A. External Financing

At $ t=0 $ , the manager issues perpetual debt with a constant coupon payment of $ c $ to exploit tax shield benefits. In addition, he sells a fraction $ \left(1-\lambda \right) $ of equity to minority shareholders, retaining only a fraction $ \lambda $ for himself. Given tax rate $ \tau $ , over a short time interval $ \left[t,t+\mathrm{d}t\right) $ , the after-tax cash flow to levered equity before investment is therefore $ \left(1-\tau \right)\left({X}_t-c\right)\mathrm{d}t $ . We assume that full loss offsets are available in terms of tax treatment.

Despite selling the securities, the manager remains the sole decision maker for the firm (i.e., he decides whether and when to exercise the growth option, as well as if and when to default).Footnote ² Following Myers’ (Reference Myers1977) argument that issuing risky debt diminishes a firm’s present market value by inducing suboptimal growth investment strategies, we assume that the growth investment is fully equity-financed.Footnote ³ Upon exercising the growth option, the investment cost $ I $ is shared between the manager and minority shareholders proportionally to their equity stakes, $ \lambda $ and $ \left(1-\lambda \right) $ , respectively.

B. Private Benefits of Control

The firm operates under imperfect corporate governance, allowing the manager to extract control-related private benefits. We assume that creditors and the government have priority over the manager’s diversion privilege (see, e.g., Morellec, Nikolov, and Schürhoff (Reference Morellec, Nikolov and Schürhoff2012), Morellec et al. (Reference Morellec, Nikolov and Schürhoff2018)). Specifically, the manager can divert a fraction $ \theta \in \left[0,1\right) $ of earnings after interest and taxes for personal consumption.Footnote ⁴ The remaining portion $ \left(1-\theta \right) $ is distributed as dividends to all shareholders on a pro rata basis. The diversion rate $ \theta $ quantifies the deficiency in corporate governance. This potential for cash diversion creates a conflict of interest between the manager and minority shareholders, which we refer to as the “owner–manager conflict.”

C. Default

While debt issuance offers tax shield benefits, it also introduces default risk. As standard in the literature, the manager optimally chooses to default when cash flow $ {X}_t $ deteriorates sufficiently. Upon default, a fraction $ \alpha $ of the unlevered firm value is lost to bankruptcy costs, while creditors receive the remaining fraction $ \left(1-\alpha \right) $ of the unlevered firm value, and shareholders get nothing.

D. Equilibrium

Let $ \overline{T} $ be the endogenous time that the manager exercises the growth option and $ \underline{T} $ be the endogenous time of default. Note that default could occur either before or after the investment. If default happens before the investment, the firm loses the growth option due to its tarnished reputation and the preemption by potential competitors.

Let $ {W}_t $ be the manager’s continuation value:

(1) $$ {\displaystyle \begin{array}{l}\hskip3.7em {W}_t={\unicode{x1D53C}}_t\left[{\int}_t^{\overline{T}\wedge \underline{T}}{e}^{-r\left(s-t\right)}\underset{\mathrm{net}\hskip0.32em \mathrm{cash}\ \mathrm{flow}\ \mathrm{before}\ \mathrm{investment}}{\underbrace{\left(\theta +\lambda \left(1-\theta \right)\right)\left(1-\tau \right)\left({X}_s-c\right)}}\mathrm{d}s-{e}^{-r\left(\overline{T}-t\right)}\underset{\mathrm{investment}\ \mathrm{cost}}{\underbrace{\lambda I}}\right.\\ {}\hskip7em \left.+{\int}_{\overline{T}\wedge \underline{T}}^{\underline{T}}{e}^{-r\left(s-t\right)}\underset{\mathrm{net}\hskip0.22em \mathrm{cash}\ \mathrm{flow}\ \mathrm{after}\ \mathrm{investment}}{\underbrace{\left(\theta +\lambda \left(1-\theta \right)\right)\left(1-\tau \right)\left({qX}_s-c\right)}}\mathrm{d}s\right].\end{array}} $$

Similarly, we define $ {J}_t $ as minority shareholders’ continuation value:

(2) $$ {\displaystyle \begin{array}{l}{J}_t={\unicode{x1D53C}}_t\left[{\int}_t^{\overline{T}\wedge \underline{T}}{e}^{-r\left(s-t\right)}\underset{\mathrm{net}\;\mathrm{cash}\ \mathrm{flow}\ \mathrm{before}\ \mathrm{investment}}{\underbrace{\left(1-\lambda \right)\left(1-\theta \right)\left(1-\tau \right)\left({X}_s-c\right)}}\mathrm{d}s-{e}^{-r\left(\overline{T}-t\right)}\underset{\mathrm{investment}\ \mathrm{cost}}{\underbrace{\left(1-\lambda \right)I}}\right.\\ {}\hskip3em \left.+{\int}_{\overline{T}\wedge \underline{T}}^{\underline{T}}{e}^{-r\left(s-t\right)}\underset{\mathrm{net}\hskip0.32em \mathrm{cash}\ \mathrm{flow}\ \mathrm{after}\ \mathrm{investment}}{\underbrace{\left(1-\lambda \right)\left(1-\theta \right)\left(1-\tau \right)\left({qX}_s-c\right)}}\mathrm{d}s\right].\end{array}} $$

Creditors receive coupon payments until the firm defaults, after which they receive a fraction $ \left(1-\alpha \right) $ of the unlevered firm value. Their valuation is therefore:

(3) $$ {D}_t={\unicode{x1D53C}}_t\left[{\int}_t^{\underline{T}}{e}^{-r\left(s-t\right)}c\mathrm{d}s+{e}^{-r\left(\underline{T}-t\right)}\left(1-\alpha \right)\left(\frac{1-\tau }{r-\mu}\right)\left({1}_{\left\{\overline{T}\le \underline{T}\right\}}{qX}_{\underline{T}}+{1}_{\left\{\underline{T}<\overline{T}\right\}}{X}_{\underline{T}}\right)\right], $$

where $ {1}_{\left\{\cdot \right\}} $ represents the indicator function. Finally, firm value is the sum of equations (1)–(3) and is given by

(4) $$ {\displaystyle \begin{array}{l}{V}_t={\unicode{x1D53C}}_t\left[{\int}_t^{\overline{T}\wedge \underline{T}}{e}^{-r\left(s-t\right)}\left[\left(1-\tau \right){X}_s+\tau c\right]\mathrm{d}s+{\int}_{\overline{T}\wedge \underline{T}}^{\underline{T}}{e}^{-r\left(s-t\right)}\left[\left(1-\tau \right){qX}_s+\tau c\right]\mathrm{d}s\right.\\ {}\hskip4.3em \left.-{e}^{-r\left(\overline{T}-t\right)}I+{e}^{-r\left(\underline{T}-t\right)}\left(1-\alpha \right)\left(\frac{1-\tau }{r-\mu}\right)\left({1}_{\left\{\overline{T}\le \underline{T}\right\}}{qX}_{\underline{T}}+{1}_{\left\{\underline{T}<\overline{T}\right\}}{X}_{\underline{T}}\right)\right].\end{array}} $$

We look for a Markov Perfect Equilibrium where $ {X}_t $ is the payoff-relevant state variable. As such, the valuations satisfy $ {W}_t=W\left({X}_t\right) $ , $ {J}_t=J\left({X}_t\right) $ , $ {D}_t=D\left({X}_t\right) $ , and $ {V}_t=V\left({X}_t\right) $ . We seek an equilibrium in which investment and default both follow threshold strategies: the firm invests when cash flows rise above a pre-determined threshold ( $ {x}_i $ ) prior to declaring default and defaults when cash flows drop below the endogenous default thresholds. Note that default may occur before or after investment, so there will be two default thresholds. Before making the investment, default happens at $ {x}_{\mathrm{\ell}} $ , whereas after investment, default happens when $ {X}_t $ falls to $ {x}_d $ .

We distinguish between two investment strategies. The firm-optimal strategy aims to maximize firm value subject to the manager’s limited liability, while the manager-optimal strategy maximizes the manager’s continuation value.Footnote ⁵ The two strategies are respectively captured by two investment thresholds $ {x}_i^F $ and $ {x}_i^S $ . Throughout the article, we define overinvestment as scenarios when $ {x}_i^S<{x}_i^F $ , and underinvestment as those when $ {x}_i^S>{x}_i^F $ . In equilibrium, creditors and minority shareholders fully anticipate the manager’s future policies and price the securities fairly at issuance. At $ t=0 $ , the manager chooses the capital and ownership structure $ \left\{c,\lambda \right\} $ to maximize the sum of his continuation value and the proceeds from selling these securities. For the remainder of the article, we denote by $ {V}^F $ and $ {V}^S $ the initial firm values under firm-optimal and manager-optimal investment strategies, respectively.

Our model integrates two types of agency conflicts: the owner––manager conflict arising from control-related private benefits and the shareholder–creditor conflict stemming from debt. Table 1 presents the baseline parameter values employed for subsequent numerical analysis.Footnote ⁶ Table 2 summarizes the key model variables in the article.

TABLE 1Baseline Parameter Values Used for Numerical Analysis

Table 1 presents the baseline parameter values adopted for the numerical analysis in the article.

TABLE 2Summary of Key Model Variables

Table 2 summarizes the symbols for key model variables in the article. Panels A, B, and C each report on two benchmarks and the general scenario. In the text, the superscripts “ $ F $ ” and “ $ S $ ” over variables indicate firm-optimal and manager-optimal values, respectively.

A. Overinvestment and Owner–Manager Conflict

We first study the scenario where the firm is all-equity financed. The model is time-homogeneous; thus, without loss of generality, we abbreviate the expectation operator under pricing measure $ \unicode{x1D53C}\left[\cdot |{X}_0=x\right] $ to $ \unicode{x1D53C}\left[\cdot \right] $ . We use “^̂” to denote the value functions and investment thresholds in this benchmark.

Define the first-passage time that $ {X}_t $ hits the endogenous investment threshold $ {\hat{x}}_i $ from below by

$$ \overline{T}\left({\hat{x}}_i\right):= \operatorname{inf}\left\{t\ge 0:{X}_t\ge {\hat{x}}_i\right\}. $$

The manager exercises the investment option once $ {X}_t $ reaches $ {\hat{x}}_i $ from below. Thus, the manager’s continuation value in equation (1) becomes

$$ \hat{W}(x)=\unicode{x1D53C}\left[{\int}_0^{\overline{T}\left({\hat{x}}_i\right)}{e}^{- rt}\left(\theta +\lambda \left(1-\theta \right)\right)\left(1-\tau \right){X}_t\mathrm{d}t+{e}^{-r\overline{T}\left({\hat{x}}_i\right)}\left[{\hat{W}}_q\left({\hat{x}}_i\right)-\lambda I\right]\right], $$

where

$$ {\hat{W}}_q(x)=\unicode{x1D53C}\left[{\int}_0^{+\infty }{e}^{- rt}\left(\theta +\lambda \left(1-\theta \right)\right)\left(1-\tau \right){qX}_t\mathrm{d}t\right]=\left(\theta +\lambda \left(1-\theta \right)\right)\left(\frac{1-\tau }{r-\mu}\right) qx $$

is the manager’s post-investment continuation value. Similarly, firm value in equation (4) becomes

$$ \hat{V}(x)=\unicode{x1D53C}\left[{\int}_0^{\overline{T}\left({\hat{x}}_i\right)}{e}^{- rt}\left(1-\tau \right){X}_t\mathrm{d}t+{e}^{-r\overline{T}\left({\hat{x}}_i\right)}\left[{\hat{V}}_q\left({\hat{x}}_i\right)-I\right]\right], $$

where

$$ {\hat{V}}_q(x)=\unicode{x1D53C}\left[{\int}_0^{+\infty }{e}^{- rt}\left(1-\tau \right){qX}_t\mathrm{d}t\right]=\left(\frac{1-\tau }{r-\mu}\right) qx $$

is the post-investment firm value.

Let $ {\hat{x}}_i^S $ and $ {\hat{x}}_i^F $ respectively denote the manager-optimal and firm-optimal investment thresholds. The solution steps follow the standard literature on investment as a real option (see McDonald and Siegel (Reference McDonald and Siegel1986), Dixit and Pindyck (Reference Dixit and Pindyck1994)). Therefore, we omit the details and leave them in Appendix A.2. By the standard smooth-pasting conditions,

$$ {\left.\frac{\partial \hat{W}(x)}{\partial x}\right|}_{x={\hat{x}}_i^S-}={\left.\frac{\partial {\hat{W}}_q(x)}{\partial x}\right|}_{x={\hat{x}}_i^S+},\hskip2em {\left.\frac{\partial \hat{V}(x)}{\partial x}\right|}_{x={\hat{x}}_i^F-}={\left.\frac{\partial {\hat{V}}_q(x)}{\partial x}\right|}_{x={\hat{x}}_i^F+}, $$

we derive the manager-optimal and firm-optimal investment thresholds:

(5) $$ {\displaystyle \begin{array}{l}{\hat{x}}_i^S=\left(\frac{\lambda }{\theta +\lambda \left(1-\theta \right)}\right)\left(\frac{1}{q-1}\right)\left(\frac{r-\mu }{1-\tau}\right)\left(\frac{\gamma_{+}}{\gamma_{+}-1}\right)I,\\ {}{\hat{x}}_i^F=\left(\frac{1}{q-1}\right)\left(\frac{r-\mu }{1-\tau}\right)\left(\frac{\gamma_{+}}{\gamma_{+}-1}\right)I,\end{array}} $$

where $ {\gamma}_{+} $ is given by equation (A.2) in Appendix A.1.

We can show that

(6) $$ \frac{\partial {\hat{x}}_i^S}{\partial \theta }=-\left(\frac{1-\lambda }{\theta +\lambda \left(1-\theta \right)}\right){\hat{x}}_i^S\le 0,\hskip2em \frac{\partial {\hat{x}}_i^S}{\partial \lambda }=\left(\frac{\theta }{\theta +\lambda \left(1-\theta \right)}\right)\left(\frac{{\hat{x}}_i^S}{\lambda}\right)\ge 0. $$

A higher $ \theta $ increases the manager’s private gains from cash diversion, thereby increasing his incentives for overinvestment. For the rest of the article, we refer to this effect as the diversion effect. Conversely, an increase in $ \lambda $ raises the investment cost borne by the manager, which increases his incentives for underinvestment. This effect is termed the cost-sharing effect. Figure 1 illustrates these patterns.

FIGURE 1Investment Thresholds in Pure-Equity Benchmark Under Only Owner–Manager Conflict

In the benchmark shown in Figure 1, the firm issues no debt ( $ c=0 $ ). Graph A varies the diversion rate $ \theta $ , with $ {\hat{x}}_i^S={\hat{x}}_i^F $ at $ \theta =0 $ . Graph B varies managerial ownership $ \lambda $ , with $ {\hat{x}}_i^S={\hat{x}}_i^F $ at $ \lambda =1 $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , and $ \lambda =0.075 $ .

It is easily seen that $ {\hat{x}}_i^S=\left(\frac{\lambda }{\theta +\lambda \left(1-\theta \right)}\right){\hat{x}}_i^F\le {\hat{x}}_i^F $ (i.e., in this benchmark, the diversion effect always dominates the cost-sharing effect). The ratio $ \frac{\lambda }{\theta +\lambda \left(1-\theta \right)} $ captures the manager’s trade-off in making the investment: he bears only a fraction $ \lambda $ of the investment cost but receives a fraction $ \theta +\lambda \left(1-\theta \right)\ge \lambda $ of the cash flows. Given that $ \theta +\lambda \left(1-\theta \right)\ge \lambda $ , the manager has incentives to overinvest, which in this context manifests as exercising the investment option too early. Note that $ {\hat{x}}_i^S={\hat{x}}_i^F $ , that is, the manager makes the investment that is firm-optimal, holds under two specific conditions: when $ \lambda =1 $ (the manager bears the full investment cost) or when $ \theta =0 $ (there is no opportunity for cash flow diversion). In either case, the manager’s interests align perfectly with maximizing firm value, eliminating his incentives to overinvest.

We compute the derivative of the initial firm value under manager-optimal strategy as:

(7) $$ {\displaystyle \begin{array}{l}\frac{\partial {\hat{V}}^S\left({x}_0;\lambda \right)}{\partial \lambda }=\left(I{\gamma}_{+}\right)\left(\frac{1-\lambda }{\lambda}\right){\left(\frac{\theta }{\theta +\lambda \left(1-\theta \right)}\right)}^2{\left(\frac{x_0}{{\hat{x}}_i^S}\right)}^{\gamma_{+}}\ge 0,\\ {}\frac{\partial {\hat{V}}^S\left({x}_0;\lambda \right)}{\partial \theta }=-\left(\theta I{\gamma}_{+}\right){\left(\frac{1-\lambda }{\theta +\lambda \left(1-\theta \right)}\right)}^2{\left(\frac{x_0}{{\hat{x}}_i^S}\right)}^{\gamma_{+}}\le 0.\end{array}} $$

Firm value increases with $ \lambda $ and decreases with $ \theta $ , and achieves the value under firm-optimal strategy when $ \lambda =1 $ . Therefore, at $ t=0 $ , the manager’s optimal ownership strategy is to set $ \lambda =1 $ , which eliminates minority shareholders and perfectly aligns managerial incentives with firm interests.

B. Underinvestment and Shareholder–Creditor Conflict

Next, we allow the firm to issue debt (i.e., $ c>0 $ ), but assume away owner–manager conflict by setting either $ \theta =0 $ or $ \lambda =1 $ . This framework follows Mauer and Ott (Reference Mauer, Ott, Brennan and Trigeorgis2000). We use “^~” to denote the value functions and thresholds in this benchmark.

The manager chooses the investment threshold $ {\tilde{x}}_i $ , the pre-investment default threshold $ {\tilde{x}}_{\mathrm{\ell}} $ , and the post-investment default threshold $ {\tilde{x}}_d $ . Let $ {\tilde{x}}_i^F $ and $ {\tilde{x}}_i^S $ represent the firm- and manager-optimal investment thresholds, respectively, maximizing firm value and the manager’s continuation value. The solution steps closely follow those in Mauer and Ott (Reference Mauer, Ott, Brennan and Trigeorgis2000), and it is a specific case of the equilibrium outlined in the subsequent Section IV. Figure 2 describes the findings.

FIGURE 2Investment and Default Thresholds in Levered Benchmark Under Only Shareholder–Creditor Conflict

In Figure 2, the benchmark scenario assumes no private benefits ( $ \theta =0 $ ) or full managerial ownership ( $ \lambda =1 $ ). Graph A displays investment thresholds as a function of coupon payment $ c $ , with $ {\tilde{x}}_i^S={\tilde{x}}_i^F={\hat{x}}_i $ at $ c=0 $ . Graph B illustrates default thresholds versus coupon payment $ c $ , with $ q\times {\tilde{x}}_d>{\tilde{x}}_{\mathrm{\ell}}^F $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , and $ q=2 $ .

Graph A of Figure 2 illustrates how investment thresholds $ {\tilde{x}}_i^F $ and $ {\tilde{x}}_i^S $ vary with coupon payment $ c $ . The solid red line indicates that the manager-optimal threshold—chosen by the manager—increases with $ c $ . This pattern aligns with the classical debt overhang argument: as the firm issues more debt, equity holders have reduced incentives to invest because they must bear full investment cost while more value accrues to debt holders. If $ c=0 $ , the debt overhang effect disappears, and the two investment thresholds are identical.

Interestingly, the dotted blue line shows that the firm-optimal investment threshold decreases with $ c $ . In other words, it is optimal for the firm to exercise the investment option earlier as it takes on more debt. This occurs because more debt leads to an earlier default, as confirmed by Graph B, where the default thresholds increase with $ c $ . Once the firm defaults, it loses the investment option. Therefore, it is optimal to exercise the option earlier as the firm takes on more debt. Note that Graph B also shows that the post-investment default cash flow threshold $ q\times {\tilde{x}}_d $ is always above the pre-investment cash flow threshold $ {\tilde{x}}_{\mathrm{\ell}}^F $ .Footnote ⁷ This difference, again, is due to the value of the investment option.

At $ t=0 $ , the manager optimally chooses the coupon payment to maximize firm value under the manager-optimal investment threshold. In this benchmark scenario, the optimal coupon level $ {\tilde{c}}^{\ast } $ is determined to be $ 1.3697 $ , which effectively balances the trade-off between the tax advantages of debt and potential bankruptcy losses. The corresponding leverage ratio $ \left({\tilde{D}}^S/{\tilde{V}}^S\right) $ and yield spread ( $ {\tilde{c}}^{\ast }/{\tilde{D}}^S-r $ ) are $ 73.05\% $ and $ 0.82\% $ , respectively.

A. Model Solution

This subsection solves the model for arbitrary $ \theta $ . The firm- and manager-optimal investment strategies are defined similarly to those in Section III, respectively, maximizing firm value and the manager’s continuation value. Let $ {x}_i $ denote the endogenous investment threshold, and $ \overline{T}\left({x}_i\right) $ represent the time to exercise the investment option. We employ a backward induction approach, beginning our analysis after the investment option has been exercised, that is, $ t>\overline{T}\left({x}_i\right) $ . The problem is identical to Leland (Reference Leland1994). Let $ {W}_q(x) $ denote the manager’s continuation value function after investment. We have the following Hamilton–Jacobi–Bellman (HJB) equation:

$$ {rW}_q(x)=\left(\theta +\lambda \left(1-\theta \right)\right)\left(1-\tau \right)\left( qx-c\right)+\mu x\frac{\partial {W}_q(x)}{\partial x}+\frac{1}{2}{\sigma}^2{x}^2\frac{\partial^2{W}_q(x)}{\partial {x}^2}, $$

which satisfies the value-matching and smooth-pasting conditions at the default boundary $ {x}_d $ :

$$ {W}_q\left({x}_d\right)=0,\hskip2em {\left.\frac{\partial {W}_q(x)}{\partial x}\right|}_{x={x}_d+}=0. $$

These conditions lead to

(8) $$ {x}_d=\left(\frac{1}{q}\right)\left(\frac{r-\mu }{r}\right)\left(\frac{\gamma_{-}}{\gamma_{-}-1}\right)c, $$

where $ {\gamma}_{-} $ is given by equation (A.2) in Appendix A.1. As argued in Leland (Reference Leland1994), the default threshold $ {x}_d $ is the same for both the firm- and manager-optimal scenarios.Footnote ⁸ Note that $ {x}_d $ is independent of the diversion rate $ \theta $ , managerial ownership $ \lambda $ , and the tax rate $ \tau $ .

Next turn to $ t<\overline{T}\left({x}_i\right) $ . Let $ W(x) $ be the manager’s continuation value function, which satisfies the following HJB equation:

$$ rW(x)=\left(\theta +\lambda \left(1-\theta \right)\right)\left(1-\tau \right)\left(x-c\right)+\mu x\frac{\partial W(x)}{\partial x}+\frac{1}{2}{\sigma}^2{x}^2\frac{\partial^2W(x)}{\partial {x}^2}. $$

There are two boundaries: the manager chooses to default as $ {X}_t\downarrow {x}_{\mathrm{\ell}}^S $ and exercises the investment option as $ {X}_t\uparrow {x}_i^S $ . Both boundaries satisfy value-matching and smooth-pasting conditions:

$$ {\displaystyle \begin{array}{l}W\left({x}_{\mathrm{\ell}}^S\right)=0,\hskip2em W\left({x}_i^S\right)={W}_q\left({x}_i^S\right)-\lambda I,\\ {}{\left.\frac{\partial W(x)}{\partial x}\right|}_{x={x}_{\mathrm{\ell}}^S+}=0,\hskip2em {\left.\frac{\partial W(x)}{\partial x}\right|}_{x={x}_i^S-}={\left.\frac{\partial {W}_q(x)}{\partial x}\right|}_{x={x}_i^S+}.\end{array}} $$

The boundary $ {x}_i^S $ characterizes the manager-optimal investment strategy. We will compare it with the firm-optimal investment strategy $ {x}_i^F $ , which maximizes firm value. Specifically, let $ {V}_q(x) $ and $ V(x) $ be the firm value functions after and before the investment, satisfying the following HJB equations:

$$ {\displaystyle \begin{array}{c}{rV}_q(x)=\left(1-\tau \right) qx+\tau c+\mu x\frac{\partial {V}_q(x)}{\partial x}+\frac{1}{2}{\sigma}^2{x}^2\frac{\partial^2{V}_q(x)}{\partial {x}^2},\\ {} rV(x)=\left(1-\tau \right)x+\tau c+\mu x\frac{\partial V(x)}{\partial x}+\frac{1}{2}{\sigma}^2{x}^2\frac{\partial^2V(x)}{\partial {x}^2}.\end{array}} $$

Similarly, the above equations satisfy the following value-matching and smooth-pasting conditions at the thresholds $ {x}_d $ , $ {x}_{\mathrm{\ell}}^F $ , and $ {x}_i^F $ :

$$ {\displaystyle \begin{array}{l}{V}_q\left({x}_d\right)=\left(1-\alpha \right)\left(\frac{1-\tau }{r-\mu}\right){qx}_d,\hskip1em V\left({x}_{\mathrm{\ell}}^F\right)=\left(1-\alpha \right)\left(\frac{1-\tau }{r-\mu}\right){x}_{\mathrm{\ell}}^F,\hskip1em V\left({x}_i^F\right)={V}_q\left({x}_i^F\right)-I,\\ {}\hskip0em {\left.\frac{\partial {W}_q(x)}{\partial x}\right|}_{x={x}_d+}=0,\hskip2em {\left.\frac{\partial W(x)}{\partial x}\right|}_{x={x}_{\mathrm{\ell}}^F+}=0,\hskip2em {\left.\frac{\partial V(x)}{\partial x}\right|}_{x={x}_i^F-}={\left.\frac{\partial {V}_q(x)}{\partial x}\right|}_{x={x}_i^F+}.\end{array}} $$

Note that the smooth-pasting conditions for the two default thresholds hold for shareholders/manager value because the firm operates under limited liability. This implies that while the investment decision aims to maximize firm value, default decisions—due to limited liability—are still made to maximize the manager’s value.

We proceed to solve the model using standard contingent-claims pricing methods detailed in Appendix A.3. The problem requires numerically solving two equations to find $ {x}_i $ and $ {x}_{\mathrm{\ell}} $ .

B. Baseline Results and Comparative Statics

At $ t=0 $ , the manager jointly chooses $ \left\{c,\lambda \right\} $ to maximize the initial firm value $ {V}^S\left({x}_0;c,\lambda \right) $ under the manager-optimal investment policy, that is,

(9) $$ \left\{{c}^{\ast },{\lambda}^{\ast}\right\}:= \underset{\left\{c,\lambda \right\}}{\arg \max}\left\{{V}^S\left({x}_0;c,\lambda \right)\right\}, $$

for given $ {x}_0 $ and $ \theta >0 $ . Under baseline parameters in Table 1, there is a unique set of internal solutions to problem (9): $ {c}^{\ast }=1.4051 $ and $ {\lambda}^{\ast }=0.0975 $ , associated with a leverage ratio of $ {D}^S/{V}^S=74.58\% $ and a yield spread of $ \left({c}^{\ast }/{D}^S-r\right)=0.84\% $ . Figure 3 shows that firm value exhibits an inverted U-shaped function of either $ c $ or $ \lambda $ . In addition to investment efficiency, the optimal coupon level $ {c}^{\ast } $ needs to balance tax shield benefits and bankruptcy losses and also depends on cash flow level $ {x}_0 $ . Accordingly, the optimal ownership $ {\lambda}^{\ast } $ needs to adjust as well.Footnote ⁹ Note that $ {\lambda}^{\ast }<1 $ , so the cost-sharing effect is weaker compared to the levered benchmark with $ \lambda =1 $ in Section III.B. Recall that both the cost-sharing and debt overhang effects contribute to investment delays. Consequently, a higher debt level ( $ {c}^{\ast }>{\tilde{c}}^{\ast } $ )—and thus a stronger debt overhang effect—is needed to balance investment efficiency, inducing a higher leverage ratio and yield spread.

FIGURE 3Firm Value Versus Coupon Payment and Managerial Ownership

Under various initial cash flow levels $ {x}_0 $ , Graph A of Figure 3 varies $ c $ while fixing $ {\lambda}^{\ast }=0.0975 $ , and Graph B varies $ \lambda $ while fixing $ {c}^{\ast }=1.4051 $ . Each curve in the two graphs exhibits an inverted U-shaped function with a unique maximum point, indicated by the vertical dashed line. Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , and $ \theta =0.01 $ .

Table 3 shows the investment and default thresholds under optimal capital and ownership structure. A few patterns emerge. First, the manager’s optimal investment threshold is lower than the firm-optimal threshold: $ {x}_i^S<{x}_i^F $ . This occurs because, under these parameter values, the diversion effect dominates the debt overhang and cost-sharing effects, leading to overinvestment. Second, before exercising the investment option, the manager-optimal default threshold is lower than the firm-optimal ( $ {x}_{\mathrm{\ell}}^S<{x}_{\mathrm{\ell}}^F $ ), while after exercising the investment, the two thresholds are identical (both are $ q\times {x}_d $ ). This result shows that the value of the investment option reduces the manager’s incentives to default.

TABLE 3Investment and Default Thresholds in a Levered Firm When There Are Both Underinvestment and Overinvestment Incentives

Table 3 presents the investment and default thresholds in a levered firm with both agency conflicts under optimal capital and ownership structure $ \left\{{c}^{\ast },{\lambda}^{\ast}\right\} $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , $ {c}^{\ast }=1.4051 $ , and $ {\lambda}^{\ast }=\mathrm{0.0975.} $

Interestingly, $ {x}_{\mathrm{\ell}}^S<q\times {x}_d $ , so that the default threshold before investment falls below the one after investment. These findings reveal that the investment option can significantly influence the manager’s default decisions.

We now examine how the model solutions—especially the investment and default thresholds, as well as the optimal ownership and capital structure—depend on the model parameters. We will compare the firm-optimal threshold with the manager-optimal one and, therefore, study how overinvestment and underinvestment are affected by these parameters.

Figure 4 illustrates how the investment and default thresholds depend on the manager’s diversion rate $ \theta $ . Graph A shows that the manager-optimal investment threshold $ {x}_i^S $ decreases with $ \theta $ , because a stronger diversion effect increases the incentives for overinvestment. By contrast, the firm-optimal threshold $ {x}_i^F $ increases with $ \theta $ because the manager’s incentives to default are diminished at higher values of $ \theta $ . This latter effect is confirmed by Graph B, which shows that a higher $ \theta $ also reduces $ {x}_{\mathrm{\ell}}^F $ , the default threshold before the investment option is exercised. Meanwhile, equation (8) shows that the default threshold $ {x}_d $ after the investment option is exercised, however, is independent of $ \theta $ . These patterns imply that the manager underinvests when $ \theta $ is low and overinvests when $ \theta $ is high. Moreover, there exists a threshold level of $ \theta $ at which the manager-optimal investment threshold coincides with the firm-optimal threshold.

FIGURE 4Investment and Default Thresholds Under Both Conflicts Versus Diversion Rate

Figure 4 Graph A highlights an intersection point between $ {x}_i^S $ and $ {x}_i^F $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ {c}^{\ast }=1.4051 $ , and $ {\lambda}^{\ast }=0.0975 $ .

Figure 5 shows how the tax rate $ \tau $ affects investment and default decisions. A higher tax rate $ \tau $ increases the tax burden of the firm, as the government claims a larger fraction of the cash flows. The reduced after-tax cash flow lowers investment incentives, leading the firms to further delay investment. This is reflected in Graph A, where both firm-optimal and manager-optimal investment thresholds increase with $ \tau $ . For the same reason, the manager chooses to default earlier, as confirmed by the higher default thresholds in Graph B. When firms default earlier, the market value of the firm’s debt decreases, which weakens the debt overhang effect. These combined effects explain why we observe underinvestment when $ \tau $ is low but overinvestment when $ \tau $ is high.

FIGURE 5Investment and Default Thresholds and Debt Value Under Both Conflicts Versus Tax Rate

Figure 5 Graph A highlights an intersection point between $ {x}_i^S $ and $ {x}_i^F $ . Parameters: $ {x}_0=1.9 $ , $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , $ {c}^{\ast }=1.4051 $ , and $ {\lambda}^{\ast }=0.0975 $ .

Figure 6 shows how cash flow volatility $ \sigma $ affects investment and default decisions. A higher volatility $ \sigma $ has two effects. First, following standard real-option logic, higher uncertainty increases the value of waiting and firm value, as confirmed by Graph C. Consistently, Graph A shows that both the firm- and manager-optimal investment thresholds increase with $ \sigma $ . For this reason, firms default at lower thresholds when uncertainty is higher, as shown in Graph B. Second, under the same coupon payment $ c $ , a higher $ \sigma $ reduces the market value of debt, as illustrated in Graph C. As a result, the debt overhang effect becomes weaker at higher levels of volatility. These two effects combined explain why there is underinvestment when $ \sigma $ is low but overinvestment when $ \sigma $ is high.

FIGURE 6Investment and Default Thresholds and Valuations Under Both Conflicts Versus Volatility

Figure 6 Graph A highlights an intersection point between $ {x}_i^S $ and $ {x}_i^F $ . Parameters: $ {x}_0=1.9 $ , $ r=0.07 $ , $ \mu =0 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , $ {c}^{\ast }=1.4051 $ , and $ {\lambda}^{\ast }=0.0975 $ .

Figure 7 illustrates how the joint optimal managerial ownership and coupon level determined by problem (9) vary with diversion rate, tax rate, and volatility. Graph A shows that the optimal managerial ownership $ {\lambda}^{\ast } $ increases with $ \theta $ . The reason is that when $ \theta $ increases, the diversion effect becomes stronger. The manager needs to share more investment costs to offset the overinvestment incentives. In fact, the ratio $ \frac{\lambda^{\ast }}{\theta +{\lambda}^{\ast}\left(1-\theta \right)} $ stays unchanged, so that the optimal coupon level $ {c}^{\ast } $ also stays unchanged.Footnote ¹⁰ Graph B illustrates that as the tax rate rises, the tax shield benefits increase, so the manager prefers to issue more debt. As a result, the manager needs to share less investment cost (lower $ {\lambda}^{\ast } $ ).Footnote ¹¹ Graph C shows that $ {c}^{\ast } $ exhibits a U-shaped relationship with $ \sigma $ , consistent with Leland (Reference Leland1994). It also shows that $ {\lambda}^{\ast } $ decreases with $ \sigma $ , which is because by selling more equity, the manager can better diversify the increased cash flow risk.

FIGURE 7Optimal Capital and Ownership Structure Versus Diversion Rate, Tax Rate, and Volatility

In Figure 7, the ratio $ \frac{\lambda^{\ast }}{\theta +{\lambda}^{\ast}\left(1-\theta \right)} $ and optimal $ {c}^{\ast } $ in Graph A remain constant across various $ \theta $ . Parameters: $ {x}_0=1.9 $ , $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.15 $ , $ \tau =0.35 $ , $ \alpha =0.5 $ , $ I=20 $ , $ q=2 $ , and $ \theta =0.01 $ .

C. Optimal Capital or Ownership Structure

In practice, firms often face constraints in simultaneously optimizing their capital and ownership structure. On the one hand, firms may be able to optimize capital structure but face limitations in adjusting ownership structure due to regulatory requirements, insider trading laws, disclosure rules, and shareholder preferences. On the other hand, firms may optimally adjust ownership structure but fail to optimize capital structure due to financial constraints, covenant restrictions, and regulatory rules. This subsection explores scenarios where firms can optimally choose either their capital structure or ownership structure, but not both.

We first investigate the firm-value-maximizing coupon level ( $ {c}^{\ast } $ ) for a given stock ownership ( $ \lambda $ ) and resource diversion ( $ \theta $ ), that is, we solve the optimization problemFootnote ¹²

(10) $$ {c}^{\ast}\left(\lambda, \theta \right)=\underset{c}{\arg \hskip0.3em \max}\left\{{V}^S\left({x}_0;\lambda, \theta \right)\right\}, $$

and examine how investment thresholds behave as functions of $ \lambda $ and $ \theta $ when $ {c}^{\ast}\left(\lambda, \theta \right) $ endogenously adjust. Table 4 presents the evolution of leverage ratios, yield spreads, and investment thresholds at the optimal capital structure across different values of $ \lambda $ and $ \theta $ . The results suggest that as $ \lambda $ increases and thus the cost-sharing effect gets stronger, the needed debt to balance investment efficiency decreases, leading to a lower optimal coupon level and the resulting leverage ratio and yield spread. Conversely, as $ \theta $ rises, more debt is necessary to mitigate the increased overinvestment incentives, so the optimal coupon level and the associated leverage ratio and yield spread increase accordingly.

TABLE 4Optimal Capital Structure and Associated Firm Characteristics

Table 4 reports the optimal capital structure and associated firm characteristics. Given managerial ownership $ \lambda $ and diversion rate $ \theta $ , the optimal coupon level $ {c}^{\ast } $ is determined as a function of $ \lambda $ and $ \theta $ : $ {c}^{\ast}\left(\lambda, \theta \right)=\underset{c}{\arg \max}\left\{{V}^S\left({x}_0;\lambda, \theta \right)\right\} $ . The associated leverage ratio is $ {L}^S:= {D}^S\left({x}_0;{c}^{\ast}\right)/{V}^S\left({x}_0;{c}^{\ast}\right) $ , and the associated yield spread is $ {YS}^S:= {c}^{\ast }/{D}^S\left({x}_0;{c}^{\ast}\right)-r $ . The corresponding pre-investment firm-optimal default threshold is $ {x}_{\mathrm{\ell}}^F $ , and the firm-optimal and manager-optimal investment thresholds are $ {x}_i^F $ and $ {x}_i^S $ , respectively. Parameters: $ {x}_0=1.9 $ , $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , and $ \lambda =\mathrm{0.0975.} $

Table 4 reveals some interesting patterns regarding the default and investment thresholds. First, as $ \lambda $ increases (and thus $ {c}^{\ast } $ decreases), the firm faces reduced default risk and can wait longer to invest, reflected by a lower $ {x}_{\mathrm{\ell}}^F $ and a higher $ {x}_i^F $ . Second, as $ \theta $ increases (and thus $ {c}^{\ast } $ increases), the default risk rises (higher $ {x}_{\mathrm{\ell}}^F $ ), prompting the firm to invest earlier (lower $ {x}_i^F $ ). Furthermore, it indicates that the manager-optimal investment threshold increases with $ \lambda $ but decreases with $ \theta $ . Recall that managerial ownership $ \lambda $ and resource diversion $ \theta $ influence investment decisions through the critical factor $ \frac{\lambda }{\theta +\lambda \left(1-\theta \right)} $ , as shown in equation (5). This ratio captures the trade-off between the cost-sharing effect and the diversion effect. As $ \lambda $ rises, the ratio increases, implying that the cost-sharing effect dominates (higher $ {x}_i^S $ ) and the firm tends to underinvest. Conversely, as $ \theta $ rises, the ratio decreases, which reflects that the diversion effect prevails (lower $ {x}_i^S $ ) and the firm has a tendency toward overinvestment. Notably, when $ \lambda $ approaches $ 0 $ , the manager bears less of the investment cost and tends to invest immediately.Footnote ¹³ In this case, the substantive results regarding the effect of diversion rate presented in Table 4 still hold.

We next study the symmetric counterpart of problem (10) to solve for the optimal managerial ownership ( $ {\lambda}^{\ast } $ ) given a coupon level ( $ c $ ) and diversion rate ( $ \theta $ ):

$$ {\lambda}^{\ast}\left(c,\theta \right)=\underset{\lambda }{\arg \hskip0.3em \max}\left\{{V}^S\left({x}_0;c,\theta \right)\right\}. $$

Figure 8 illustrates the optimal ownership, investment thresholds, and key ratio, $ \frac{\lambda^{\ast }}{\theta +{\lambda}^{\ast}\left(1-\theta \right)} $ , as $ c $ or $ \theta $ varies. First, note that when $ c=0 $ , the model reduces to the pure-equity benchmark, where $ {\lambda}^{\ast }=1 $ and $ {x}_i^F={x}_i^S $ . As debt increases and the debt overhang effect gets stronger, the required cost-sharing effect to balance investment efficiency decreases, as reflected by a lower $ {\lambda}^{\ast } $ in Graph A. Graph B shows that the firm-optimal investment threshold $ {x}_i^F $ decreases due to the increased default risk. With a reduced $ {\lambda}^{\ast } $ as $ c $ rises, the ratio $ \frac{\lambda^{\ast }}{\theta +{\lambda}^{\ast}\left(1-\theta \right)} $ decreases, implying the diversion effect dominates (lower $ {x}_i^S $ ) and the firm tends to overinvest. Finally, as $ \theta $ rises, the optimal ownership also increases to mitigate the increased diversion effect, as depicted in Graph C. Accordingly, the equilibrium key ratio $ \frac{\lambda^{\ast}\left(c,\theta \right)}{\theta +{\lambda}^{\ast}\left(c,\theta \right)\left(1-\theta \right)} $ , which captures the manager’s trade-off between investment cost and cash flow revenue, remains unchanged as $ \theta $ varies. As a result, the investment and default thresholds also stay constant, as evidenced by equations (A.5), (A.6), and (A.7).

FIGURE 8Optimal Ownership Structure and Associated Firm Characteristics

In Figure 8, given coupon level $ c $ and diversion rate $ \theta $ , the optimal managerial ownership level $ {\lambda}^{\ast } $ is determined as a function of $ c $ and $ \theta $ : $ {\lambda}^{\ast}\left(c,\theta \right)=\underset{\lambda }{\arg \max}\left\{{V}^S\left({x}_0;c,\theta \right)\right\} $ . The corresponding firm-optimal and manager-optimal investment thresholds are $ {x}_i^F $ and $ {x}_i^S $ , respectively. The ratio $ \frac{\lambda^{\ast}\left(c,\theta \right)}{\theta +{\lambda}^{\ast}\left(c,\theta \right)\left(1-\theta \right)} $ in Graph C remains constant across various $ \theta $ . Parameters: $ {x}_0=1.9 $ , $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , and $ c=1.4051 $ .

D. Investment Efficiency Versus Ownership and Capital Structure

This subsection assumes that the firm has no flexibility in choosing its ownership or capital structure, and examines how investment efficiency responds to exogenous changes in $ \lambda $ or $ c $ . Recall that the manager’s equity ownership $ \lambda $ affects investment decisions through two effects: cost-sharing and diversion. These effects are captured separately by $ \lambda $ and $ \theta +\lambda \left(1-\theta \right) $ , with their ratio $ \frac{\lambda }{\theta +\lambda \left(1-\theta \right)} $ being the key factor, as shown in the case of the pure-equity firm in Section III.A. When $ \lambda $ increases, both terms increase, but the cost-sharing effect dominates since the ratio $ \frac{\lambda }{\theta +\lambda \left(1-\theta \right)} $ increases with $ \lambda $ . This explains why the manager-optimal investment threshold $ {x}_i^S $ increases with $ \lambda $ , as shown in Graph A of Figure 9. In contrast, the firm-optimal investment threshold $ {x}_i^F $ decreases with $ \lambda $ . Note that a stronger cost-sharing effect caused by increased $ \lambda $ reduces the attractiveness of investment to the manager, which reduces his opportunity cost of pre-investment default and thereby incentivizes earlier default, as confirmed by Graph B. Therefore, it is optimal for the firm to exercise the investment option earlier. These patterns imply that there is underinvestment when $ \lambda $ is high and overinvestment when $ \lambda $ is low.

FIGURE 9Investment and Default Thresholds Under Both Conflicts Versus Managerial Ownership

Figure 9 Graph A highlights an intersection point between $ {x}_i^S $ and $ {x}_i^F $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , and $ {c}^{\ast }=1.4051 $ .

Turning to the effect of the coupon payment $ c $ , Figure 10 illustrates how the coupon payment $ c $ affects investment and default decisions. A higher coupon payment $ c $ increases the debt overhang effect, so that the manager finds it optimal to delay investment. This is reflected by the higher manager-optimal investment threshold in Graph A. A larger debt burden also induces earlier default, as confirmed by the higher default thresholds in Graph B. Given that firms default earlier under higher coupon payments, the firm-optimal strategy is to exercise the investment option earlier, which explains why the firm-optimal investment threshold decreases with $ c $ in Graph A. These patterns imply that there is underinvestment when $ c $ is high and overinvestment when $ c $ is low.

FIGURE 10Investment and Default Thresholds Under Both Conflicts Versus Coupon Payment

Figure 10 Graph A highlights an intersection point between $ {x}_i^S $ and $ {x}_i^F $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , and $ {\lambda}^{\ast }=0.0975 $ .

1. Investment Efficiency

Previous figures have shown that firm-optimal and manager-optimal investment thresholds can coincide so that underinvestment and overinvestment effects exactly offset each other. In this case, the investment policy is efficient. Figure 11 plots the combinations of ownership ( $ \lambda $ ) and capital structure ( $ c $ ) that achieve this efficiency, which is referred to as the Firm-Optimal Investment (FOI) curve. Several interesting patterns emerge. First, there exists a negative relationship between $ c $ and $ \lambda $ along the efficiency curve. This substitutability arises because both debt overhang and cost-sharing effects contribute to underinvestment. Second, the curve shifts upward with the diversion rate $ \theta $ , as higher diversion increases incentives for overinvestment. For any given $ c $ , a higher $ \lambda $ is needed to curb the overinvestment incentives. Similarly, the curve shifts upward with tax rate $ \tau $ because higher default thresholds reduce the market value of debt and weaken the debt overhang effect. For a given $ \lambda $ , a higher $ c $ is needed to curb overinvestment. Interestingly, Graph C shows that the substitutability between $ c $ and $ \lambda $ decreases with volatility ( $ \sigma $ ). This occurs because higher $ \sigma $ , by reducing the market value of debt, reduces the incremental debt overhang effect from an increase in $ c $ , requiring a smaller reduction in $ \lambda $ to maintain investment efficiency.

FIGURE 11Firm-Optimal Investment Curve Versus Diversion Rate, Tax Rate, and Volatility

In Figure 11, combination $ \left\{c,\lambda \right\} $ on each curve achieves the firm-optimal investment (FOI). Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , and $ \theta =0.01 $ .

2. Measuring Efficiency Loss in Investment

Recall that $ {V}^F $ and $ {V}^S $ are the initial firm values under firm-optimal and manager-optimal investment policies, respectively. We measure agency cost as the percentage efficiency loss in initial firm value due to suboptimal investment policies (i.e., $ \left({V}^F-{V}^S\right)/{V}^F $ ). Under $ \left\{{c}^{\ast },{\lambda}^{\ast}\right\} $ and other parameters in Table 1, agency costs are $ 0.02\% $ . This is lower than $ 0.07\% $ in the levered benchmark, where $ \theta =0 $ or $ \lambda =1 $ , and considerably lower than $ 0.90\% $ reported by Mauer and Ott (Reference Mauer, Ott, Brennan and Trigeorgis2000) and $ 0.61\% $ by Hackbarth and Mauer (Reference Hackbarth and Mauer2012). The relatively low agency costs in our model arise because the diversion effect introduces overinvestment incentives, which partially offset the debt overhang effect. Figure 12 illustrates how agency costs vary with parameters $ \left\{\theta, \tau, \sigma, \lambda, c\right\} $ . All graphs exhibit U-shaped patterns with a minimum value of 0.

FIGURE 12Agency Costs Under Both Conflicts

In Figure 12, agency costs are the efficiency loss in initial firm value, measured as $ \left({V}^F-{V}^S\right)/{V}^F $ and expressed in percentage (%). The five graphs respectively vary i) diversion rate, ii) tax rate, iii) volatility, iv) managerial ownership, and v) coupon payment, each exhibiting a U-shaped curve with a minimum value of 0. Parameters: $ {x}_0=1.9 $ , $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , $ q=2 $ , $ \theta =0.01 $ , $ {c}^{\ast }=1.4051 $ , and $ {\lambda}^{\ast }=0.0975 $ .

A. Extension: Ownership-Dependent Resource Diversion

This section extends the model by assuming $ \theta $ is a decreasing function of $ \lambda $ .Footnote ¹⁴ This extension is motivated by the rationale that as $ \lambda $ increases, managers are less inclined to divert resources because they share with minority shareholders the loss in firm value as resources are diverted (see Jensen and Meckling (Reference Jensen and Meckling1976)).Footnote ¹⁵

Specifically, we assume $ \theta =f\left(\lambda \right) $ , satisfying $ {f}^{\prime}\left(\lambda \right)<0 $ , $ f(1)=0 $ , and $ f(0)=b $ , where $ b\in \left(0,1\right) $ stands for the maximum diversion rate. This formulation implies that when the manager retains all the equity, he has no incentive to divert. The derivative $ {f}^{\prime}\left(\lambda \right) $ reflects the sensitivity of the manager’s diversion rate to his equity stake. In the numerical analysis, we model $ f\left(\cdot \right) $ as an exponential function,Footnote ¹⁶ that is,

$$ \theta =f\left(\lambda \right):= {\left(1-b\right)}^{\lambda }+b-1, $$

where $ b=0.01 $ . The solution steps remain unchanged.

Compared to the case of fixed diversion rate, when $ \theta $ decreases with $ \lambda $ , an increase in ownership has an additional effect of diminishing the diversion effect, which further reduces the manager’s incentives to overinvest. We can see this effect clearly in the case of a pure-equity firm. By substituting $ \theta =f\left(\lambda \right) $ into equation (5), we derive that

(11) $$ \frac{\mathrm{d}{\hat{x}}_i^S}{\mathrm{d}\lambda }=\left(\frac{f\left(\lambda \right)\overset{\mathrm{d}\mathrm{iminished}\ \mathrm{diversion}}{\overbrace{-\lambda \hskip0.1em \left(\hskip0.1em 1-\lambda \hskip0.1em \right){f}^{\prime}\left(\lambda \right)}}}{f\left(\lambda \right)+\lambda \left(1-f\left(\lambda \right)\right)}\right)\left(\frac{{\hat{x}}_i^S}{\lambda}\right)\ge 0. $$

The incremental delay in investment is captured by the additional positive term $ \left(-\lambda \left(1-\lambda \right){f}^{\prime}\left(\lambda \right)\right) $ in equation (11), which is equal to $ 0 $ in equation (6).

For a levered firm, the qualitative results continue to hold when the diversion rate depends on ownership. Table 5 further compares two scenarios: one where the diversion rate $ \theta $ is fixed, and another where $ \theta $ decreases as ownership $ \lambda $ increases. When $ \theta $ decreases with ownership, we find that a lower ownership stake $ \lambda $ is needed to achieve the firm-optimal investment level compared to the case of fixed $ \theta $ . Moreover, this difference in needed ownership levels becomes less pronounced as coupon payment $ c $ increases.

TABLE 5The Ownership Needed to Achieve Firm-Optimal Investment

Table 5 reports the needed ownership to achieve firm-optimal investment for various coupon levels. The upper panel presents the case where the diversion rate is fixed at $ \theta \equiv 0.01 $ , while the lower panel presents the case where the diversion rate is modeled as $ \theta =f\left(\lambda \right):= {\left(1-b\right)}^{\lambda }+b-1 $ , with $ b=0.01 $ . Parameters: $ r=0.07 $ , $ \mu =0 $ , $ \sigma =0.1 $ , $ \tau =0.35 $ , $ \alpha =0.6 $ , $ I=20 $ , and $ q=2. $

Finally, the diminished diversion effect due to higher ownership contributes to the increase in firm value. We can see this effect from the pure-equity case. By equations (5) and (A.3), and applying $ \theta =f\left(\lambda \right) $ , we derive that

(12) $$ \frac{\mathrm{d}{\hat{V}}^S\left({x}_0;\lambda \right)}{\mathrm{d}\lambda }=\left(I{\gamma}_{+}\right)\left(\frac{1-\lambda }{\lambda}\right)\left(\frac{{\left[f\left(\lambda \right)\right]}^2\hskip0.1em \overset{\mathrm{d}\mathrm{iminished}\ \mathrm{diversion}}{\overbrace{-\lambda \left(1-\lambda \right){f}^{\prime}\left(\lambda \right)f\left(\lambda \right)}}}{{\left[f\left(\lambda \right)+\lambda \left(1-f\left(\lambda \right)\right)\right]}^2}\right){\left(\frac{x_0}{{\hat{x}}_i^S}\right)}^{\gamma_{+}}\ge 0. $$

The additional positive term $ \left(-\lambda \left(1-\lambda \right){f}^{\prime}\left(\lambda \right)f\left(\lambda \right)\right) $ in equation (12) reflects the incremental firm value from the reduced diversion effect as $ \lambda $ increases, which equals $ 0 $ in equation (7).

B. Empirical Relevance

Our model integrates two types of conflicts, which allows us to identify the diversion effect and cost-sharing effect that impact investment efficiency alongside the debt overhang effect. We show that the diversion effect incentivizes overinvestment, while the cost-sharing effect more closely aligns managerial incentives with firm interests, thereby reducing the tendency to overinvestment. These findings are consistent with Morck et al. (Reference Morck, Shleifer and Vishny1989), who assert that there is a natural tendency for the manager to allocate company resources in ways that maximize his own interests, often conflicting with the interests of external shareholders. Moreover, their empirical research indicates that granting a certain level of equity to the manager can effectively mitigate this conflict of interest.

We have demonstrated that higher managerial ownership mitigates overinvestment in a pure-equity firm. This result is supported by empirical evidence indicating that pure-equity family firms are generally reluctant to get access to external equity financing, as reported in Romano et al. (Reference Romano, Tanewski and Smyrnios2001), Blanco-Mazagatos, De Quevedo-Puente, and Castrillo (Reference Blanco-Mazagatos, De Quevedo-Puente and Castrillo2007), Wu et al. (Reference Wu, Chua and Chrisman2007), and Croci, Doukas, and Gonenc (Reference Croci, Doukas and Gonenc2011). Further, we have shown that in a levered firm, there is underinvestment when $ \lambda $ is high and overinvestment when $ \lambda $ is low. This finding is consistent with Hermalin and Weisbach (Reference Hermalin and Weisbach1991), Griffith (Reference Griffith1999), and Davies, Hillier, and McColgan (Reference Davies, Hillier and McColgan2005), who provide empirical evidence of a significant nonlinear relationship between corporate value and managerial ownership, which is also supported by Morck et al. (Reference Morck, Shleifer and Vishny1988) and McConnell and Servaes (Reference McConnell and Servaes1990), who document an inverted U-shaped relationship between firm performance and insider ownership, as illustrated in Graph B of Figure 3.

Our model generates several new testable predictions. First, we have shown how agency conflicts and market conditions, described by the parameters $ \left\{\theta, \tau, \sigma, \lambda, c\right\} $ , affect investment timing and agency costs. Second, we provide predictions on how ownership structure and capital structure vary with firm characteristics and the institutional environment represented by $ \left\{\theta, \tau, \sigma \right\} $ .

C. Relation to the Literature

Our model builds upon the growth option framework of Mauer and Ott (Reference Mauer, Ott, Brennan and Trigeorgis2000) and Hackbarth and Mauer (Reference Hackbarth and Mauer2012), who emphasize the underinvestment problem and the agency costs of debt. Beyond the traditional trade-off between tax benefits and bankruptcy costs (see Leland (Reference Leland1994)), we enhance their framework by incorporating the manager’s private benefits and show how different forces interact and shape the firm’s investment and financing policies.

Our work extends existing work on agency frictions (see, e.g., Jensen and Meckling (Reference Jensen and Meckling1976), Jensen (Reference Jensen1986), Stulz (Reference Stulz1990), and Morellec (Reference Morellec2004)). Jensen and Meckling (Reference Jensen and Meckling1976) suggest that increasing managerial ownership can align the interests of managers and shareholders, thereby reducing agency costs. Jensen (Reference Jensen1986) argues that debt can mitigate the agency costs of free cash flow by limiting the cash flow available for managerial discretion. In a two-period model, Stulz (Reference Stulz1990) examines how optimal financing policies can mitigate agency frictions and investment inefficiencies. The friction in Stulz (Reference Stulz1990) is the managers’ free cash flow problem, where managers may invest in negative-NPV projects to derive perquisites from investment. In that model, shareholders cannot observe cash flows or investment decisions; therefore, managers tend to overinvest. Debt serves as a useful tool to force managers to pay out cash and curb incentives for overinvestment. However, this hard-claim nature of debt can lead to underinvestment when the realized cash flows are low. Morellec (Reference Morellec2004) further extends Stulz (Reference Stulz1990) to a continuous-time framework. In his model, the manager tends to overinvest in negative-NPV projects due to his ability to derive utility from investment. He then shows how debt financing can mitigate managerial diversion by restricting free cash flow.

Our framework differs in several key ways. Unlike Stulz (Reference Stulz1990), there is no asymmetric information in our model. Most fundamentally, Jensen (Reference Jensen1986), Stulz (Reference Stulz1990), and Morellec (Reference Morellec2004) all focus on managers’ free cash flow problem, where managers may invest in negative-NPV projects; however, our model has no free cash flow problem since investments always have positive NPV and the manager (along with equity holders) has deep pockets, following the standard approach as in the Leland (Reference Leland1994) model. In our model, overinvestment incentives arise from managers’ ability to divert cash flows, and firms issue debt to exploit the tax shield benefit of debt.Footnote ¹⁷ Indeed, we show that when the tax rate is 0, the firm optimally never issues any debt.

Moreover, the dynamic nature of our model reveals important interactions between investment and default decisions that are absent from the Stulz’s (Reference Stulz1990) static framework. Specifically, the uncertain timing between investment and default highlights how investment options influence default incentives, similar to the results in Mauer and Ott (Reference Mauer, Ott, Brennan and Trigeorgis2000) and Hackbarth and Mauer (Reference Hackbarth and Mauer2012). For instance, we find that the post-investment default threshold is independent of agency frictions ( $ \theta $ , $ \lambda $ ) and tax shield benefits ( $ \tau $ ). However, the pre-investment default threshold varies with these parameters, highlighting how investment options interact with default incentives and agency friction severity.

Our dynamic model also generates different predictions from Stulz (Reference Stulz1990) regarding the effects of cash flow volatility. While Stulz (Reference Stulz1990) finds that higher volatility universally reduces firm value across debt levels, our model, consistent with Dixit and Pindyck (Reference Dixit and Pindyck1994), shows that increased volatility can enhance the value of investment options and potentially increase firm value. As shown in Figure 6, higher volatility mitigates underinvestment but exacerbates overinvestment.

Morellec et al. (Reference Morellec, Nikolov and Schürhoff2012), (Reference Morellec, Nikolov and Schürhoff2018) study the impact of managerial diversion on leverage decisions. Our work has a different focus: we study how joint ownership and capital structure decisions interact to mitigate and eliminate investment inefficiency associated with the timing of investment. Kanagaretnam and Sarkar (Reference Kanagaretnam and Sarkar2011) study how managerial ownership addresses underinvestment in a growth option model, but there is no incentive for overinvestment, and they do not study optimal financing structures.