse managers who face projects; see Christensen et al. (2002) and Dutta and Reichelstein (2002). In multi-divisional ?rms aggregate ?rm risk also becomes a coordination issue. To that end, Stoughton and Zechner (2004) explore negative externalities across divisions as higher aggregate risk may necessitate a higher proportion of relatively expensive equity (rather than debt) capital. In contrast, Homburg and Scherpereel (2004) emphasize positive externalities across divisions due to possible diversi?cation e?ects. Either way, incremental risk must be charged for as part of the capital budgeting process.
33
See, for instance, Arya and Glover (2001), Friedl (2003) and Pfei?er and Schneider (2006)
35
Appendix A: Proofs
Proof of Proposition 1 We ?rst prove that the PMN mechanism is a satisfactory mechanism. Under the PMN mechanism, division i’s performance measure in period t is given by: ˜i , ??i | ?i ) = [xit · ?i ? zit (? ˜i , ??i ) · ?i (? ˜i , ??i ) · b] · I ? (? ˜i , ??i ) ?it (? b ˜i , ??i ) · I ? (? ˜i , ??i ) = xit · ?i ? · ?i (? ? ˜ ? (?i , ??i ) · Xi ˜i , ??i ) = Since ?i (? takes the form: ˜i , ??i | ?i ) = xit · [?i ? ?? (??i )] · I ? (? ˜i , ??i ) ?it (? i (32)
˜i ,??i )·Xi ·?? (??i ) ?? (? i Pn ? ? ˜ j =1 ? (?i ,??i )·Xj ·?j (??j )
and the denominator is just equal to b, by the
de?nition of the internal rate of return, manager i’s performance measure in period t
From (32), we note that in each period, manager i’s payo? depends on his own report ˜i only through its impact on the decision rule I ? (·). Furthermore, (32) shows that ?
? each manager has strong incentives to induce I ? = 1 when ?i ? ?i (??i ), and I ? = 0 ? when ?i < ?i (??i ). Consequently, the PMN mechanism satis?es our requirement of
strong incentive compatibility ˜i , ??i | ?i ), ?it (?i , ??i | ?i ) ? ?it (? ˜i , and ??i . for all i, t, ?i , ? We now prove the necessity part, i.e., the PMN mechanism is the unique satisfactory mechanism in the class described in (5). We ?rst show that for, a given asset allocation rule {?i }n i=1 , any strongly incentive compatible mechanism must have the
? (??i ) property that for all ?i > ?i ? (??i ) xit · ?i (33) ?i · b If ui can vary in some open neighborhood, strong incentive compatibility implies ? ˜i > ?? (??i ): (??i ) and ? directly that for any ?i > ?i i
zit (?i , ??i ) =
˜i , ??i ) = zit (?i , ??i ). zit (? 36
Suppose now that for some period t,
? xit · ?i (??i ) + ?t (??i ). ?i · b ˜i = ?? (??i ) + i , incentive compatibility If division i’s pro?tability parameter is ? i ? zit (?i (??i ), ??i ) =
requires that:
T
uit · [ i · xit ? ?t (??i ) · ?i · b] ? 0.
t=1
˜i = ?? (??i ) ? Conversely, for ? i
T
i
incentive compatibility requires:
uit · [? i · xit ? ?t (??i ) · ?i · b] ? 0.
t=1
Since both of the above inequalities have to hold for any
i
> 0 and for all ui in some
open neighborhood, (33) must hold. Finally, by Lemma 1 there exist a unique capital charge rate r ˆ and a unique depreciation schedule, d, implementing the intertemporal cost allocation
? ?i ?? (??i ) (??i ) , ..., xiT · i . ?i (?) · b ?i (?) · b It remains to demonstrate the uniqueness of the asset sharing rule, (?i (?), ..., ?n (?)).
zi (?) =
xi1 ·
Relative bene?t depreciation ensures that
? xit · ?i (??i ) = zit (?) · ?i (?) · b xit · ?i (?) · b. = ? ? (?) · Xi ? ?? (?) · Xi · ?i (??i ) ?? ?i (?) = b ? ? ? (?) · Xi · ?i (??i ) = . n ? ? j =1 ? (? ) · Xj · ?j (??j )
Proof of Lemma 1 For any given z ? (z1 , · · · , zT ) ? RT + , the mapping
t?1
ft (d, r ˆ) ? dt + r ˆ· 37
1?
? =1
d?
= zt
(34)
de?nes a set of T non-linear equations in (d, r ˆ). We show that the above system of equations has a unique solution in D × (?1, ?) for each z ? RT + . Solving the ?rst T ? 1 equations in (34) recursively for (d1 , · · · , dT ?1 ), we get dt = zt + r ˆ · zt?1 + r ˆ(1 + r ˆ)zt?2 + ... + r ˆ(1 + r ˆ)t?2 z1 ? r ˆ(1 + r ˆ)t?1 , (35)
for t = 1, ..., T ? 1. Substituting this solution into the last component of equation (34) gives zT = (1+ˆ r)· (1 + r ˆ)T ?1 ? (1 + r ˆ)T ?2 · z1 ? (1 + r ˆ)T ?3 · z2 ? ... ? (1 + r ˆ) · zT ?2 ? zT ?1 . Multiplying both sides by ? ˆ T ? (1 + r ˆ)?T and simplifying, this last equation reduces to:
T
? ˆ t · zt = 1 .
t=1
(36)
T t=1
For any z ? (z1 , · · · , zT ) ? RT + , the polynomial
? ˆ t · zt is increasing in ? ˆ and
therefore (36) will be satis?ed for a unique r ˆ. In conjunction with (35) this shows that there is a unique solution to the system of T equations.
Proof of Corollary 2 Since
n j =1 ? ?? (?) · Xj · ?j (??j ) = b, the asset sharing rule in the PMN mechanism
? (? ) ?? (? )·Xi ·?i ?i . b
simpli?es to ?i (?) =
To prove the result, therefore, it su?ces to show
that r? (?) is decreasing in ?i . This follows because
? ? r? (?) = ro (?1 (??1 ), · · · , ?n (??n )), ? (??j ) is decreasing in ?i for all ro (·) is increasing in each of its arguments, and ?j
i = j.
38
Proof of Proposition 2. We again begin with the su?ciency part. Under the competitive hurdle rate mechanism, period t performance measure is given by: ˜i , ??i | ?i ) = ?it (? xit · ?i ? xit
T ? =1 [1
+
o ? ri (?i (??i ))]??
· xi?
· bi
˜i , ??i ) · Ii? (? (37)
? ˜i , ??i ), = xit · [?i ? ?i (??i )] · Ii? (?
It follows from (37) that each manager has strong incentives to induce Ii? = 1 when
? ? ?i ? ?i (??i ), and Ii? = 0 when ?i < ?i (??i ). This proves the su?ciency part of
Proposition (2). To prove the necessity part, we note that manager i’s performance measure in period t is given by ?( ?i , ??i ) = [xit · ?i ? zit (?i , ??i ) · bi ]. · Ii (?i , ??i ) Using arguments parallel to the ones in the proof of Proposition 1, we can show that zit (?i , ??i ) =
xit bi ? ? · ?i (??i ) for any ?i > ?i (??i ). By Lemma 1, there exists a
unique depreciation schedule and a unique capital charge rate that implements the intertemporal cost allocation scheme: zi (?) = xiT ? xi1 ? · ?i (??i ), ..., · ?i (??i ) . bi bi
As argued in the text, one solution is provided by the capital charge rate r ˆ = r? (?) combined with the relative bene?t depreciation rule and this must therefore be the unique solution.
Proof of Proposition 3 ˜i , ??i | ?i ) denote type For a given pro?le of types other than division i, ??i , let Uit (? ˜i . Therefore, the manager’s ?i manager’s utility payo? in period t when he reports ?
39
˜i , ??i | ?i ) ? total utility payo?s are given by Ui (? competitive hurdle rate mechanism,
T t=1
˜i , ??i | ?i ). Under the ? t · Uit (?
˜i , ??i | ?i ) = ?(? ˜i , ??i ) + ? (? ˜i , ??i ) · [ait + I ? (? ˜i , ??i ) · (xit · ?i ? zit (? ˜i , ??i ))] ? vit (ait ), Uit (? i where Ii? (·, ·) denotes the optimal investment decision rule and zit (·, ·) denotes the sum of depreciation and interest charges, as de?ned in (6). For each division, the central o?ce chooses the compensation parameters {?it , ?it }T t=1 ˜ ˜ ˜ such that ?it (?) = v and ?it (?) = vit (¯ ait ) ? v · a ¯it for any report pro?le ?. This
it it
choice of bonus coe?cients ensures that ait = a ¯it for each i and t. Furthermore, the relative bene?t depreciation schedule corresponding to the agency-adjusted compet˜ ??i ) = xit · ??? (??i ). As a consequence, itive hurdle rate r?? (??i ) implies that zit (?,
i i
manager i’s total utility payo? becomes: ˜i , ??i | ?i ) = ?i · I ? (? ˜i , ??i ) · [?i ? ??? (??i )] Ui (? i i where ?i ?
T t=1
(38)
? t · vit · xit .
It is clear from (38) that manager i’s participation constraint holds for each ??i , and each manager has a strong dominant strategy incentive to report his information truthfully. Furthermore, a comparison with (40) reveals that manager i’s interim utility payo?s E??i [Ui (?i , ??i | ?i )] from (38) coincide with those of the optimal revelation mechanism.
Proof of Corollary 4
?? (??i |?1 , ?2 ) is de?ned For given ?1 and ?2 , division i’s cuto? pro?tability parameter ?i
by the equation:
?? VNP Vi (?i (??i | ?1 , ?2 )) = max{VNP Vj (?j | ?j ), 0}, j = i.
Since the virtual npv is given by VNP Vi (?i | ?i ) = NP Vi (?i ) ? ?i · Hi (?i ), it follows that VNP Vi (?i | ?i ) is uniformly decreasing in ?i . This immediately implies that 40
?? ?i (??i | ·) is uniformly increasing in ?i and uniformly decreasing in ?j , j = i. The ?? result follows since the agency-adjusted competitive hurdle rate ri (? | ?1 , ?2 ) is equal o ?? o to ri (?i (??i | ?1 , ?2 )), and ri (·) is an increasing function.
Proof of Proposition 4 Under the PMN mechanism, manager i’s utility payo?s in period t are given by: ˜i , ??i | ?i ) = ?(? ˜i , ??i ) + ? (? ˜i , ??i ) · [ait + I ? (? ˜i , ??i ) · (xit · ?i ? zit (? ˜i , ??i ))] ? vit (ait ). Uit (? As in the proof of Proposition 3, high managerial e?orts can be induced by ˜) = v for each report ? ˜. Combined with the PMN asset allocation setting ?it (? it ˜i , ?i )}N , relative bene?t depreciation corresponding to the hurdle rate rule {?i (?
i=1
˜i , ??i ) ensures that zit (?, ˜ ??i ) = xit · ??? (??i ). When the ?xed payments are r (? i ˜ chosen such that ?it (?) = vit (¯ ait ) ? vit · a ¯it , manager i’s total utility payo?s become:
??
˜i , ??i | ?i ) = ?i · I ? (? ˜i , ?i ) · [?i ? ??? (??i )]. Ui (? i
(39)
Clearly, the above PMN mechanism ensures that manager i’s participation constraint holds for each ??i , and each manager has a strong dominant strategy incentive to report his information truthfully. Furthermore, each manager’s interim utility payo?s E??i [Ui (?i , ??i | ?i )] from (39) coincide with those of the optimal revelation mechanism as given in (40). Proof of Corollary 5 Since ?i (?i ) = Vi (?i ) ? ?i · Hi (?i ), equation (28) implies that the cuto?-pro?tability
?? (??i | ?1 , ..., ?n ) is uniformly increasing in ?j for all 1 ? j ? n. The parameter ?i
result follows because:
?? ?? r?? (? | ?1 , ..., ?n ) = ro (?1 (??1 | ?1 , ..., ?n ), · · · , ?n (??n | ?1 , ..., ?n )) ,
and ro (·) increases in each of its arguments. 41
Appendix B: Derivation of Relaxed Program P
The participation constraint E??i [Ui (?i , ??i | ?i )] ? 0 in the principal’s original program P will hold with equality for the lowest type ?i . This boundary condition combined with the fact that ait (?i , ??i | ?i ) = cit (?i , ??i ) ? xit · ?i · Ii (?i , ??i ) and the “local” incentive compatibility condition implies that manager i will earn the following informational rents:
?i T
E??i [Ui (?i , ??i | ?i )] = E??i
?i
? t · vit (ait (qi , ??i | qi )) · xit · Ii (qi , ??i ) dqi .
t=1
(40)
Integrating by parts yields that manager i’s expected informational rents are given by:
¯i ? T
E? [Ui (?i , ??i | ?i )] = E??i
?i
? t · [vit (ait (?)) · xit · Ii (?) · Hi (?i )] fi (?i )d?i ,
t=1
(41)
where ait (?) ? ait (?i , ??i | ?i ). Since Ui (?i , ??i | ?i ) ?
T t=1
? t · [sit (?i , ??i | ?i ) ? vit (ait (?))], substituting (41) into
n
the objective function in P yields: max E?
i=1
(ai (? ),Ii (?))n i=1
?i (?) ? B (?)
subject to(ie ), (is ), where
T
?i (?) ?
t=1
? t · {ait (?) ? vit (ait (?)) + xit · [?i ? vit (ait (?)) · Hi (?i )] · Ii (?)} .
For any given investment rule Ii (?), the central o?ce will choose ait to maximize: ait ? vit (ait ) ? vit (ait ) · xit · Hi (?i ) · Ii (?) 42
Condition (19) implies that 1 ? vit (¯ ait ) ? vit (¯ ait ) · xit · Hi (?i ) · I (?) > 0 for all ?i and each Ii (?) ? {0, 1}. It is therefore optimal to induce the highest level of e?ort a ¯it for all ?i . Consequently, the optimization program in P simpli?es to the program in P , and the optimal investment rules for the exclusive and shared asset settings are as given by (21) and (22), respectively. To complete the proof, we need to show that the resulting scheme is globally incentive compatible. As shown in Mirrlees (1971), a mechanism is incentive compatible ˜ ˜i . provided it is locally incentive compatible, and ?Ui (?i ,??i |?i ) is weakly increasing in ?
??i
For the above mechanism: ˜i , ??i | ?i ) ?Ui (? = ??i
T
˜i , ??i | ?i )) · xit · Ii (? ˜i , ??i ), ? t · vit (ait (?
t=1
˜i since v (ait (·, ??i | ?i )) is increasing in ? ˜i and the optimal which is increasing in ? it Ii (·, ??i ) is an upper-tail investment policy in both settings.
43
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