Capital structure as optimal contracts

North American Journal of Economics and Finance 10 (1999) 363–385

Capital structure as optimal contracts Chun Chang* Department of Finance, Carlson School of Management, University of Minnesota, Minneapolis, MN 55455, USA Received 1 August 1999; received in revised form 1 August 1999

Abstract Motivated by the existence of audited accounting income, this paper introduces observable income into an otherwise costly-state-verification model. It is shown that the optimal contract between the corporate insider and the outside investors can be interpreted as a combination of debt and equity. Testable implications are derived. © 1999 Elsevier Science Inc. All rights reserved.

1. Introduction One of the most important yet unresolved issues in the theory of financial economics concerns itself with corporate capital structure. Since the influential paper of Modigliani and Miller (1958) that showed the irrelevancy of firm’s financial structure in a perfect economy, many attempts have been made to introduce market imperfections, such as taxes, bankruptcy, information and agency costs, into the theory.1 These models yield a number of important insights. However, corporate capital structure remains a puzzle (Myers, 1985). One prominent future of most previous work on capital structure is that debt and equity are assumed to be the only feasible financial contracts available to firms. Firms can only issue debt and equity contracts. These two forms of financial contracts are exogeneously given and they are not allowed to use other forms of financial contracts. This assumption is unsatisfactory for three reasons. First, imposing contractual forms may produce “artificial” distortions that would not be there if the contracts were endogenously determined. Second, only by deriving financial contracts endogenously, can we understand the variety and complexity * Corresponding author. Tel.: ⫹1-612-624-8305; fax: ⫹1-612-626-1335. E-mail address: [email protected] (C. Chang) 1062-9408/99/$ – see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S 1 0 6 2 - 9 4 0 8 ( 9 9 ) 0 0 0 2 8 - 5

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of existing financial contracts. Third, by deriving financial contracts endogenously, we can better identify the underlying factors that give rise to the use of a particular form of financial contracts. And we can better understand how the changes in these factors will change the firm’s financial structure. The derivation of optimal financial contracts has been the subject of a number of papers (Townsend, 1979; Diamond, 1984; Gale and Hellwig, 1985). In these models, the ownermanager privately observe the cash flows of the firm. Outside investors can only verify the cash flows ex post at a cost. The optimal contract in this setting can be interpreted as a debt contract. These models have been criticized for not being able to portrait the behavior of publicly-held corporations where some information about their cash flows are publicly available and where outside equity is also issued. This paper represents a preliminary step toward establishing a theory of capital structure that imposes no ad hoc restrictions on the contractual forms.2 I present a model in which the optimal financial contract is derived from tastes, technologies and endowments. In the model a corporate insider (manager) needs funds to finance a project. The funds are to be raised from outside investors. In contract to Townsend-type models, I assume that the project’s future payoff consists of two parts. One of which is readily observable (through disclosure, for example). The other part is only verifiable ex post at a cost. Without a verification, the use of the second part is subject to the manager’s discretion. She can use it to pursue her own interest. The manager is assumed to maximize what is called “corporate wealth”,3 total cash flow minus the amount paid to the investors, subject to the capital supply constraint that the outside investors are given the opportunity cost of their capital. With higher “corporate wealth”, the manager can enjoy less capital market scrutiny, better relationship with the employees by paying them higher wages or bonus, and higher salary and perquisites themselves. In the environment just described, the ex ante optimal contract between the manager and the investors is derived. Under the assumption that the bankruptcy cost is increasing in the value of the firm’s assets, bankruptcy is optimally designed to take place when the observable part of the firm’s return is low. The optimal contract is such that the manager precommits herself to pay a fixed amount. If the observable part of the firm’s return is below the fixed amount, the manager will have to sacrifice some or all of the discretionary part to make up for the precommitted amount. Bankruptcy will occur if the total payment to the investors is below this amount. If the observable part is above the fixed amount, the investors are only paid the observable part of the firm’s return. The optimal contract derived can be interpreted as a combination of debt and equity contracts. However, the interpretation is exact when a specific distribution assumption is made. The equityholders are entitled to the observable part of the firm’s cash flow that is in excess of the precommitted amount. They are, however, not endowed with the right of declaring bankruptcy. The debtholders are endowed with this right and the priority to claim the firm’s cash flow when a bankruptcy occurs. In addition to a more rigorous formulation, this model identifies some important determinants of corporate capital structure. Among them are: the firm’s profitability, the riskiness of its observable earnings, its level of disclosure, its tax rate and nondebt tax shields, the

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default-free rate of return, and the magnitude of bankruptcy cost. For each of these factors, comparative static results are derived. Some of these implications, such as those of disclosure and default-free rate, to my knowledge, have not been studied in the literature. Some of them are different from those of existing models. For example, most signalling models predict that more profitable firms use more debt. This model predicts the opposite. For the past two decades or so, the formal economic study of contractual relationship has made substantial progress. However, the emphasis is on the issue of how to align efficiently the interest of agents with that of principals, where the agents are the ones who work or make decisions on the behalf of the principals. The contracts thus derived are interpreted as compensation or labor contracts (for a review see Hart and Holmstrom, 1985). At the same time, there are only a few attempts to study financial contracts formally. Townsend (1979), Diamond (1984), and Gale and Hellwig (1985) have shown debt contract is the only optimal contract when lenders cannot observe borrowers’ income without costs. Because the optimal contracts do not exhibit features of equity contracts, their models are not rich enough to address the issue of capital structure in general. They do, however, serve as an important starting point of my model.4 Since this paper was initially written, the literature of financial contracting has grown substantially (for reviews see Harris and Raviv, 1992; Allen and Winton, 1995). Most of the new work, however, has not followed the costly state verification setting with the exception of Boyd and Smith (1998). Boyd and Smith (1998) have independently derived debt and equity as optimal contracts when the firms have two types of projects: one whose return is publicly observable and one whose return is only observable at a cost. They also edogenize the firm’s investment choice between the two types of projects. Because their setup is somewhat different, they derive quite different results. The plan of the paper is as follows. In Section 2, the model and the assumptions are presented and the optimal contract is derived. In Section 3 the optimal contract is interpreted as a combination of debt and equity. The comparative static results concerning the optimal amount of debt are derived in Section 4. In Section 5, we extend our basic model: first we allow the firm to choose its level of disclosure, and a theory of corporate ownership structure is presented; then we introduce corporate taxes into the model. Other extensions are also considered. The paper concludes with some empirical observations and possible future research directions. All proofs are in the Appendix. Also included in the Appendix is a numerical example that shows the model can generate reasonable debt ratios for relatively small bankruptcy cost and small managerial discretion.

2. The model and the characterization of optimal contract Our model is two date economy. An agent named manager is endowed with an investment project, but has no money. The project requires an investment of $1 on date one and yields Y (dollars) on date two, where Y is a random variable. Other agents are called investors and each of them is endowed with a fraction of a dollar on date one. Since those agents who invest are identical, they will be referred to as the outsider or the investor. All agents are risk neutral and consumption takes place at the end of date two (the case of risk averse manager

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will be considered in Section 6). The investor is indifferent between $1 on date one and $(1 ⫹ i) on date two, where i is an exogenously given default free rate. We’ll be interested in characterizing the agreement between the manager and the investor on the division of the ex post profit y (lowercase letters are used to denote the realization of a random variable). I assume that Y can be decomposed into two parts: Y ⫽ X ⫹ Z, where X and Z are both random variables. The realization of X, x, is observed by both parties without cost and the realization of Z, z, is observed only by the manager and it is only verifiable by the investor at a cost. In our model, x may be interpreted as audited earnings of the firm. The amount that the manager “consumes” is the difference between Y and whatever is paid out to the investor. This amount is the sum of the manager’s compensation, the perquisite they consume and the money they spends pursuing goals other than the investor’s wealth maximization. For example, the manager is interested in maximizing the size of the firm, they want to retain and reinvest as much earnings as possible. Z may represent the portion of the firm’s cash flow that can be potentially invested in the projects that have no benefit to the investor.5 The decomposition of Y into X and Z captures the notion that outside investors have some ideas about what is going on in a firm, but cannot control everything. Suppose that Y is distributed on [L, ⬁), where L ⱖ 0. I assume that the discretionary term z is never negative, namely, Z has a support of [0, ⬁). As a result X is also distributed on [L, ⬁). Let F( ) and G( ) be the distribution functions for X and Z, respectively. For the moment X and Z are assumed to be statistically independent. Note that the verification cost is paid ex post. One may think that the firm can also change the ex ante decomposition of Y by incurring some (say, disclosure) cost. This case is studied in Section 5 where we’ll investigate how the firm chooses its ownership structure. The verification cost is assumed to be a function of the total return y, and is denoted by b( y). One can think that the firm’s assets generating X and Z are intertwined in a complicated way, the verification cost also depends on x. To avoid degeneracy, I shall assume that F⬘( x) ⫽ f( x) ⬎ 0 for all x in [L, ⬁), though the case of f( x) ⫽ 0 can be readily incorporated into the analysis. I also assume that F( x) and G( x) are sufficiently smooth. To guarantee the existence and the uniqueness of an optimal contract, we need some joint assumptions about the verification cost b( y) and the hazard rate for Z, g( z)/[1 ⫺ G( z)], where g( z) is the density function of Z. A.1. 1 ⱖ b⬘( y) ⬎ 0 for all y in [L, ⬁) and g( z)/[1 ⫺ G( z)] is nondecreasing. A.2. b⬘( y) ⬍ 0 for all y in [L, ⬁) and g( z)/[1 ⫺ G( z)] is nonincreasing. From the empirical evidence presented in Warner (1977) and Altman (1984), it seems reasonable to assume that bankruptcy cost is increasing in the firm’s value. Most commonly encountered distribution functions such as normal, uniform, exponential, LaPlace, Gamma, and Weilbull with degree of freedom greater than one all have the property that the hazard rate is nondecreasing. Therefore, A.1. seems to be a quite plausible assumption. In some other cases, however, one might also argue that the lower is the firm’s return, the more likely is it to incur agency costs (indirect bankruptcy costs). This situation may be described by A.2. Under either assumption, the same results are derived. The order in which the events take place is as follows: On date one, a contract between

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the manager and the investor is designed and it is sold to the investor for $1 needed for the investment; on date two, X and Z are realized and the manager reports earnings; then based on the report, the investor decides whether to verify or not, and the payment is made to the investor; finally the investor consumes what is paid to them, and the manager consumes the realized Y net of what is paid to the investor and the verification cost (if a verification occurs). For the moment, I assume, as other works on contracting, that the investor can ex ante commit themself to an verification schedule. That is, if he is called upon to verify ex post according to what is specified in the contract, he will do so regardless of the cost. Later in Section 6, I shall discuss how this assumption may be relaxed. An ex ante agreement (contract) on the division of y can be made contingent on x without verification and can only be made contingent on z with verification. Therefore, a contract can be represented by two schedules R( x, z) and D( x, z). R( x, z) specifies the payment made by the manager to the investor net of possible verification cost, and D( x, z) is the verification schedule. When x occurs and z is the reported realization of Z, we denote by D( x, z) ⫽ 1 a decision to verify and by D( x, z) ⫽ 0 a decision not to verify. Randomized verification strategies are not considered here. Ex ante the manager will choose schedules that maximize their expected utility subject to the constraint that the expected payment to the investor is at least 1 ⫹ i (in date $2), the opportunity cost to the investor. Note, that for an optimal contract, the individual rationality constraint will always be binding. That is, the expected payment to the investor will be 1 ⫹ i for the optimal contract. Because the expected utility of the manager is the expected value of Y minus the expected payment to the investor and the expected verification cost, the maximization problem is equivalent to the problem of minimizing the expected verification cost subject to the same individual rationality constraint. For a given x, the situation is similar to that in Townsend (1979) or Gale and Hellwig (1985). Because the investor does not know anything more beyond the fact that y is at least x, to prevent the manager from under-reporting the true earnings, a fixed payment R( x), independent of the reported z, is demanded by the investor as long as the reported earnings x ⫹ z are above it. An verification occurs if and only if the reported earnings fall below it (i.e., the fixed payment is not met). Consequently, the schedule D( x, z) is completely determined by this R( x). Note that because X is publicly observable, the contract can be made contingent on x. This establishes: P 1: For any x in [L, ⬁), there exists a number R( x) depending on x such that an optimal contract satisfies: i. R( x, z) ⫽ R( x) for y ⫽ x ⫹ z ⱖ R( x); R( x, z) ⫽ x ⫹ z ⫺ b( x ⫹ z) otherwise. ii. D( x, z) ⫽ 0 for y ⫽ x ⫹ z ⱖ R( x); D( x, z) ⫽ 1 otherwise. According to Proposition 1, if the observable x is less than R( x) but the realized z is high such that x ⫹ z ⱖ R( x), the manager will give up part of the unobservables z to meet the obligation R( x) instead of undergoing an verification, in which case they would get nothing. If both the observable x and unobservables z are low, namely, x ⫹ z ⬍ R( x), the manager has no way to pay R( x) and an verification will occur, the investor will get y ⫺ b( y). x)⫺x By the proposition, for a given x the expected payoff to the investor is 兰 R( [x ⫹ z ⫺ 0

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b( x ⫹ z) dG( z) ⫹ R( x)[1 ⫺ G(R( x) ⫺ x)]. So the ex ante individual rationality constraint is:

冕冕 ⬁

L

R共 x兲⫺x

关 x ⫹ z ⫺ b共 x ⫹ z兲兴 dG共 z兲 ⫹ R共 x兲关1 ⫺ G共R共 x兲 ⫺ x兲兴 dF共 x兲 ⱖ 1 ⫹ i

0

(1) Because the realized return y ⫽ x ⫹ z is at least x and a payment of x can be verified without any cost, if E(X) ⱖ 1 ⫹ i, a contract specifying R( x) ⫽ x will provide the investor opportunity cost and no verification will be needed. If E(X) ⬍ 1 ⫹ i, then R( x) has to be raised above x for at least some x in [L, ⬁) to compensate for the investor, and verification will occur with positive probability. Suppose that R( x) ⬍ x for some x in [L, ⬁), we can raise R( x) to x. The cost to verify this is zero because it is common knowledge that the firm’s true return is at least x. But now the increased expected payment can be used to reduce the payment in states for which R( x) ⬎ x. The expected verification cost can be reduced, and the ex ante utility of the manager is increased. This result is stated in 2.1. Lemma 1 Suppose E(X) ⬍ 1 ⫹ i, then there is a positive probability that a verification will occur, and an optimal contract will always entail R( x) ⱖ x. Now we are ready to characterize the optimal schedule R( x). The manager’s optimization problem can be formally written as follows:

冕冕 ⬁

Min R共 兲

L

R共 x兲⫺x

b共 x ⫹ z兲 dG共 z兲 dF共 x兲

(2)

0

s.t. 共1兲 and R共 x兲 ⱖ x By Lemma 1, R( x) ⱖ x can be imposed as a constraint in Eq. (2). This problem can be solved by using variational method. Suppose that function R( x) solves the problem Eq. (2). The necessary condition for R( x) to satisfy can be derived as follows. Let r( x) be a feasible perturbation on R( ), that is, for all e sufficiently close to 0, R( x) ⫹ er( x) satisfies the constraints. Define: J共e兲 ⫽

冕冕 ⬁

L

R共 x兲⫹er共 x兲⫺x

关b共 x ⫹ z兲 ⫺ ␭ 共 x ⫹ z ⫺ b共 x ⫹ z兲兲兴 dG共 z兲 ⫺ ␭ 关R共 x兲

0

⫹ er共 x兲兴关1 ⫺ G共R共 x兲 ⫹ er共 x兲 ⫺ x兲兴} dF共 x兲 ⫹ ␭

(3)

This is the Lagrangian formed from Eq. (2) by ignoring R( x) ⱖ x and by replacing R( x) with R( x) ⫹ er( x). Denote the density function of Z by g( z), then a necessary condition for R( ) to be optimal is:

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Fig. 1.

J⬘共0兲 ⫽

冕

⬁

兵共1 ⫹ ␭ 兲b共R共 x兲兲 g共R共 x兲 ⫺ x兲 ⫺ ␭ 关1 ⫺ G共R共 x兲 ⫺ x兲兴其r共 x兲 dF共 x兲 ⱖ 0

L

(4) For the portion of R( ) such that R( x) ⬎ x, r( ) can be any function. Eq. (4) implies that for R( x) ⬎ x 共1 ⫹ ␭ 兲b共R共 x兲兲 g共R共 x兲 ⫺ x兲 ⫺ ␭ 关1 ⫺ G共R共 x兲 ⫺ x兲兴 ⫽ 0

(5)

By differentiating Eq. (5) with respect to x, we can verify (see Lemma 2) that 0 ⱕ R⬘( x) ⬍ 1 for any x such that x ⬍ R( x) under either A.1. or A.2. So eventually the portion of the curve for which R( x) ⬎ x will hit the portion for which (if there is one). Furthermore, this also implies that R( x) ⬎ x can happen only when x is low, that is, in an interval [0, B) where B is the first x such that R( x) ⫽ x. Therefore, any verification can only occur when x is less than B and the reported z is less than R( x) ⫺ x. When x ⱖ B the manager pays x and no verification occurs. Thus we know the general shape of an optimal R( x). It is shown in Fig. 1. An interesting case is when the hazard rate of Z is a constant d. That is, Z is exponentially distributed with parameter d. In this case R( x) is a constant when R( x) ⬎ x. This constant is B ⫽ b ⫺1 ( ␭ /[(1 ⫹ ␭ )d]) where b ⫺1 ( ) is the inverse function of b( ). This situation is illustrated in Fig. 2. If the verification cost is proportional to the return, i.e., b( y) ⫽ yb where b is a constant in (0, 1), then B ⫽ ␭ /[(1 ⫹ ␭ )bd]. Now the only thing left is the determination of B. Once B is determined, since R(B) ⫽ B, Eq. (5) is reduced to: b共R共 x兲兲 g共R共 x兲 ⫺ x兲 ⫺ g共0兲b共B兲关1 ⫺ G共R共 x兲 ⫺ x兲兴 ⫽ 0

(6)

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Fig. 2.

Eq. (6) has a unique solution for R( ) under A.1. or A.2., hence R( x) can be exactly located. R( ) determined from Eq. (6) depends on B, let me denote by RB( x) the partial derivative of R( x) with respect to B. Note that by Eqs. (5) and (6), b(B) ⫽ ␭ /[(1 ⫹ ␭ ) g(0)]. Because ␭ ⬎ 0, 0 ⬍ b(B) ⬍ 1/g(0). When b is monotone, this implies that B is in some interval. Let’s denote the intersection of this interval and the interval [L, ⬁) by I F , meaning the feasible interval for B. Intuitively, if B were extremely high, verification would occur in almost all the cases. A further increase in B would not raise the expected payoff to the investor. So the optimal B must be in a bounded interval. 2.2. Lemma 2 Under A.1. or A.2., for each B in I F , there is a unique solution R( x) to Eq. (6) and it satisfies 0 ⱕ R⬘( x) ⬍ 1 and RB( x) ⬎ 0 for all x in [L, B]. Fig. 1 shows the required payment schedule to the investor as a function of the observable x when the hazard rate of Z is strictly monotone. Fig. 2 shows the required payment schedule to the investor as a function of the observable x when the hazard rate of Z is a constant. RB( x) ⬎ 0 for all x in [L, B] means that a decrease in B will lower the schedule R( x) uniformly. So to minimize the expected cost of verification, B should be chosen as low as possible provided that the investor’s opportunity cost is recovered. Therefore, the optimal B is the minimum B such that the investor’s individual rationality constraint is binding. Note that the expected return to the investor will not always be raised by an increase in B. In fact, when B is raised beyond the interval I F , the expected payoff to the investor will decline. If no B in I F can meet the investor’s expected rate of return, then no contract is feasible and the project is not profitable enough to be undertaken. The characterization is thus completed. We summarize these results in the following proposition. P 2: Suppose that the project is sufficiently profitable and E(X) ⬍ 1 ⫹ i, then under either A.1. or A.2., there exists a unique optimal contract. Its payment schedule R( x) is charac-

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terized as follows: there exists a number B in [L, ⬁) such that R( x) ⬎ x for x in [L, B) and R( x) ⫽ x otherwise; in the interval [L, B), R( x) is determined by equation (5); and the optimum is achieved at the minimum B such that the investor’s individual rationality constraint Eq. (1) is binding.

3. Interpretations I shall interpret a verification as bankruptcy. At the heart of this interpretation is the bankruptcy-declaring party’s rights to demand their claims be completely paid and to deprive any managerial control and any managerial discretion by liquidating the firm’s assets. Although bankruptcy occurred in the real world is much more complicated, this function of bankruptcy is a major one. Bankruptcy cost in practice may involve more than the cost paid to the lawyers, the court and the accountants. The verification cost in our model can be thought of as consisting of these direct bankruptcy costs and the agency costs associated with adverse incentives when the firm is in a financial distress. An equity contract is defined to be a payment schedule that is contingent on the nondiscretionary part of the firm’s cash flow and that is subordinate to any debt contract. The equityholders are not endowed with the right to declare a bankruptcy. Therefore the payoff to the equityholders is subject to the managerial discretion within the constraints of auditing, disclosing, and monitoring. To an ordinary outside shareholder, the most significant change he can hope for or impose is to replace the current management with a new, perhaps equally discretionary, management. First, let us consider the case of exponentially distributed Z, the optimal contract derived in the last section can be interpreted as a convertible bond in which the manager promises to pay B (principal plus coupon), the bondholders has the option to surrender the bond and to become the firm’s sole equityholders (there are no equityholders before they convert). The optimal contract can also be viewed as a combination of debt and equity. The debtholders have the right to demand a fixed payment of B and when x is above B, the equityholders have the right to claim the reported residual x ⫺ B which is the “true” residual y ⫺ B net of the manager’s discretionary consumption z. If Z is not exponential, R( x) will be increasing for R( x) ⬎ x. Therefore, as x becomes lower, the required payment is reduced accordingly. In practice, when a firm has been having a difficult time, very often the bondholders of the firm are willing to make concessions and to accept a payment that is below the originally promised payment without forcing the firm into bankruptcy. To the extent that the contracting parties understand this when the contract is signed, we may still interpret the contract as a combination of debt and equity. Note that although R( x) can decrease as x decreases for x ⬍ B, it does not decrease as fast as x does. That is, the investor is willing to make small concessions, so R( x) is still strictly above x for x ⬍ B and bankruptcy is still possible. Therefore interpreting B as the level of debt is appropriate. In the remainder of the paper, the results will be stated in terms of equity-debt-mixture interpretation, they can be easily reinterpreted in terms of convertible bonds. To compare the magnitudes of leverage in market value, we need to know the market values of the equity and debt contracts on date one. Since the shareholders get x ⫺ B whenever x is above B, the

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market value of total shares is V e ⫽ 兰 ⬁ B ( x ⫺ B) dF( x)/(1 ⫹ i). Because the total market value of the firm’s financial package on date one equals to its investment, which is one, the market value of the bonds is V d ⫽ 1 ⫺ V e ⫽ 1 ⫺ 兰 ⬁ B ( x ⫺ B) dF( x)/(1 ⫹ i) and V d is also the firm’s debt to asset ratio. Note that V e is decreasing in B and V d is increasing in B. That is, when the promised payment on the debt B increases, the amount of debt increases relative to the amount of equity. At the end of the Appendix, an example with uniformly distributed X and exponentially distributed Z is worked out. For various parameter values of the distributions and the bankruptcy cost, and for different face value B, the default-free rate i that can be achieved by B is calculated by using Eq. (1). Then the optimal debt ratio V d is calculated. The results are tabulated. We can see even when the bankruptcy cost is small and/or when the manager’s discretionary power is small (less than 10% of the total expected return), the model generates significant debt to equity ratios. 4. Comparative statics The benefit usually attributed to the use of debt is the tax advantage. In our model the cost associated with debt is the cost of an verification (bankruptcy) and there is no tax so far in the model. The reason that the debt is used is its ability to curb the discretionary power of the manager. If the manager has a large discretionary power beyond the publicly observable information, an all equity financing, which is preferable because of no dead weight loss, may not be viable (this is the case where E(X) ⬍ 1 ⫹ i). By committing ex ante to pay a fixed amount or else being subject to a bankruptcy, the manager’s ability of getting away with a large amount of discretionary power ex post is limited. As a result, she is better off ex ante. Of course, using debt is costly because of the bankruptcy cost, it should be used as little as possible provided that the investor gets his opportunity cost of capital. The most significant factor in explaining the cross-sectional behavior of firms’ capital structure seems to be the firms’ profitability (see Kester, 1986; Raymar, 1986). In our model, if the project is more profitable in the sense that either X or Z or both are stochastically larger (in the first order stochastic dominance sense), then it is easier to meet the investor’ required rate of return. The investor is willing to supply the capital even faced with less precommitment from the manager. P 3: Under A.1. or A.2., an increase in either X or Z or both in the first order stochastic dominance sense, whereas holding everything else constant, will lower B, hence V d . The following result says that under A.1., more risky is the observable X (represented by a mean-preserving-spread of X), whereas holding Z the same, the less debt the firm should issue, provided that both the verification cost and the hazard rate of Z are concave. P 4: Under A.1., suppose that b⬘ ⱕ 1, b⬙ ⱕ 0, and the hazard rate of Z is increasing and concave, then other things being equal, the firm with more risky X has lower B. The interpretation of this proposition deserves some caution. In equilibrium, the firm’s reported earnings are not exactly x because the manager may have to give up part of z and report them together with x to meet the required debt payment when x is low. The actual earnings reported may be higher than x when x is less than B. When Z is held the same, however, an increase in the volatility of X is likely to correspond to an increase in the

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volatility of the reported earnings. If this is indeed the case, we can say that the firms with more risky reported earnings have less debt. Now let’s hold the total return Y constant and see how different decompositions of Y into X and Z affect the optimal debt ratio. More precisely, we want to know how an increasing (in the first order stochastic dominance sense) in the manager’s discretionary power (namely, a decrease in X) changes the debt ratio V d . P 5: An increase in the manager’s discretionary power Z, whereas holding the total return Y the same, will increase the optimal amount of debt V d . This result is not surprising since the reason debt enters into this model is to limit the manager’s discretionary power. This result may be used to compare capital structures in different countries that have different institutional arrangements such as different disclosure law. P 6: An increase in the default-free rate i will raise the firm’s debt ratio V d . If there are many managers who are competing for the right to control corporations and the market for corporate control is competitive, then the firms in which the managers have high discretionary power will be the takeover targets of other managers. These other managers can give investors a higher i through tender-offers. If the market for corporate control is more competitive than the market for capital, i will rise so that the managers earn a competitive return in terms of discretionary power. According to Proposition 6, the firms’ debt ratio will be higher than it would be otherwise. (see Harris and Raviv, 1986, for a similar result). This result can also be used to compare the periods in which the real rate of return is low with those in which the real rate of return is high. People have questioned the validity of tax advantage-bankruptcy cost models of capital structure. It has been argued that the observed magnitudes of bankruptcy costs do not seem to be high enough to explain the low levels of debt actually used by some firms (see Miller, 1977). Our results, on the contrary, indicate that if the firm’s disclosure policy (the division of Y into X and Z) is held constant, then lower bankruptcy cost is associated with less debt. P 7: Ceteris paribus, if the verification cost function b( ) is replaced with c( ) and b( y) ⬍ c( y) for all y, then the resulting B, hence the amount of debt V d , is higher. This seemingly counter-intuitive result can be understood as follows. In the tax advantagebankruptcy cost trade-off model, debt is substituted for equity to the point where the marginal cost of debt equals that of equity. An increase in the bankruptcy cost would raise the marginal cost of debt, so more equity is substituted for debt. In this model, contractual forms are constrained by the information (contracting) cost, the firm cannot freely issue any contract it wants to. For example, if the firm discloses nothing and if it is prohibitively costly for the outside investors to monitor (X ⫽ 0), then the only feasible outside financing will be through a debt contract, the cost of issuing equity will be extremely high. Faced with an increase in the bankruptcy cost, the firm cannot substitute equity for debt. To raise the capital necessary for the investment, the firm has to promise a higher face value on the debt to compensate the investor for the lowered expected payoff in the bankruptcy states. 5. Disclosure, corporate ownership structure, taxes, and other extensions So far we have taken the decomposition of Y into X and Z, namely, the discretionary power of the manager, as given. To certain extent, the discretionary power is determined by

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the firm’s disclosing (auditing) and/or by the investors’ monitoring activities (hereafter I’ll refer to these activities as disclosure for simplicity). For example, a private firm can decide whether to go public, hence to comply with more stringent disclosure requirements. A public firm can decide to do the opposite. In these cases, the managerial discretion is affected by corporate ownership structure. To the extent that corporations change their capital structures more often than their ownership structures, taking the managerial discretion as given is a good assumption to begin with. Note that strictly speaking, it does not make much sense talking about who owns the firm if the firm is viewed as a nexus of contracts as we do (see Klein, Crawford, and Alchian, 1978; Fama, 1980). Here by “corporate ownership structure”, I refer to the extent to which information concerning the firm’s operation is disclosed. Of course, disclosure (auditing and monitoring included) is not free. To reduce the managerial discretion, more efforts are needed. Perhaps more significantly, as more information about the firm is disclosed, more proprietary information may become available to the firm’s competitors. The firm may suffer more loss as a result. To simplify the notations, we assume that Z is exponentially distributed here so that it can be represented by the parameter d. Let’s denote by C(d) the total costs of disclosure, auditing and monitoring. More disclosure means higher d or lower Z. Assume C⬘(d) ⬎ 0, namely, it costs more to disclose more. Previously since d was fixed, this cost could be included in i, the cost of capital, without loss of generality. Now since d is a choice variable, the total capital needed by the project is 1 ⫹ i ⫹ C(d), where d is the level of disclosure the manager controls. For simplicity I have assumed the auditing or disclosure cost C(d) is paid in advance, i.e., before the realization of Y. (For this to be true, the lowest realization of Y, L, has to be above C(⬁).) For every d, by applying the previous results, there is an optimal amount of debt B(d) associated with it. Now overall optimum is achieved at a d* which minimizes the expected bankruptcy cost plus the monitoring cost C(d). d* can be higher than the disclosure level required by law. This justifies why a firm may voluntarily release more information than what is required. If d* is lower than the level required by law, imposing disclosure law may be inefficient. To state it formally, the overall optimization problem faced by the manager is:

冕 冕 B共d兲

Min C共d兲 ⫹ d⬎0

0

B共d兲

b共 y兲 dH共 y兲 dG共 z兩d兲

z

冕 再 冋冕 冕 ⬁

s.t.

B共d兲

I 兵 z⬍B其

0

z

⫹ I 兵 zⱖB其

⬁

z

ⱖ 1 ⫹ i ⫹ C共d兲

关 y ⫺ b共 y兲兴 dH共 y兲 ⫹

冎

冕

⬁

B共d兲

R共 y ⫺ z兲 dH共 y兲

册

R共 y ⫺ z兲 dH dG共 z兩d兲 (7)

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where the left hand side of the constraint is the payoff function to the investor written in terms of the distribution functions H( y) and G( z兩d) of Y and Z, respectively (see the proof of Proposition 6 for the derivation of this constraint). R( ) is determined by Eq. (6). We have assumed that the manager has no money at all. If the manager had some initial wealth w, then it is easy to show that she would invest all her wealth w and raise 1 ⫺ w, from the outside investors (see Gale and Hellwig, 1985). This is because the manager in this model is the “real residual claimant” and bears the expected bankruptcy cost, a dead-weight loss. If the expected rate of return from investing in the project is at least as good as the market rate i, by investing all her money in the project, they would reduce the expected bankruptcy cost. In fact if w was more than 1, they could internalize the problem by not having to raise any external capital. When the manager’s wealth is relatively high with respect to 1, it can provide a large cushion for the funds raised externally so that the expected cost of bankruptcy is low. If it is low enough relative to the cost of disclosure, it’s optimal for the firm not to release any information and to be a private firm. In terms of our model, this is the case when Y ⫽ Z. Note that in this case the optimal external financial contract consists only of debt contract, the manager’s position here is the same as that of an inside shareholder (owner-manager). On the other hand, when the manager’s wealth is insignificant relative to the total funds needed, disclosing no information, hence using an all-debt financing would cause the expected bankruptcy cost relative high with respect to the cost of disclosure. Therefore there are incentives to make some information public so that external equity financing can become feasible. When the incentives are strong enough, the firm will go public. The outside financing will be divided into equity and debt. The equity ownership and the control are generally separated, but the use of debt as a precommitment overcomes the incentive problem. Thus we have provided a theory of corporate ownership structure. Another interesting extension is to see how the optimal contract changes when the taxes are introduced. For simplicity I assume the firm is taxed at a fixed rate t, there is no personal tax and the interest payment on debt is tax-deductible. If only X is publicly observable, the firm can only be taxed on the amount of X in excess of the coupon payment on the bond. Let r be the coupon payment. Note that the model does not say how the required payment B is divided into principal and coupon as long as they add up to B. The only assumption we shall make about the coupon payment is that r is increasing in B, namely, more debt implies more coupon payment. It is assumed that the discretionary term z is not observable by the IRS, so it is not taxed. When the corporate taxes are imposed, let’s denote by W(t, B) the earnings of the firm after taxes, where t is the tax rate and B is the level of debt. If the realized x is above B after taxes, namely, x(1 ⫺ t) ⫹ tr ⱖ B, then W(t, B) ⫽ x(1 ⫺ t) ⫹ tr. If x is low such that x(1 ⫺ t) ⫹ tr ⬍ B, but the unobservable z is large enough so that ( x ⫹ z)(1 ⫺ t) ⫹ tr ⱖ B, then the manager can avoid bankruptcy by reporting part of z such that the after tax earnings are exactly B, W(t, B) ⫽ B. If both x and z are low such that ( x ⫹ z)(1 ⫺ t) ⫹ tr ⬍ B, but y is above the interest deduction r, then W(t, B) ⫽ y(1 ⫺ t) ⫹ tr and the firm is bankrupt after taxes are paid. Finally, if y ⬍ r, then the firm dose not pay any taxes and it defaults, W(t, B) ⫽ y. To summarize:

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W共t, B兲 ⫽

冤

x共1 ⫺ t兲 ⫹ tr B y共1 ⫺ t兲 ⫹ rt y

x ⱖ 共B ⫹ rt兲/共1 ⫺ t兲 y ⱖ 共B ⫹ rt兲/共1 ⫺ t兲 共B ⫹ tr兲/共1 ⫺ t兲 ⱖ y ⱖ r r⬎y

冥

(8)

With taxes, W is the payment to the investor after taxes (before bankruptcy cost). Now if the tax rate is increased from t to t⬘, we shall show that the resulting optimal debt level is higher. Suppose the resulting level of debt B⬘ is not higher, B⬘ ⱕ B, then r⬘ ⬍ r and it is straightforward to verify from above formula that W(t⬘, B⬘) ⬍ W(t, B) for all possible values of x and y, such that y ⱖ x. This will imply that the expected return to the investor is less than 1 ⫹ i, a contradiction. The kind of non-debt tax shields considered by DeAngelo and Masulis (1980) can also be incorporated into the function W above. It can be shown that an increase in the firm’s nondebt deduction or in its tax credit will increase W, hence lower the optimal amount of debt. P 8: Suppose that the coupon payment increases with the amount of debt, then, other things being equal, an increase in the corporate tax rate will increase the optimal amount of debt whereas an increase in the firm’s nondebt tax shields will reduce it. Although this result is consistent with other models of capital structure, the reason is different. Other things being equal, an increase in the corporate tax rate or a decrease in the firm’s nondebt tax shields represent an erosion on the firm’s profitability. As a result, the level of debt has to be raised to compensate the investor. If the manager is risk averse, for a given x the optimal will still be a debt contract except that the manager may keep a constant amount when an verification occurs (see Townsend, 1979). In our framework, this constant amount will also be independent of x since the investor can give the manager complete insurance when a bankruptcy occurs. So risk aversion on the part of manager can be readily incorporated into the analysis. Of course, the optimal debt to equity ratio will depend on the manager’s attitude towards risk. This will substantially complicate the comparative statics. We have assumed that X and Z are independent for simplicity. In general, both of them can depend on the realization of Y. Because Y ⫽ X ⫹ Z, we can equivalently write Y and Z conditioned on X. Therefore without loss of generality, we can assume that the distribution of Z depend on the realization of X, G( z兩x). Let h( z, x) denote the hazard rate function g( z兩x)/[1 ⫺ G( z兩x)]. Now if instead of A.1. we assume that b⬘ ⬎ 0 and 0 ⱕ h 1 ⫺ h 2 ⱕ b⬘h ⫹ bh 1 , where hi stands for the partial derivative of h with respect to the i-th independent variable; and instead of A.2., we assume that b⬘ ⬍ 0 and 0 ⱖ h 1 ⫺ h 2 ⱖ b⬘h ⫹ bh 1 ; then the main characterization results are essentially unchanged. Under either of these assumptions, other results in the paper can be retained after some minor adjustments. In the above analysis, the investor were assumed to be able to commit themselves ex ante to the verification schedules. But if the manager always tells the truth as expected, why should the investor engage in a costly verification ex post. On the other hand, if the manager knows that the investor cannot commit to the verification schedules, she may have incentives to lie. Therefore we have an issue of time-consistency here. This is a common problem associated with a lot of contracting models. Fortunately, in our model, the investor’s commitment to the verification schedule can be rationalized by the recent work on nonco-

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operative bargaining or durable good monopoly. Suppose that the manager fails to meet the required payment (either on date one or on date two) specified by the optimal contract. However, the investor still does not know the firm’s realized return except that it is below some fixed amount. The investor has two options: either to propose a division on the reported realized return without verifying it; or to take an verification and get the true return net of verification cost. If the first option is adopted, the situation is like that of bargaining with incomplete information. The investor can be viewed as the player who does not know the true valuations of the other player (the manager). The investor proposes a payment, the manager can either accept or reject. If the proposal is rejected, then the investor makes another proposal until the manager finally accepts. As shown by Fudenberg et al. (1985), Grossman and Perry (1986), and Gul et al. (1986), the equilibrium behavior in this kind of situations typically involves bluffing and delaying in reaching an agreement, hence a loss of efficiency due to delayed consumptions. If the bankruptcy cost is not too large, the investor would be better off enforcing a bankruptcy than bargaining with the manager and trying to reach an agreement. So the second option will be taken. In equilibrium, the manager with a realized return that is below the minimum payment levels will always face an verification and will receive nothing. Nevertheless, in such a situation the manager can be viewed as always reporting L, the lowest possible realization, hoping to get some edge in case an offequilibrium event occurs.

6. Discussion and conclusion Economists have noticed that firms in Japan have been systematically using more debt than their counterparts in the US, despite no significant difference in the tax codes between the two countries. People have attributed the difference in the leverage ratios to the difference in the institutional framework: A strong tie between firms and banks in Japan. If we take a closer look at the existing evidence, however, one can see that a major difference between the two countries is in the disclosure practice. It is a general practice in Japan to point someone within the company as the company auditor. (see Matsumoto, 1982, for a detailed account of Japanese corporate systems). As Kester (1986) puts it: “Japanese corporations face less stringent disclosure requirements than do US companies, and are well known for their secrecy. The Tokyo Stock Exchange seems to exhibit characteristics symptomatic of information asymmetries that might further foster a reluctance on the part of companies to raise equity capital publicly.” Another factor is that US firms in general are more profitable than Japanese firms (e.g., Choi, Hino, Min, Nam, Ujiie, and Stonehill, 1983), according to our theory they should use less debt. Therefore Japanese firms’ reliance on bank loans may be a result of these two underlying factors. The traditional tax advantage-bankruptcy cost models have been criticized for not being able to explain the wide range of debt to asset ratios that is actually observed. Estimated bankruptcy costs do not seem high enough to cause some firms like IBM to borrow so little (e.g., Miller, 1977). In our model a high bankruptcy cost is not needed to generate a low debt to asset ratio. On the contrary, a higher bankruptcy cost implies a higher debt to asset ratio.

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Our model also implies that more profitable firms borrow less other things being equal. This might have been the case for IBM. Of course, above discussions only involve some casual observations and do not constitute a complete empirical test of the model. The validity of the implications of the model should be verified by such a test. On the theoretical level, the model is still too simple to address some of the important questions concerning corporate finance. Because most existing firms are going concerns, perhaps the most interesting extension is to make the model dynamic and address the issues such as reinvestment, dividends and bond structures. Another possible extension is to introduce ex ante information asymmetry, hence the issue of signalling. In summary, the approach taken in this paper is rooted in the economic tradition that nothing should be taken for granted except preferences, technologies and endowments. It also takes the view that firm’s behavior can be better understood when the firm is regarded as a set of contracts. The advantage of this approach besides having purely theoretical satisfaction is that we can have a well-specified model in which the underlying factors that determine the firm’s financial structure can be identified and ceteris paribus conditions can be precisely stated in postulating hypotheses. In this model, corporate capital structure is viewed as a mechanism to alleviate the problem caused by “the separation of ownership from control”. The comparative statics have some interesting implications. The extant empirical evidence seems to be consistent with the implications of the model. Much more work needs to be done before we can completely understand the financial structure of the firm.

Notes 1. See, among others, Jensen and Meckling (1976), Leland and Pyle (1977), Ross (1977), Grossman and Hart (1982), and Bradley, Jarrell, and Kim (1984). 2. Like many models in economics, certain features of this model are quite special. No claim has been made about the robustness of the optimal contract with respect to all the changes in the environment. Our goal here is to look at a plausible environment in which firm’s financial structure can be endogenously determined and gain some insight as to what will happen when equity and debt contracts are not taken as given. 3. In a comprehensive and in-depth study of twelve large manufactory firms from the first half of Fortune 500 list, a “study attempts to describe the world as it is, and it adopts the perspective of the individual manager-an—an appropriate starting point even for those who wish to bring ‘is’ and ‘ought’ together” (Donaldson, 1984, p. 2), Gordon Donaldson has concluded that instead of maximizing the shareholders’ value, the financial objective that guided the top managers of the firms studied is: “the maximization of corporate wealth. Corporate wealth is that wealth over which management has effective control and which is an assured source of funds, at least within the limits of meaningful strategic planning. In practical terms it is cash, credit, and other corporate purchasing power by which management commands goods and services.” (Donaldson, 1984, p. 22, emphasis original.) My formulation of the manager’s preference is in the same spirit (see also Jensen, 1986). Murphy (1985) has presented evidence that the growth in sales and the managerial positions (hence the resources

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under control) seem to be more important than the return to shareholders in explaining the changes in managers’ compensation. So corporate wealth maximization can be consistent with managers’ personal wealth maximization. 4. In a recent paper, Williams (1986) has shown that some familiar financial contracts can be optimally derived in the presence of asymmetric information between corporate insiders and outsiders. One of the major differences between his model and this one is that he assumes the managers (insiders) always act in the best interest of the outside equity-holders, whereas in my model the conflicts of interest between the inside managers and all outside investors are modeled explicitly and outside investors are not exogenously divided into stockholders and bondholders. 5. Our interpretation here is very similar to the “free cash flow” arguments made by Jensen (1986). In a sense this model may be viewed as a formalization of his arguments.

Acknowledgment I thank George Constantinides, Doug Diamond, Ron Dye, Mike Fishman, Kose John, Bob McDonald, Bill Rogerson, Nancy Stokey, and especially, Milt Harris for helpful discussions and suggestions. I am also grateful to the seminar participants at Boston University, UBC, Brown, Chicago, Duke, Indiana, Iowa, Minnesota, Northwestern, and Ohio State for their comments. All remaining errors are my own responsibility.

Appendix

1. Proof of proposition 1 For the case that Y ⫽ Z, that is, the case when everything is unobservable, we want to show that for an optimal contract the verification schedule is lower-tailed regardless of the functional form of b( y). The proposition then follows as a corollary by indexing everything by x. Suppose the contrary is true, that is, suppose we have an optimal contract whose verification schedule is not lower-tailed. Let’s divide the interval [L, ⬁) into sub-intervals according to D( x, z) ⫽ 1 or 0 so that adjacent intervals have different verification status. Let us number the sub-intervals from right to left by I 1 , I 2 , I 3 , and so on. Without loss of generality let us suppose that no verification is called upon when y is reported in I 1 so D ⫽ 0 for y in I 1 . As a result, D ⫽ 1 for y in I 2 and D ⫽ 0 for y in I 3 . Let y i denote a generic y in I i . By the Revelation Principle (see Harris and Townsend, 1981; Myerson, 1979), we can restrict our attention to the contracts that are incentive compatible, i.e., the contracts under which the manager does not have incentive to lie. It’s easy to see that the payment schedule must be constant on I 1 and on I 3 due to the incentive compatibility constraint. Since y 3 ⬍ y 1 , for the manager with y 1 not to underreport it as y 3 , we should have y 1 ⫺ R( y 1 ) ⱖ y 1 ⫺

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R( y 3 ). This implies that R( y 1 ) ⱕ R( y 3 ). Similarly, because y 3 ⱕ y 2 , for the manager with y 2 not to lie, it must be true that y 2 ⫺ R( y 2 ) ⫺ b( y 2 ) ⱖ y 2 ⫺ R( y 3 ). This implies that R( y 3 ) ⱖ R( y 2 ) ⫹ b( y 2 ) ⱖ R( y 2 ). Now let’s define a new contract (denoted by *) as follows: let R*( y) ⫽ R( y 3 ) and D*( y) ⫽ 0 for all y in I 1 , I 2 , and I 3 ; R*( y) ⫽ R( y) and D*( y) ⫽ D( y) otherwise. It’s easy to see that the new contract is incentive compatible. The expected bankruptcy cost is less because no verification is called for when y is in I 2 . The expected payoff to the investor is possibly higher than 1 ⫹ i. This excess expected payoff can be eliminated by uniformly subtracting a constant form the payoff schedule R*. So we ended up with a contract that involves less expected cost of bankruptcy (dead weight loss) while providing the investor the same expected rate of return. Therefore the contract we started with must not have been an optimal.

2. Proof of lemma 2 I shall prove the lemma under A.1. The case under A.2. is analogous and will be indicated as we go along. Since b( ) is increasing and the hazard rate is nondecreasing, it is clear from Eq. (6) that for each x there is only one solution for R( x). Denote the left hand side of Eq. (6) by A(R, x, B). We are going to show that A R , the partial of A( ) w.r.t. R is positive at R that satisfies Eq. (6). First note that, by differentiating, nondecreasing g( z)/[1 ⫺ G( z)] implies g⬘( z) ⱖ ⫺g 2 ( z)/[1 ⫺ G( z)]. Then differentiating Eq. (6) with respect to R, we have A R共R, x, B兲 ⫽ b⬘共R兲 g共R ⫺ x兲 ⫹ b共R兲 g⬘共R ⫺ x兲 ⫹ g共0兲b共B兲 g共R ⫺ x兲 ⱖ b⬘共R兲 g共R ⫺ x兲 ⫺ b共R兲 g 2共R ⫺ x兲/共1 ⫺ G共R ⫺ x兲兲 ⫹ g共0兲b共B兲 g共R ⫺ x兲 ⫽ 兵b⬘共R兲 g共R ⫺ x兲共1 ⫺ G共R ⫺ x兲兲 ⫺ b共R兲 g 2共R ⫺ x兲兴 ⫹ g共0兲b共B兲 g共R ⫺ x兲关1 ⫺ G共R ⫺ x兲兴}/关1 ⫺ G共R ⫺ x兲兴 ⫽ 兵b⬘共R兲 g共R ⫺ x兲共1 ⫺ G共R ⫺ x兲兲 ⫺ 关b共R兲 g共R ⫺ x兲 ⫺ g共0兲b共B兲共1 ⫺ G共R ⫺ x兲兲兴 g共R ⫺ x兲兲/共1 ⫺ G共R ⫺ x兲兲 ⬎0 The last inequality obtains because the expression in the square brackets is the first order condition Eq. (5), which equals 0. (Under A.2., A R ⬍ 0). By differentiating Eq. (6) w.r.t. x, we get R⬘( x) ⫽ ⫺A x /A R Since: ⫺A x ⫽ b共R兲 g⬘共R ⫺ x兲 ⫹ g共0兲b共B兲 g共R ⫺ x兲 ⬍ A R ⫽ b⬘共R兲 g共R ⫺ x兲 ⫹ b共R兲 g⬘共R ⫺ x兲 ⫹ g共0兲b共B兲 g共⫺R ⫺ x兲, ⫺A x ⫽ b共R兲 g⬘共R ⫺ x兲 ⫹ g共0兲b共B兲 g共R ⫺ x兲 ⱖ ⫺b共R兲 g 2共R ⫺ x兲/关1 ⫺ G共R ⫺ x兲兴 ⫹ g共0兲b共B兲 g共R ⫺ x兲

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381

⫽ ⫺兵b共R兲 g共R ⫺ x兲 ⫺ g共0兲b共B兲关1 ⫺ G共R ⫺ x兲兴其 g共R ⫺ x兲/关1 ⫺ G共R ⫺ x兲兴 ⫽0 and A R ⬎ 0 (under A.2., A R ⬍ ⫺A x ⱕ 0), we have 0 ⱕ R⬘( x) ⬍ 1. By differentiating Eq. (6) w.r.t. B, we get: R B共 x兲 ⫽ ⫺A B/A R ⫽ 关1 ⫺ G共R ⫺ x兲兴 g共0兲b⬘共R兲/A R ⬎ 0 Q.E.D. 3. Proof of proposition 2 All is left to show is the existence and the uniqueness of the optimal contract. For each B, we can solve for R( x) from Eq. (6). Rewrite the left hand side of the investor’s individual rationality constraint Eq. (1) as:

冕 再冕 冕 B

L

⫹

R共 x兲⫺x

冎

关 x ⫹ z ⫺ b共 x ⫹ z兲兴 dG共 z兲 ⫹ R共 x兲关1 ⫺ G共R共 x兲 ⫺ x兲兴 dF共 x兲

0

⬁

x dF共 x兲

B

Where R( x) also depends on B. Differentiate this expression with respect to B. We have:

冕

B

关1 ⫺ G共R共 x兲 ⫺ x兲 ⫺ b共R共 x兲兲 g共R共 x兲 ⫺ x兲兴R B共 x兲 dF共 x兲

L

Since by Eq. (4), 1 ⫺ G(R( x) ⫺ x) ⫺ b(R( x)) g(R( x) ⫺ x) ⫽ b(R( x)) g(R( x) ⫺ x)/ ␭ ⬎ 0 and by Lemma 2, R B ( x) ⬎ 0, the left hand side of Eq. (1) is increasing in B. Also by Lemma 2, we know that minimizing the expected verification cost is equivalent to minimizing B. Therefore the optimal B is the one that makes Eq. (1) binding and this B is unique. Since R( x) is uniquely determined by B, the optimal contract is unique. Q.E.D. 4. Proof of proposition 3 The left hand side of the individual rationality constraint in Eq. (1) is the payoff function to the investor. The derivative of the integrand of this payoff function with respect to x, after some simplification, is: ⫺b共R共 x兲兲 g共R共 x兲 ⫺ x兲关R⬘共 x兲 ⫺ 1兴 ⫹ R⬘共1 ⫺ G共R共 x兲 ⫺ x兲兲 ⫺b共R共 x兲兲 g共R共 x兲 ⫺ x兲关R⬘共 x兲 ⫺ 1兴 ⫹ R⬘共1 ⫺ G共R共 x兲 ⫺ x兲兲

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⫹

冕

R共 x兲⫺x

关1 ⫺ b⬘共 x ⫹ z兲兴 dG共 z兲

(a.1)

0

The sum of the first two terms is positive. The third term is nonnegative if b⬘( y) ⱕ 1. Therefore the integrand is increasing in x. An increase in F( ) in the first order stochastic dominance sense will increase the payoff above 1. As a result, the optimal amount of debt will be lowered. For a fixed x, the payment schedule R( x, z) is nondecreasing in z. Therefore an increase in Z will increase the integrand in Eq. (1) for any x, hence will raise the expected payoff to the investor above 1 ⫹ i. As a result, the optimal B will be lowered. An increase in X and a decrease in B will lower V e ⫽ 兰 ⬁ B ( x ⫺ B) dF( x)/(1 ⫹ i). Hence V d ⫽ 1 ⫺ V e will be raised. Q.E.D.

5. Proof of proposition 4 If we can show that the integrand in the payoff function to the investor, i.e., (a.1) in the proof of Proposition 3, is convex, then a mean preserving spread of F(x) will increase the payoff to the investor. As a result the optimal B will be reduced. By Eq. (5) b(R)g(R ⫺ x) ⫽ k[1 ⫺ G(R ⫺ x)], where k ⫽ ␭/(1 ⫹ ␭) and 0 ⬍ k ⬍ 1. Now (a.1) can be rewritten as: 关共1 ⫺ k兲 R⬘ ⫹ k兴关1 ⫺ G共R ⫺ x兲兴 ⫹

冕

R⫺x

关1 ⫺ b⬘共 x ⫺ z兲兴 dG共 z兲

(a.2)

0

Differentiate (a.2) with respect to x noting that R is a function of x. We find: 共1 ⫺ k兲关1 ⫺ G共R ⫺ x兲兴R⬙ ⫹ 关k ⫹ 共1 ⫺ k兲 R⬘兴 g共R ⫺ x兲共1 ⫺ R⬘兲 ⫹ 共1 ⫺ b⬘共R兲兲 g共R ⫺ x兲 ⫺ 兰b⬙共 x ⫹ z兲dG

(a.3)

Since k ⬍ 1 and R⬘ ⱕ 1, (a.3) is positive if b⬙ ⱕ 0 and R⬙ ⱖ 0. So all we need is to prove R⬙ ⱖ 0. When x ⱖ B R( x) ⫽ x, so R⬙ ⫽ 0. When x ⬍ B from Eq. (6): R共 x兲 ⫽ b ⫺1共k关1 ⫺ G共R ⫺ x兲兴/g共R ⫺ x兲兲 ⫽ v共w共R ⫺ x兲兲,

(a.4)

where v( ) is the inverse function of b( ) and w( ) is the reciprocal of the hazard rate function of Z multiplied by k. It is straightforward to show that v⬘ ⬎ 0, v⬙ ⱖ 0, w⬘ ⬍ 0, and w⬙ ⱖ 0. As a result by differentiating (a.4) twice we have: R⬙ ⫽ 兵v⬙关共R⬘ ⫺ 1兲w⬘兴 2 ⫹ 共R⬘ ⫺ 1兲 2v⬘w⬙其/关1 ⫺ v⬘w⬘兴 ⱖ 0 Q.E.D.

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383

6. Proof of proposition 5 Let H( y) be the distribution function of Y. For simplicity, let L ⫽ 0. In terms of H( ) and G( ), the payoff function to the investor can be written as:

冕 再 冋冕 冕 ⬁

T共 z兲

I 兵 z⬍B其

0

⫹ I 兵 zⱖB其

关 y ⫺ b共 y兲兴 dH共 y兲 ⫹

Z

⬁

冎

冕

⬁

T共 z兲

R共 y ⫺ z兲 dH共 y兲

册

R共 y ⫺ z兲 dH共 y兲 d ⫺ G共 z兲

z

where I { } is the indicator function and T( z) ⫽ Max{ y兩z ⱕ y ⱕ R( y ⫺ z)} is the critical value of y, for a given z, below which a verification will take place. At original G( ) and B, the above expression equals 1. For a fixed B, the integrand inside the scripted brackets is decreasing in z. Therefore, a first order increase in Z will decrease the payoff to the investor at the original B. As a result, B should be raised to increase the payoff back to 1 ⫹ i. Q.E.D.

7. Proof of proposition 6 An increase in i will raise the right hand side of the individual rationality constraint, namely, raise the opportunity cost of capital. The optimal B has to be raised to compensate the investor. Since V e is decreasing in both B and i, it will be lower. Hence, V d ⫽ 1 ⫺ V e will be higher. Q.E.D.

8. Proof of proposition 7 In the individual rationality constraint in Eq. (1), if b( y) is replaced with c( y), the payoff to the investor will be reduced below 1 at the original B. To compensate the investor, the optimal B associated with c( y) will have to be raised, because the original B is the smallest which satisfied the individual rationality constraint when bankruptcy cost were given by b( ). Q.E.D.

9. Proof of proposition 8 See the arguments in the text.

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10. A numerical example Suppose that X is uniformly distributed on [0, H] and Z is exponentially distributed with parameter d (the density function g( z) ⫽ de ⫺dz and E(Z) ⫽ 1/d). Suppose that b( y) ⫽ b*y. For various values of H, d, and b, following tables list the debt to asset ratio V d corresponding to different risk-free rate i. H ⫽ i Vd H ⫽ i Vd H ⫽ i Vd H ⫽ i Vd H ⫽ i Vd

2, d ⫽ 10, b ⫽ 10%. E(Y) ⫽ 1.1 0.2% 20.2% 2, d ⫽ 5, b ⫽ 10%. E(Y) ⫽ 1.2 0.7% 20.6% 2, d ⫽ 5, b ⫽ 20%. E(Y) ⫽ 1.2 0.3% 20.2% 2.1, d ⫽ 10, b ⫽ 10%. E(Y) ⫽ 1.15 5.4% 24.1% 1.9, d ⫽ 4, b ⫽ 10%. E(Y) ⫽ 1.2 ⫺1.7% 43%

1.4% 44.8%

1.7% 51.8%

2.1% 75.5%

1.3% 29%

2.7% 45.5%

5.7% 76.3%

2.6% 45.4%

3.2% 65.1%

6.4% 47.4% 1% 64.4%

2.6% 75.6%

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