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Journal of Economics and Business

Internal capital markets, empire building, and capital structure Alexandros P. Prezas Suffolk University, Sawyer Business School, Finance, 8 Ashburton Place, Boston, MA 02108-2770, United States

a r t i c l e

i n f o

Article history: Received 16 August 2006 Received in revised form 4 September 2008 Accepted 8 September 2008 JEL classiﬁcation: G31 G32 G34 Keywords: Capital structure Internal capital markets Asset allocation Empire building

a b s t r a c t Internal funds generated by assets in place are available to ﬁnance the bulk of new investment by nonﬁnancial ﬁrms. Self-interested management has incentives to misallocate these funds in order to increase their control rents. There are two ways to impact future discretionary investment. First, using debt diverts funds to creditors and away from management. Second, having in place more assets that do not provide internal ﬁnancing reduces the funds subject to managerial discretion. Investment in such assets and debt ﬁnancing are inversely related in controlling self-interested management. As a result, ﬁrms borrow more and own proportionally more assets that provide internal funds as the average proﬁtability of these assets, or that of future investment, increases. Firms may borrow less while increasing investment in the less valuable assets that do not supply internal ﬁnancing as the expected proﬁtability of these assets increases. © 2008 Elsevier Inc. All rights reserved.

There is evidence that U.S. nonﬁnancial ﬁrms rely on internal funds for investment. Fama and French (1999) ﬁnd that in 1951–1996 sample ﬁrms could exclusively ﬁnance their new investments using internal funds; it is because of their substantial dividend and interest payments that these ﬁrms ﬁnanced internally only about 70% of such investment. Lamont (1997) ﬁnds that in 1981–1991 more than 75% of the investment by sample ﬁrms was ﬁnanced internally. Further, he points to the liquidity spillovers associated with the 1986 oil price shock as evidence for the ﬁnancial interdependencies the allocation of internally generated funds creates across corporate segments.1

E-mail address: [email protected] Williamson (1975) suggests that the allocation of cash ﬂows across competing investment projects is an important managerial choice. Stein (1997) provides a rationale for internal capital markets where credit-constrained headquarters creates value using winner-picking to reallocate scarce funds across projects it oversees. 1

0148-6195/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jeconbus.2008.09.001

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As Jensen (1986) indicates, internal funds enable ﬁrms to avoid the monitoring of the capital markets. When there is separation of ownership and control, management may have incentives to engage in empire building, i.e., invest the funds in projects that maximize managerial control rents, even if these projects have negative net present value (NPV). Donaldson (1984) documents such behavior. Servaes (1994) provides evidence of overinvestment by oil and gas ﬁrms and large ﬁrms that were takeover targets or went private. Berger and Ofek (1995) ﬁnd that, compared to undiversiﬁed ﬁrms, diversiﬁed ﬁrms tend to overinvest in and subsidize money-losing divisions. Shin and Stulz (1996) ﬁnd that investment spending by small divisions of diversiﬁed ﬁrms strongly depends on the cash ﬂows of other divisions. Jensen (1986) argues that empire building can be reduced if the ﬁrm issues debt but does not retain the proceeds of the issue (e.g., uses them to buy back equity or pay dividends). The debt service obligation incurred forces management to disgorge future cash ﬂows instead of investing them in new negative NPV projects. Jensen (1986) argument is formalized in Stulz (1990) for short-term debt which matures when the new investment decision is made, and in Hart and Moore (1995) for long-term debt maturing after the new investment decision is made. This paper suggests a way of controlling empire building for ﬁrms that ﬁnance new investment internally that has not been considered in the literature. It proposes that investment in assets that do not provide internal funds for future investment may help mitigate this managerial agency problem.2 Examples of such assets are new product or market development, investment in start up companies, mine development, etc. Empirical literature provides evidence of the existence of these assets. Poterba and Summers (1995) ﬁnd in their survey of CEOs at 1000 of the largest American ﬁrms that, on average, 21.1% of the respondent ﬁrms’ budget was devoted to projects with no expected payoffs in the ﬁrst ﬁve years. Reinhardt (1973) calculates it took about $1 billion and a gestation period of 42 months between the beginning of the development effort and the onset of production for Lockheed’s L-1011 Tri Star Airbus aircraft.3 Essentially, such investment amounts to a managerial commitment to refrain from future empire building and is rational even if these assets have lower cash ﬂows than competing ones that provide internal funds for new investment. Given debt’s well established role in curbing empire building, it follows that investment in assets that do not provide internal ﬁnancing for new investment could work along with debt in controlling overinvestment costs. However, managers apply effort to manage the projects from which they derive control rents. When proﬁtable new projects cannot prevent bankruptcy, control rents are impaired so that management has no incentive to exert the effort required to manage these projects. Therefore, underinvestment results if management (and shareholders) do not beneﬁt from positive NPV investment. For ﬁrms ﬁnancing future investment internally, increasing either debt or investment in assets that do not provide internal funds can mitigate overinvestment but exacerbate underinvestment. Nevertheless, the origin of this effect differs depending on the tool used. Debt diverts funds to creditors and thus away from selfinterested management or shareholders. Assets that do not supply internal funds reduce the cash ﬂow from assets in place thereby leaving less in the hands of self-interested management or shareholders. It then follows that different pairs of debt and investment can be used to control suboptimal (i.e., overand under-) investment.4 In these pairs, employing more assets that preclude future investment is associated with lower debt levels and vice versa, leading to a link between the two corporate policies.

2 This is consistent with Lamont (1997) who also suggests overinvestment declines when internal ﬁnancing falls, although he does not provide a formal analysis. 3 Similarly, recent ﬁnancial press reports it took Airbus almost $20 billion of investment and seven years prior to delivering its ﬁrst B380 airplane. At the same time, Boeing after years of investment has not yet delivered its ﬁrst competing 787 Dreamliner aircraft. Finally, it took more than a decade and staggering costs for Pﬁzer Inc. to make and market Exubera, an inhaled insulin drug. 4 Other models that explore the role of debt in preventing self-interested management from undertaking unproﬁtable investments account for a trade-off between overinvestment and underinvestment. In Stulz (1990), short-term debt reduces the probability of overinvestment, increases the probability of underinvestment, and leads to the loss of investment opportunities in the event of bankruptcy. In Hart and Moore (1995), low levels of long-term debt allow management to subsequently borrow against cash ﬂows from assets in place and overinvest, but high levels of debt overmortgage these cash ﬂows leading to underinvestment.

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Given this link, the objective is to study the impact of internal capital markets and empire building on corporate investment and ﬁnancing. Speciﬁcally, this paper examines a ﬁrm’s ﬁnancing and allocation of capital across two projects when subsequent investment is ﬁnanced internally, and management derives control rents from investment only if the ﬁrm is solvent. Although both projects provide long-term funds for debt repayment, only one supplies internal funding for new investment. It is shown that ﬁnancing and allocation of resources across the two projects are interrelated through the cost of future suboptimal investment. Optimal long-term debt is chosen to prevent suboptimal investment on average. Optimal allocation to the project that supports new investment is positively correlated with its cash ﬂow advantage over the alternate project, the proﬁtability of future investment, and the variability of the alternate project cash ﬂow, but negatively correlated with its own cash ﬂow variability. Comparative statics show that ﬁrms borrow more and invest proportionally more in the project that supplies internal funds when the expected proﬁtability of the project or new investment increases. Also, ﬁrms invest more in the alternate project as project proﬁtability increases; at the same time, they may borrow less. In this paper, future suboptimal investment depends on the ﬁrm’s current ﬁnancing and assets in place. These investment and ﬁnancing decisions anticipate subsequent managerial discretion and impact it through the future funds being available internally. Essentially, the internal capital market affects ﬁrm value by providing a link between the ﬁrm’s current and future choices. In this respect, this paper differs from existing internal capital market literature which provides insights about the internal market efﬁciency of ﬁnancially constrained ﬁrms in allocating available funds.5 In contrast, this paper examines the role the internal market plays in determining the impact of managerial discretion on the ﬁrm’s decisions and value. Also, the paper differs from existing literature that studies corporate investment and debt policies under managerial discretion. Unlike the present paper, existing literature allows management to raise funds needed for empire building externally, and does not consider the role of assets in place in controlling managerial discretion. In what follows, Section 1 introduces the model. Section 2 determines optimal allocation and debt. Section 3 provides comparative statics of the optimal choices with respect to the problem parameters. Section 4 summarizes the paper’s ﬁndings. 1. The model Consider a model with risk-neutral investors, zero risk-free rate, and three dates, t = 0, 1, 2. At t = 0, management allocates available resources (normed to one) to two uncorrelated scalable projects labeled P1 and P2. A portion w is invested in P1, the balance, 1 − w, in P2. Both projects last for two periods, but have different cash ﬂow patterns. A unit investment in P1 yields two uncertain, independent and identically distributed cash ﬂows Xt at both t = 1 and t = 2 equaling C > 1 with probability p ∈ (0,1), and 0 otherwise. A unit investment in P2 provides an uncertain t = 2 cash ﬂow Y that equals D > 1 with probability q ∈ (0,1), and 0 otherwise. Hence, the expected value and the standard deviation of the cash ﬂows of the two projects are E(X) = pC;

X =

E(Y ) = qD;

Y =

p(1 − p)C

(1)

q(1 − q)D

where the subscript t of the P1 cash ﬂows has been suppressed for simplicity. Further, it is assumed that the expected cash ﬂow of P2 is higher than that of P1 in the second period, but less than that over

5 Regarding the availability of funds, in Stein (1997), headquarters raises funds for the internal market externally and passes them along to the ﬁnancially constrained segments via winner-picking. In Hubbard and Palia (1999), funds originate from ﬁnancially unconstrained bidders that acquire ﬁnancially constrained target ﬁrms. Regarding their effectiveness in allocating funds, in Gertner, Scharfstein, and Stein (1994), and Stein (1997), internal capital markets are assumed to be more efﬁcient than external capital markets because corporate headquarters are or have incentives to become more informed than outside capital suppliers about investment opportunities. By comparison, in Sharfstein and Stein (2000), and Rajan, Servaes, and Zingales (2000), headquarters are induced by divisional managers to allocate capital excessively to those divisions that have poor investment opportunities but where rent-seeking incentives are the strongest.

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both periods; i.e.,6 pC < qD < 2pC.

(2)

Finally, it is assumed that either project has zero salvage value at any date. At t = 1 a third scalable project, P3, appears. A unit investment in it yields a t = 2 cash ﬂow s. This cash ﬂow is independent of those from P1 and P2 and at t = 0 its value is uncertain and uniformly distributed on the interval [0, S]; S > 1. Cash ﬂow uncertainty for all projects is resolved at t = 1.7 The ﬁrm liquidates at t = 2. It is assumed that management derives a constant positive amount of control rents per unit of investment, but contributes ﬁxed effort to manage it. Further, these control rents are assumed to increase the manager’s utility, net of effort disutility, if, following investment in P3, the ﬁrm is solvent at t = 2. As a result, management invests in P3 even if it has negative NPV and is assumed that no incentive scheme can prevent such behavior.8 However, if investment in P3 cannot prevent bankruptcy at t = 2, control rents are impaired (or even vanish) and, after taking into account effort, management suffers a net loss of utility from t = 1 investment. Hence, management does not invest in P3 even if the project’s NPV is positive. The above assumption says that the manager prefers to carry out all investment in P3 at t = 1 unless it is infeasible (i.e., as indicated below, no internal funds are available to invest in P3) or it prevents the manager from capturing the beneﬁts of the control rents associated with P3 (i.e., the ﬁrm is bankrupt at t = 2). This is a common assumption in the literature – e.g., Stulz (1990), Hart and Moore (1995) and Li and Li (1996). Following this literature, does not require to explicitly model managerial control rents and utility characteristics. Nevertheless, the impact of such managerial behavior on the ﬁrm’s choices is discussed in the following section.

6 Unlike the present model, Thakor (1993) assumes that “late bloomers,” like P2, are intrinsically more valuable than “early winners,” like P1. Using this assumption, Thakor (1993) aims at showing that, despite the value disadvantage of the projects that start paying early, managers behave “myopically” and invest in them to take advantage of the timing of their cash ﬂows (i.e., to minimize the need for future external ﬁnancing). Similarly, Narayanan (1985) and Holmstrom and Ricart i Costa (1986) suggest that, due to personal incentives, managers make myopic investment choices that stockholders do not prefer. On the other hand, the present model aims at showing that, despite the value disadvantage of projects that pay late like P2, investing in them is desirable because of their ability to control empire building. Hence, if it was assumed that P2 is more valuable than P1, the overand under-investment problems (described in Section 2) will still be possible, but it would not be clear whether investing in P2 is desirable due to its higher productivity or its ability to control managerial discretion. Further, the paper’s assumption is not critical in deriving the interior optimal allocation to the two projects or the optimal debt ﬁnancing at t = 0. These results will hold even if it was assumed that P2 is more valuable than P1 as long as the expected payoff of future (t = 1) investment is sufﬁciently large. 7 This assumption relates to the leveraged ﬁrm’s option to invest in P3. As in Myers (1977), the decision whether to exercise this option requires the value of the ﬁrm’s assets is known (i.e., uncertainty about the cash ﬂows of the assets is resolved) at the decision point. In Myers, ﬁnancing for P3 is exogenous and the decision point is at the end of a single-period model. However, this paper accounts for the ﬁnancing of P3; as indicated later in this section, it comes from the t = 1 cash ﬂow of P1. Like existing papers that explicitly account for the ﬁnancing of P3 (see Hart and Moore, 1995; Li and Li, 1996), a two-period model is used in this paper. As common in those models, ﬁnancing for P3 is obtained and the option decision point is (i.e., cash ﬂow uncertainty is resolved) at t = 1, the end of the ﬁrst period. However, given P3 is not ﬁnanced with end-of-horizon cash ﬂows and, as assumed in Section 2, debt matures after the option to invest in P3 expires (what Myers calls the interesting case) and is paid from the total t = 2 cash ﬂows of the ﬁrm, the analysis will not change if uncertainty is resolved and the option decision point is at t = 2. 8 This is a standard assumption in the literature. For example, Hart (1991) suggests that incentive schemes may be insufﬁcient to curb overinvestment because managers may require a very large incentive payment to give up the control rents they derive from empire building. Also, Hart and Moore (1995) consider the “extreme case in which the empire-building motive is so strong that no feasible ﬁnancial incentive payment can persuade the manager not to invest at date 1.” Further, they assume that “the manager has no (or little) initial wealth and so cannot be charged up front for the empire building beneﬁts.” Similarly, Stulz (1990) assumes that no monetary rewards at t = 2 can dominate the managerial beneﬁts from empire building at t = 1. Grossman and Hart (1982) suggest that adjusting ﬁnancial structure can be cheaper in curbing managerial empire building tendencies. Jensen (1986) discusses an alternative mechanism to prevent managers from empire building. This alternative calls for leaving managers with control over the use of future cash ﬂows, but at the same time announcing a “permanent” increase in the dividend as a way of promising to pay out future cash ﬂows. However, Jensen suggests such promises are weak because future dividends can be reduced. Stulz (1990) also rules out dividend policies that result in the automatic removal of managers who do not pay a promised dividend as a way of controlling the empire building problem. He suggests that such policies can only be enforced through shareholder action. Further he states that, if shareholders can remove managers who do not pay an appropriate dividend, the managerial commitment to pay the dividend is superﬂuous.

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177

Table 1 Combined cash ﬂows from P1 and P2 and their probabilities. wX1

wX2

(1 − w)Y

Fi = w(X1 + X2 ) + (1 − w)Y

i

wC wC wC wC 0 0 0 0

wC wC 0 0 wC wC 0 0

(1 − w)D 0 (1 − w)D 0 (1 − w)D 0 (1 − w)D 0

F1 = 2wC + (1 − w)D F2 = 2wC F3 = wC + (1 − w)D F4 = wC F5 = wC + (1 − w)D F6 = wC F7 = (1 − w)D F8 = 0

1 = p2 q 2 = p2 (1 − q) 3 = p(1 − p)q 4 = p(1 − p)(1 − q) 5 = p(1 − p)q 6 = p(1 − p)(1 − q) 7 = (1 − p)2 q 8 = (1 − p)2 (1 − q)

The use of external funds for empire building has been studied in existing literature. To concentrate on internal ﬁnancing, it is assumed that investment in P3 may proceed only if P1 pays off at t = 1. Such reliance on internal ﬁnancing for new investment can be viewed as a form of soft capital rationing. It then follows that there is no investment in P3 when P1 does not pay off at t = 1. Finally, the t = 2 cash ﬂows of P1 and P2 cannot be invested in P3 as they accrue after the new investment decision is made. 2. Optimal allocation and debt choices Debt is chosen at t = 0. Prior to that, the ﬁrm is completely ﬁnanced with equity. To justify the use of debt, as in Stulz (1990), Hart and Moore (1995) and Li and Li (1996), the ﬁrm is viewed as being subject to a takeover bid at t = 0, but not at t = 1.9 This prevents management from value-reducing investment at t = 0. To prevent shareholders from tendering their shares and thus retain control, self-interested management voluntarily exchanges debt for equity to raise ﬁrm value.10 Debt requires a repayment B at t = 2, otherwise the ﬁrm is bankrupt. As shown in Appendix A, one-period debt is optimally set equal to zero at t = 0. In addition, it is assumed that the ﬁrm does not issue one-period debt at t = 1 as this option has already been explored in Hart and Moore (1995). Further, assuming t = 1 short-term debt away allows one to concentrate on the case where new investment is ﬁnanced internally. Given the assumptions about the cash ﬂows of P1 and P2, t = 0 investment results in a cash ﬂow realization Fi , i ∈ ˚ ≡ {1,2, . . ., 8}. The t = 1 values of these cash ﬂow realizations and their corresponding probabilities, i , i ∈ ˚ are reported in Table 1. Management may invest in P3 at t = 1 whenever P1 has a positive payoff; i.e., when i ∈ ˚ ≡ {1,2,3,4}. The probability of this happening is p. There are two possible ways the manager can invest suboptimally at t = 1. First, the manager invests the amount wC in P3 even if its NPV is negative (the overinvestment problem). Such overinvestment is possible when the NPV of P3 is negative but in combination with the cash ﬂow from t = 0 investment exceeds the promised debt repayment. In other words, when the cash ﬂow from t = 0 investment exceeds the contractual debt obligation B, the excess Fi − B > 0 creates a cushion of cash. As a result, the manager can avoid bankruptcy and capture the beneﬁts of the control rents associated with investment in P3 even if its net present value, (s − 1)wC, is negative but does not exceed the cash cushion provided by the t = 0 investment. All else the same, exogenously increasing the promised debt repayment reduces overinvestment by reducing the cash cushion. Ceteris paribus, overinvestment may also decline as the allocation to P2 increases. Given B and w, management may overinvest for the realizations Fi for which i ∈ O ≡ {i ∈ ˚ |Fi > B}. Speciﬁcally, for i ∈ O, overinvestment in P3 takes place only when its cash ﬂow satisﬁes si ≡ 1 + (B − Fi )/wC ≤ s < 1. When this happens, bonds receive their promised payment while stocks receive the leftovers, Fi + (s − 1)wC − B, at t = 2. However, if i ∈ O but the cash ﬂow of P3 satisﬁes 0 ≤ s < si , the negative NPV of P3 exceeds the cash cushion provided by t = 0 investment and bankruptcy cannot be avoided. Since managerial control rents are impaired

9 As in Li and Li (1996), the paper’s qualitative results carry through if the ﬁrm’s t = 0 debt level is positive but lower than the optimal level determined later, or if the t = 1 probability of a successful takeover bid is less than 1. 10 However, other interpretations are also possible. For instance, it can be assumed that shareholders – through their board of directors – make the debt decision to maximize equity value. Alternatively, the use of debt can be viewed as a commitment by management in response to the managerial agency problem as in Harris and Raviv (1990).

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Table 2 Managerial action regarding investment in P3 and payoffs to stocks and bonds. s

Payoff to bonds

Payoff to stocks

Manager’s action

Panel A: Fi ; i ∈ O [1, S] [si , 1) [0, si )

B B B

Fi + (s − 1)wC − B Fi + (s − 1)wC − B Fi − B

Invests in P3 Overinvests in P3 Does not invest in P3

Panel B: Fi ; i ∈ U [si , S] [1, si ] [0, 1]

B Fi Fi

Fi + (s − 1)wC − B 0 0

Invests in P3 Underinvests in P3 Does not invest in P3

in bankruptcy, the manager does not invest in P3. In such a case, bonds still receive their promised payment B while stocks receive Fi − B at t = 2. Finally, if i ∈ O and 1 ≤ s ≤ S, the manager invests in P3, bonds are paid in full and stocks receive Fi + (s − 1)wC − B at t = 2. The second way self-interested management may proceed at t = 1 is to forgo investment in P3 even if its NPV is positive (the underinvestment problem).11 This is possible when internal funds are available but the positive NPV of P3 combined with the cash ﬂow from t = 0 investment does not sufﬁce to cover promised debt repayment. Alternatively, when the cash ﬂow Fi from assets in place falls short of debt repayment B, it creates a cash shortage Fi − B < 0. As a result, the manager cannot avoid bankruptcy and capture the control rents associated with investment in P3 if its net present value, (s − 1)wC, is positive but less than this cash shortage. Ceteris paribus, this cash shortfall increases with debt repayment, leading to more underinvestment. All else the same, underinvestment may also increase as the allocation to P2 increases. Given B and w, underinvestment in P3 may occur at t = 1 for the cash ﬂow realizations Fi for which i ∈ U ≤ {i ∈ ˚ |Fi < B}. Speciﬁcally, for i ∈ U, underinvestment in P3 takes place only when its cash ﬂows satisfy 1 < s ≤ si ≡ 1 + (B − Fi )/wC. When this happens, bonds receive the whole Fi while stocks receive nothing at t = 2. However, if i ∈ U but the cash ﬂow of P3 satisﬁes si < s ≤ S, the positive NPV of P3 exceeds the cash shortage from the t = 0 investment and the manager invests in it. In such a case, bonds receive their full promised payment while stocks receive the leftovers, Fi + (s − 1)wC − B, at t = 2. Finally, if i ∈ U and 0 ≤ s ≤ 1, the manager does not invest in P3, the bonds receive the whole Fi and the stocks receive nothing at t = 2. Table 2 summarizes the managerial action taken and the payoffs to bonds and stocks under the possible values of the t = 2 cash ﬂow of P3 for the Fi realizations which may lead to overinvestment (Panel A) or underinvestment (Panel B). The paper’s results require that O and U are nonempty.12 It follows from the bond payoffs in Table 2 that creditors will be willing to supply the required debt ﬁnancing at t = 0 as long as their bonds are priced to provide their required rate of return, i.e., as long as they pay a price equal to the value of the bonds given by

i∈O

i B +

i∈U

⎡ i ⎣B

S si

1 dS + Fi S

si 0

⎤ 1 dS ⎦ + S

i min(B, Fi ),

i ∈ ˚

where ˚ ≡ {5,6,7,8} and the last term indicates that when P1 does not pay off at t = 1, bonds receive the minimum of their promised payment (if the ﬁrm is solvent) or the payoff of the t = 0 investment

11 This differs from the underinvestment problem in Hart and Moore (1995) where management cannot raise the required ﬁnancing for investment at t = 1. 12 For example, assume p = 0.182, C = 4.00, q = 0.11, D = 13.20, and S = 1.2. Using (9), developed later in this section, determines w, the allocation to P1. Then, the four relevant cash ﬂow realizations are: F1 = 8.22, F2 = 7.66, F3 = 4.39, and F4 = 3.83. Under the ﬁrst cash ﬂow realization the ﬁrm overinvests in states s ∈ (0.053, 1), while under the second cash ﬂow realization it overinvests in states s ∈ (0.198, 1). Under the third cash ﬂow realization, the ﬁrm underinvests in states s ∈ (1, 1.053), as it does in states s ∈ (1, 1.198) under the fourth cash ﬂow realization. Hence, the sets O and U are not empty.

A.P. Prezas / Journal of Economics and Business 61 (2009) 173–188

179

(if the ﬁrm is bankrupt). On the other hand, the t = 0 value of equity is given by

⎧S ⎨ i

i∈O

⎩

i

i∈U

[Fi + (s − 1)wC − B]

1 ds + (Fi − B) S

si

S

+

si

⎫ ⎬

1 ds S ⎭

0

[Fi + (s − 1)wC − B]

1 ds + S

si

i max(0, Fi − B),

i ∈ ˚

where the last term provides the payoff to stocks when investment in P3 is not feasible at t = 1. Adding the above values of debt and equity and then subtracting 1 (the total amount invested in P1 and P2 at t = 0), provides the following expression for V, the expected value of the ﬁrm

p V = 2pwC + q(1 − w)D − 1 + S

Z≡

⎧ ⎪ Si ⎨ 1 S

⎪ ⎩i∈U

i

S (s − 1)wCds − Z; 1

(s − 1)wCds −

1

i

i∈O

1

Si

(s − 1)wCds

⎫ ⎪ ⎬ ⎪ ⎭

=

(3)

i (Fi − B)2

i ∈ ˚

2SwC

> 0.

Eq. (3) provides the ﬁve component parts of the ﬁrm’s value. The ﬁrst part, 2pwC, is the expected present value of the investment in P1 over its life. Similarly, the second term, q(1 − w)D, is the expected present value of the investment in P2 over its life. The third term, −1, denotes the total amount invested in P1 and P2 at t = 0. Hence, the ﬁrst three terms taken together give the NPV of the t = 0 investment.

S

The fourth part of the value, p/S 1 (s − 1)wCds, is the expected NPV of P3 when funds are available to invest in it (with probability p) and it is proﬁtable (i.e., 1 < s ≤ S). The last term, Z, is the cost of suboptimal investment and, as suggested in the second line of Eq. (3), it has two parts of its own. The ﬁrst one is the expected cost of underinvestment. Intuitively, when underinvestment occurs for a realization Fi ; i ∈ U, the ﬁrm’s total cash ﬂow is lower than what it would had been if the ﬁrm had invested in P3 in each state 1 < s ≤ si . This value reduction represents the cost of underinvestment for any realization Fi ; i ∈ U, and is equal to 1/S

1 si

(s − 1)wCds. The weighted

average of these costs for all the underinvestment realizations (each weight being the corresponding probability of Fi ) is the expected cost of underinvestment. The second component of Z is the expected cost of overinvestment. Intuitively, when overinvestment takes place for a realization Fi ; i ∈ O, the ﬁrm’s total cash ﬂow is lower than what it would had been if the ﬁrm had not invested in P3 in each state si ≤ s <1. This value reduction is the cost of overinvestment for any realization Fi ; i ∈ O, and is equal to s −1/S 1 i (s − 1)wCds. The weighted average of these costs for all the overinvestment realizations is the expected cost of overinvestment. Overall, then Eq. (3) says that ﬁrm value equals the expected NPV of the t = 0 investment in P1 and P2, plus the expected NPV of P3 over its proﬁtable states when funds are available internally for investment in it at t = 1, minus the cost of suboptimal investment, Z (which combines the cost of underinvestment and overinvestment). At t = 0, optimal debt repayment level, B* , and optimal allocation to P1, w* , are chosen to maximize the expected value of the ﬁrm. The ﬁrst order conditions for the value maximization problem are VB = −ZB = 0; ZB = −

p [(1 + p)wC + q(1 − w)D − B] SwC

(4)

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and Vw = (2pC − qD) + −1 Zw = SwC

i ∈ ˚

p S

S (s − 1)Cds − Zw = 0; 1

B − Fi ∂Fi + i (B − Fi ) 2w ∂w

(5) ,

where the second lines in Eqs. (4) and (5) give the marginal cost of suboptimal investment associated with debt repayment and allocation to P1, respectively. The intuition behind the ﬁrst order conditions is straightforward. Eq. (4) states that, given an allocation to P1, debt repayment optimally increases until its impact on Z vanishes at the margin. This is not an unexpected result. From (3), B affects ﬁrm value only through the cost of suboptimal investment Z since all the other terms of V are independent of leverage. As noted earlier, increasing B has a dual effect on Z; it intensiﬁes underinvestment but reduces overinvestment. Hence as (4) indicates, B optimally increases until the marginal cost of increased underinvestment equals the marginal beneﬁt of reduced overinvestment. On the other hand from (3), the marginal allocation to P1 affects V via the expected NPV of t = 0 and t = 1 investment, as well as the cost of suboptimal investment. The impact of a marginal increase in w on t = 0 investment is captured by the ﬁrst term of (5), while its impact on the t = 1 investment is captured by the second term. The third term provides the marginal effect of w on Z; as noted earlier, allocating more to P2 may intensify underinvestment but reduce overinvestment. Then (5) says that, given debt repayment, more is allocated to P1 until the resulting beneﬁt from t = 0 and t = 1 investment equals the cost of suboptimal investment at the margin.13 Like the selection of debt, the choice of investment in P2 by self-interested management is equivalent to a commitment to reduce overinvestment because allocating more to P2 reduces the funds internally available at t = 1 when P1 pays off. Inspection of (4) and (5) suggests that B* and w* must be chosen simultaneously at t = 0; i.e., they are not independent of each other. Speciﬁcally, from (4), allocation to P1 impacts optimal debt repayment because it affects ZB , the cost of suboptimal investment associated with the marginal unit of debt. Similarly from (5), debt repayment affects optimal allocation through Zw , the cost of suboptimal investment associated with the marginal allocation to P1. This means that investment and debt ﬁnancing are interrelated through the cost of suboptimal investment. Intuitively, the amount wC management may invest at t = 1 in their interest depends on the t = 0 capital allocation choice w. Whether the manager will invest that amount suboptimally or not (i.e., what the cost of suboptimal investment is) depends on how the NPV of the new investment compares with the cash ﬂow from the t = 0 investment net of debt repayment. In turn, this net cash ﬂow depends on the t = 0 resource allocation and contractual debt repayment. Thus, when determining the values of B and w that maximize V, the necessary ﬁrst order conditions (4) and (5) must be solved simultaneously.14 Solving (4) for optimal debt repayment gives B∗ = (1 + p)wC + q(1 − w)D.

(6)

As noted earlier, given the assets in place, optimal debt repayment minimizes the cost of t = 1 suboptimal investment. However, this cost arises only if internal funds are available; i.e., if the t = 1 payoff of P1 is positive. If P1 does not pay at t = 1, management has no funds to misinvest, the cost of suboptimal investment vanishes, and with it the controlling role debt plays. As a result, debt repayment should be chosen in a way that prevents management from suboptimal investment when funds are available at t = 1. Recalling that cash ﬂow uncertainty is resolved at t = 1, the choice of debt repayment would had been straightforward if it were made at t = 1. Speciﬁcally, B should be set equal to the revealed

13 From (2) and the requirement V = 0, it follows from (5) that Z > 0 at the optimum. Further, the existence of an interior w w w* requires that initially Z is inversely related with investment in P1. 2 14 The second order conditions sufﬁcient for this maximization require V BB < 0, Vww < 0, and Vww VBB − VwB > 0. From (4) and

(5), VBB = (−p/SwC) < 0 and Vww = (−pY2 /SCw3 ) < 0, while it is assumed that the third condition also holds.

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cash ﬂow Fi to prevent suboptimal investment and maximize ﬁrm value. Nevertheless, Fi is uncertain when debt is chosen at t = 0. In such a case, B should be chosen to prevent suboptimal investment on average. To accomplish that, Eq. (6) requires optimal debt repayment is equal to the total of wC plus the expected t = 2 cash ﬂow from t = 0 investment. Setting optimal debt repayment equal to this total implies that the cash ﬂow from assets in place expected when P1 pays off at t = 1 (henceforth, the conditional cash ﬂow from assets in place) just covers the debt obligation and precludes suboptimal investment on average.15 Further, (6) points to the interdependence of debt repayment and t = 0 resource allocation. Using (2) and (6), an exogenous increase in w enhances the conditional cash ﬂow from assets in place, thereby increasing B* by ∗ = (1 + p)C − qD > 0. Bw

(7)

Intuitively, (7) says that allocating more to P1 increases the cash ﬂow that is available internally at t = 1 if this project pays off; hence, more is available for suboptimal investment. To counter this, the promised debt repayment is increased. Conversely, allocating more to the less valuable P2 reduces the size of the internal capital market, management has less to misinvest, and less debt is required to control managerial discretion. Substituting (6) into (5), results in Vw = 2pC − qD +

pC(S − 1)2 1 p X2 + Y2 1 − 2 − 2S 2SC w

=0

(8)

where the last term is Zw , the marginal cost of suboptimal investment.16 Solving (8) for the optimal allocation to P1 yields17 w∗ =

Y [(X2

+ Y2 ) − (2SC/p)(2pC

− qD) − C 2 (S − 1)2 ]

1/2

.

(9)

Substituting (9) in (6) gives B* as a function of model parameters. Intuitively, (8) provides the determinants of the optimal allocation to P1. First, due to its expected cash ﬂow advantage over P2, the marginal investment in P1 enhances ﬁrm value by 2pC − qD. Ceteris paribus, exogenously increasing this advantage results in a bigger value increase for the marginal allocation to P1. Hence, allocation to P1 increases until the beneﬁt of this bigger value increment is offset by the ensuing higher marginal cost of suboptimal investment. Similarly, more is invested in P1 as the expected NPV of P3 (over its proﬁtable states) increases exogenously. Third, all else the same, allocation to P1 is directly related to the variance Y2 of the cash ﬂow from P2. The rationale is that the higher variance reduces the cost of suboptimal investment associated with the marginal allocation to P1.18 The ﬁrm takes advantage of this cost reduction by allocating more to P1. Finally, allocation to P1 is inversely related to the variance X2 of its own cash ﬂow. A higher variance increases the cost of suboptimal investment from the marginal allocation to P1; to prevent that, investment in P1 declines.19

15 Indeed, when P1 pays off at t = 1, the expected value of the cash ﬂow realization F net of debt repayment is zero; i.e., E(F − i i (Fi − B∗ ) = 0. Further, the variance of this net cash ﬂow realization is (Fi − B∗ )2 . Then, an alternative B∗ ) = i i i∈˚

i∈˚

interpretation of B* follows from (4) and the second equality in the deﬁnition of Z in (3). Speciﬁcally given w, B* minimizes the of debt repayment. variance of the cash ﬂow realization Fi net 16 This follows from (Fi − B∗)2 = p[w2 X2 + (1 − w)2 Y2 ] and the second equality in the deﬁnition of Z in (3). It suggests i i∈˚

that when P1 pays off at t = 1, the variance of the Fi realizations net of debt repayment is determined by the variance of the t = 2 payoff from the t = 0 investment. Given the t = 1 payoff of P1 equals wC for all these realizations, this is an intuitively pleasing result. 17 From (9), an interior optimal allocation to P1 (i.e., 0 < w ∗ < 1) requires that ≡ 2 − (2SC/p)(2pC − qD) − C 2 (S − 1)2 > 0. X 18 To see this, consider the deﬁnition of Z in (3) in conjunction with footnote 16. It then follows that an increase in w reduces 2 2 the contribution of Y to Z. The higher Y is, the larger the reduction. 19 The effects of these four determinants on w* should be viewed as “constrained” comparative statics. For instance, the direct relationship between w* and Y2 discussed in the text is not about any change in the variance of P2 but only about those changes

which do not alter the expected value of the project’s cash ﬂow – i.e., mean-preserving changes in Y2 . Since from (1) the expected value and variance of the cash ﬂows of P2 depend on q and D, the discussion here is concerned with changes in variance resulting

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In sum, the preceding analysis shows that optimal asset allocation and debt are determined simultaneously and are inversely related in controlling managerial discretion. Speciﬁcally, at t = 0 the ﬁrm allocates its ﬁxed resources (normed to one) to P1 and P2. For an ongoing ﬁrm, one can view these two projects as the ﬁrm’s assets in place representing its ﬁxed resources; P1 stands for the ﬁrm’s “cashcows” which are expected to pay off annually, while P2 stands for the ﬁrm’s assets in place with long gestation periods. For a ﬁrm which has currently allocated more of its resources to P1, more internal funds are expected to be subject to managerial discretion when P1 pays off in the future. Then, a testable implication of the model is that among ongoing ﬁrms which have in place proportionally fewer assets with long gestation periods, those which employ higher levels of debt to control managerial discretion (i.e., minimize the cost of suboptimal investment) should be expected to display better operating performance than those with lower levels of debt. The activity in the oil industry in the 1980s lends support to the paper’s conclusions. Due to the tenfold increase of oil prices during the best part of the 1970s, ﬁrms in the oil industry invested large amounts of internal funds in oil exploration and development (E&D) projects, several of them with negative NPV. By the early 1980s, several such ﬁrms had high levels of oil reserves which like P1 were expected to generate internal funds annually. At the same time due to a decline in oil consumption these ﬁrms were left with excess reﬁning and distribution capacity which like P2 was not expected to provide internal funds in the short run. The ensuing mergers, takeover threats, and voluntary restructuring, induced changes. In particular, existing excess reﬁning and distribution capacity was sold, while large amounts of debt were raised with the proceeds swapped for equity and/or distributed as dividends. Further, oil ﬁrms kept investing new internal funds in E&D projects (the equivalent of P3), but such new investment was cut back to the most proﬁtable projects. The above changes were consistent with the present model. Speciﬁcally, the increased debt level of the oil ﬁrms and the simultaneous reduction in the proportion of assets paying off with some delay resulted in more proﬁtable future investment of their free cash ﬂow; i.e., reduced suboptimal investment. The gains in efﬁciency and in value resulting from those changes reportedly provided huge gains to the shareholders of oil ﬁrms that merged (e.g., Gulf/Chevron, Getty/Texaco, and Dupont/Conoco), or oil ﬁrms that restructured voluntarily (e.g., Phillips, Uncola, and Arco). To introduce a standard of comparison for the model’s solution, assume there are no over- and under-investment. Then, the cost of suboptimal investment vanishes and (4) and (5) reduce to VB = 0 and Vw = 2pC − qD +

p S

S (s − 1)Cds > 0. 1

Intuitively, if there is no suboptimal investment, debt repayment plays no role and its level is irrelevant. Further, the t = 0 allocation is driven by the proﬁtability of P1 and P2 and the ability of these projects to provide internal ﬁnancing for P3; hence, investment is exclusively in P1. This differs from the interior allocation obtained when managerial suboptimal investment was possible. In that case, some resources were invested in the less valuable P2. This was rational because it reduced internal resources, thereby reducing the cost of managerial discretion by more than the value of the ﬁrm’s t = 0 and t = 1 investment.20

when both q and D change in such a way as to leave expected cash ﬂow unchanged. For example, assume that initially q = 0.11 and D = 13.20 and then they change to q = 0.108 and D = 13.44. The expected cash ﬂow of P2 remains unchanged at 1.452, but Y2

increases from 17.06 to 17.40. As indicated in the text, this increase in Y2 results in higher w* . Of course, q or D could change

exogenously in a way that alters both Y2 and E(Y). The impact of such changes in the model’s parameters on w* and B* is discussed in Section 3. 20 Further, the solution to the value maximization problem in Eqs. (3)–(5) does not always prevent management from investing at t = 0 in P2 when its NPV is negative. However, such investment in P2 should not be viewed as t = 0 empire building as long as the sum of its NPV plus the value it creates through its impact on the investment in P1 and on the cost of t = 1 managerial discretion is positive. On the other hand, if this sum is always negative, management does not optimally invest in P2.

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The optimal allocation derived in the presence of debt is also contrasted with that for an all-equity ﬁnanced ﬁrm. With no debt repayment required, management can still overinvest the t = 1 cash ﬂow generated by P1 in P3, but there is no underinvestment. Then, optimal allocation depends on the sign of Vw , given by p Vw = 2pC − qD + S pC Zw = > 0. 2S

S (s − 1)Cds − Zw ;

(10)

1

In other words, now the marginal cost of overinvestment is a positive constant and Z steadily increases with the marginal allocation to P1. Thus, Vw is now independent of w, implying a corner solution. This is in contrast to the interior optimal allocation in the presence of debt. To compare the optimal allocations obtained under the two ﬁnancing scenarios, recall from (8) that at the optimum Zw = (P/2SC)[X2 + Y2 (1 − (1/w2 ))] when debt is used. This implies that Zw is lower when the ﬁrm uses debt than with all-equity ﬁnancing. Thus, the change in value due to the marginal allocation, Vw , is greater with debt ﬁnancing, implying that optimal allocation to P1 is at least as high as that under all-equity ﬁnancing.21 3. Comparative statics As shown above, a leveraged ﬁrm optimally allocates funds to both P1 and P2. From (8), this interior optimal allocation is determined by the cash ﬂow variability of the two projects, the advantage of P1 over P2, and the potential of P3. Since the values of these determinants hinge on model parameters, it is now shown how the optimal allocation is affected by exogenous changes in such parameters. Differentiating (8) with respect to a parameter j ∈ {S, q, D, p, C} provides its impact on optimal allocation. Then, differentiating (6) with respect to j and recognizing that optimal allocation changes with j, provides the effect of model parameters on optimal debt repayment. First, consider the impact of S. Using (8), it is shown in Appendix A that as P3 becomes more proﬁtable, the additional cash ﬂow from the marginal allocation to P1 is expected to provide more value, net of suboptimal investment cost, when invested in P3; thus, more is apportioned to P1. From (6), the larger allocation to P1 resulting from the higher S increases the conditional cash ﬂow from assets in place and, to control that, debt repayment increases. Second, consider the effect of the characteristics of P2. When the payoff D of P2 increases, the marginal allocation to P1 becomes less proﬁtable (as the advantage of P1 over P2 declines) and the marginal cost of suboptimal investment declines. As shown in Appendix A, the combined effect is reduced investment in P1. Like D, as the probability q of the payoff for P2 increases, the advantage of P1 over P2 declines; however, the marginal cost of suboptimal investment may increase or decrease. Overall, as shown in Appendix A, allocation to P1 is inversely related with q. From (6), a higher q or D directly enhances the conditional cash ﬂow from assets in place via the increased proﬁtability of P2 but reduces it due to the lower investment in P1. Depending on the relative sizes of these two effects, debt repayment may increase or decline. Finally, consider the impact of the characteristics of P1. When the payoff C of P1 increases, the marginal allocation to P1 becomes more proﬁtable (as the advantage of P1 over P2 increases), the

21 If model parameters are such that V > 0 for all w in (10), then w* = 1 under all-equity ﬁnancing. Further for the same w parameters Vw > 0 for all w in (5), implying w* = 1 when the ﬁrm uses debt. If model parameters are such that Vw = 0 for all w in (10), then optimal allocation is irrelevant under all-equity ﬁnancing. But for the same parameters Vw > 0 for all w in (5), implying w* = 1 when the ﬁrm uses debt. Finally, if model parameters are such that Vw < 0 for all w in (10), then w* = 0 under all-equity ﬁnancing. However, for the same parameters Vw in (5) is initially positive because, as noted in footnote 13, the cost of suboptimal investment is initially inversely related with w when the ﬁrm uses debt. Hence, ﬁrm value initially increases with w, implying that, unlike the all-equity scenario, w* > 0. It then follows from these three cases that an interior optimal allocation to P1 under debt may exist only if model parameters are such that Vw < 0 in (10) for all w; i.e., if pC(1+(S/2)) < qD. Hence, (2) becomes pC(1+(S/2)) < qD < 2pC, requiring S < 2.

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expected value from investing the marginal P1 cash ﬂow in P3 (in its proﬁtable states) increases, as does the marginal cost of suboptimal investment. As shown in Appendix A, the overall result is higher investment in P1. Like C, as the probability p of the payoff for P1 increases, the advantage of P1 over P2 and the expected value from investing the marginal P1 cash ﬂow in P3 (in its proﬁtable states) increase; however, the marginal cost of suboptimal investment may increase or decrease. Overall, as shown in Appendix A, allocation to P1 is directly related with p. From (6), increasing p or C increases the conditional cash ﬂow from assets in place directly via the increased proﬁtability of P1 and indirectly due to the higher investment in P1; to counter that, B* increases. Overall, the statics suggest that ﬁrms borrow more and invest proportionally more in “cashcows” like P1 as their expected payoffs increase, or new investment becomes more valuable. By contrast, ﬁrms invest proportionally more in projects which pay late like P2 as their expected payoffs increase; at the same time they may borrow less. Alternatively, these statics indicate that longterm debt is positively correlated with the proﬁtability of new investment or that of “cash-cows” in place.22 However, debt may be inversely related with the proﬁtability of assets in place with “longterm” payoffs. This is reminiscent of Myers’ (1993) discussion of a negative correlation between debt and asset proﬁtability and Kester’ (1986) empirical evidence of such relation for U.S. and Japanese ﬁrms. 4. Summary Existing theoretical literature often assumes that management may invest suboptimally. Overinvestment occurs when managers enjoy private beneﬁts of control that make negative NPV projects enticing. Underinvestment occurs when the beneﬁts of new investment accrue to bondholders. Existing empirical literature documents that internal funds are available to ﬁnance the majority of new investment by nonﬁnancial ﬁrms. This paper investigates how both long-term debt and the judicious selection of projects that do not provide internal funds can jointly reduce the problems of overand under-investment while at the same time ensuring adequate internal resources are available for follow-on investment. The analysis is based on the idea that as ﬁxed debt obligations reduce the cash ﬂow available for empire building, so does investment in assets that do not provide ﬁnancing for future investment via the internal capital market. Investment in these assets amounts to a managerial commitment to refrain from empire building and may be desirable even if such assets are expected to be less valuable than competing ones whose cash ﬂow is subject to managerial discretion. All else the same, higher investment in assets whose cash ﬂows are not available for future investment may reduce overinvestment but exacerbate underinvestment. But, so does increased use of debt. Hence, investment and ﬁnancing are inversely related in controlling suboptimal investment cost. A testable implication of the model is that among ﬁrms which have in place proportionally fewer assets with long gestation periods, those using more debt should be expected to have better operating performance than those with lower levels of debt. Also, it is shown that optimal debt choice precludes future suboptimal investment on average. Concurrently, optimal allocation to the project that provides funds for new investment via the internal market is shaped by several factors. Speciﬁcally, it increases with its value advantage over the alternate project that does not provide internal funds, the cash ﬂow variability of the alternate project, and the NPV of the project in which suboptimal investment may occur, but declines as the variability of its own cash ﬂow increases. Comparative statics suggest that ﬁrms which ﬁnance future investment internally and their management may invest suboptimally borrow more and have in place more “cash-cows” when the expected proﬁtability of these assets or future investment increases. The statics also suggest that such ﬁrms may borrow less while having in place more assets with “long-term” payoffs as the average proﬁtability of these assets increases.

22 Note that increased average proﬁtability due to a higher C also implies higher variability of P1; higher proﬁtability due to a higher p may imply higher (lower) variability of P1 if p > (<) 50%. Analogous arguments also apply to the proﬁtability of P2.

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Acknowledgements I am thankful for comments from an anonymous reviewer. I am indebted to George Kanatas and Robyn McLaughlin for useful discussions and comments. The encouragement of the Executive Editor, Kenneth Kopecky, and the guidance and input of the Editor, Robert Taggart, are greatly appreciated. Appendix A It is shown that short-term debt is optimally set equal to zero at t = 0. The proof parallels Hart and Moore’s (1995) proof of their Proposition 1. Speciﬁcally, assume the ﬁrm issues short-term debt at t = 0 which requires a payment b at t = 1. Then using the paper’s notation, the ﬁrm invests at t = 1 if Fi + (s − 1)wC ≥ B + b

(A1)

If (A1) holds, the total return to t = 0 claimholders, denoted with R, is R = Fi + (s − 1)wC

(A2)

of which t = 0 bondholders receive B + b and the rest goes to the shareholders. If (A1) does not hold and the manager cannot invest, his objective becomes to keep the ﬁrm as a going concern. This can happen as long as wC ≥ b

(A3)

in which case R = Fi

(A4)

If (A1) and (A3) do not hold, the manager liquidates the ﬁrm at t = 1 and, since P1 and P2 have zero salvage value, R = wC

(A5)

At t = 1, the manager’s ﬁrst choice is to invest if he has the money to do so. His second choice is to maintain the ﬁrm as a going concern if he cannot invest in P3 but can pay b. His third choice is to liquidate the ﬁrm. A low b is good because it reduces the probability of liquidation in (A3). Liquidation is undesirable because (A4) and (A5) imply that R is higher when the ﬁrm survives than when it liquidates. Hence, b is optimally set equal to zero. The comparative statics of w* and B* with respect to the problem parameters j ∈ {S, q, D, p, C} are now derived. For j = S, (8) gives wS∗ =

−VwS Vww

where, VwS =

pC 1 pC(S − 1)2 [2S(S − 1) − (S − 1)2 ] + 2pC − qD + 2 S 2S 2S

=

1 [pC(S − 1) + 2pC − qD] > 0. S

Since from footnote 14 Vww < 0, it follows that wS∗ > 0. Then from (6) and (2), BS∗ = [(1 + p)C − qD] wS∗ > 0 For j = q, (8) gives wq∗ =

−Vwq Vww

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where, recalling the deﬁnition of in footnote 17 and using (1), Vwq

p = −D − (1 − 2q)D2 2SC

1−

Y2 +

Y2

=

For 1 − 2q < 0, Vwq < 0. For 1 − 2q > 0, p 2SCq(1 − q)

Vwq =

p 2SCDq(1 − q) (1 − 2q) − . p 2SCq(1 − q)

(1 − 2q)[p(1 − p) − 4S − (S − 1)2 ]C 2 −

2q2 SCD p

<0

since, using footnote 21, the term in the squared brackets is always negative. Hence, wq∗ < 0. Then from (6), Bq∗ = (1 − w)D + [(1 + p)C − qD]wq∗ which can be either positive or negative. For j = D, (8) gives ∗ = wD

−VwD Vww

where, recalling the deﬁnition of in footnote 17 and using (1),

2 + Y2 − (2SC/p)(2pC − qD) − C 2 (S − 1)2 pq(1 − q)D 1− X 2 SC Y p qSD = p(1 − p)C − 4SC + − C(S − 1)2 . SD p

VwD = −q −

Given footnote 21, the following two cases are considered. First, if qD < 2pC, then VwD <

pC [p(1 − p) − 2S − (S − 1)2 ] ≡ < 0 SD

since the term in the squared brackets is always negative. Second, if qD > pC(1+(S/2)), then

VwD >

pC S 2 + 2S + 2 p(1 − p) − SD 2

≡<0

since the term in the squared brackets is always negative. Then, overall, 0 > > VwD > , implying ∗ < 0. Then from (6), wD ∗ ∗ = q(1 − w) + [(1 + p)C − qD]wD BD

which can be either positive or negative. For j = p, (8) gives wp∗ =

−Vwp Vww

where using (1), 2 C(S − 1)2 (2 − 3p)pC − − Y Vwp = 2C + 2S 2S 2SC 2SqD −p (1 − 2p)C − . = 2S p2

1−

X2 + Y2 − (2SC/p)(2pC − qD) − C 2 (S − 1)2 Y2

For 1 − 2p < 0, Vwp < 0. For 1 − 2p > 0, there are again two cases. First, if qD < 2pC, then

Vwp <

−pC 4S (1 − 2p) − 2S p

≡E>0

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since the term in the squared brackets is always negative. Second, if qD > pC(1 + (S/2)), then

Vwp >

2S −pC (1 − 2p) − 2S p

1+

S 2

≡H>0

since the term in the squared brackets is always negative. Then, overall, 0 < H < Vwp < E, implying wp∗ > 0. Then from (6), ∗ > 0. Bp∗ = wC + [(1 + p)C − qD]wD

Finally, for j = C, (8) gives wC∗ =

−VwC Vww

where using (1),

2 + Y2 − (2SC/p)(2pC − qD) − C 2 (S − 1)2 pY2 p(S − 1)2 p2 (1 − p) 1− X − + 2S 2S 2SC Y2 −p S = p(1 − p)C − (4pC − qD) − C(S − 1)2 . SC p

VwC = 2p +

Given footnote 21, the following two cases are considered. First, if qD < 2pC, then VwC >

−p p(1 − p) − 2S − (S − 1)2 ≡ K > 0 S

since the term in the squared brackets is always negative. Second, if qD > pC(1 + (S/2)), then

VwC <

−p S p(1 − p) − S 3 − S 2

− (S − 1)2 ≡ > 0

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