# Equilibrium Defaultable Corporate Debt and Investment111We thank Lorenzo
Garlappi, Bob Goldstein, Zhiguo He, David D. Yao and Xunyu Zhou for helpful
comments. We alone are responsible for any errors.

Hong Chen Shanghai Advanced Institute in Finance, Shanghai Jiao Tong
University, Shanghai, China. E-mail<EMAIL_ADDRESS>Murray Z. Frank
Department of Finance, University of Minnesota, Minneapolis, MN 55455 and
SAIF. Email<EMAIL_ADDRESS>
(February 11, 2022)

In dynamic capital structure models with an investor break-even condition, the
firm’s Bellman equation may not generate a contraction mapping, so the
standard existence and uniqueness conditions do not apply. First, we provide
an example showing the problem in a classical trade-off model. The firm can
issue one-period defaultable debt, invest in capital and pay a dividend. If
the firm cannot meet the required debt payment, it is liquidated. Second, we
show how to use a dual to the original problem and a change of measure, such
that existence and uniqueness can be proved. In the unique Markov-perfect
equilibrium, firm decisions reflect state-dependent capital and debt targets.
Our approach may be useful for other dynamic firm models that have an investor
break-even condition.

Keywords: corporate investment, defaultable debt, trade-off theory,
equilibrium existence and uniqueness proof

##  1 Introduction

We consider a discrete-time model of a firm that can issue one-period
defaultable debt, invest in capital and pay a dividend. The firm has a concave
production function of the capital and production is subject to an exogenous
random shock. In each period, the firm starts with a capital level, a bond
payment that is due on old debt. The exogenous state is revealed so production
revenue becomes known. The firm decides the amount of the new debt to be
issued or the savings, investment in capital, and dividends to pay. The firm
must make the promised payment on old debt if it can. If the production
revenue and new debt are not sufficient to make the promised payment to old
debt, the firm is in default. In default the firm is liquidated and the
proceeds go to the debt holder. Due to possible default the debt is risky. The
price of debt is determined by the equilibrium requirement that the investor
must expect to break-even relative to the risk-free interest rate.

The firm maximizes its expected total discounted dividend over an infinite
horizon or to the time epoch when it enters default (whichever comes first).
The Bellman equation for the firm problem does not generate a contraction
mapping. Accordingly, the usual existence and uniqueness proof as in Stokey
and Lucas Jr. (1989) and Puterman (1994) does not apply. This is due to the
connection between default risk and debt pricing. Specifically, the current
debt price depends on whether the firm will default in the next period or the
value function of the next period. We provide an alternative approach by
introducing a dual problem. With a change of measure, the dual problem is
shown to be a contraction mapping and there exists a unique solution. This
shows that the Markov-perfect equilibrium (Definition 2.3) exists and is
unique for the original problem. Under the equilibrium, we show that the price
of debt decreases with the amount of debt issued, and the firm never saves.
There exists an optimal capital target and bond target. These depend on the
exogenous state. When capital is below target and the cash flow is sufficient,
the firm issues debt to reach the target.

Our model is a basic capital structure trade-off theory model. It is widely
believed that trade-off theory provides a good account of the debt and
investment decisions of large public firms, see Ai et al. (2021). That belief
is largely grounded in a sequence of influential papers including, Cooley and
Quadrini (2001); Hennessy and Whited (2007); Nikolov et al. (2021). These
papers show that numerical simulations of trade-off models with investment and
financing frictions can generate empirically appealing solutions when the
parameters are reasonably calibrated or estimated. However, the equilibrium
theory underpinning that empirical success is less well established than
generally understood. Without establishing the existence and uniqueness for
these models, any numerical simulations may not be a good reflection of the
economic forces at work. If the Markov-perfect equilibrium is not unique, then
interplay among the equilibria may be important for an understanding of the
driving forces. If however, the equilibrium exist and is unique, then more
traditional approaches may suffice (Strebulaev and Whited, 2012).

Why has the difficulty that we address not already been dealt with in the
literature? Previous studies have generally sidestepped the issue by making
specialized assumptions. For example Gomes (2001) does not require that debt
investors expect to break-even. Hennessy and Whited (2007) impose a debt
break-even condition on the debt market, but they have a reduced form equity
market without a break-even requirement. Conditional on a given interest rate,
they provide conditions such that the firm’s value function exists and it
satisfies a contraction mapping. However, they also caution that there are
technical issues such that in both Cooley and Quadrini (2001) and Hennessy and
Whited (2007) the firm’s problem is not necessarily concave, and their overall
equilibrium is not necessarily unique. Nikolov et al. (2021) impose an
investor break-even condition, and find that the model provides a good account
of the decisions of large public firms in Compustat. But they do not prove
equilibrium existence or uniqueness. Our model does not have specialized
assumptions to sidestep the technical difficulties. Instead we stick with
classical assumptions and show how to deal with the associated challenges.

The conceptual difficulty is not a mathematical or a technical quirk. It is
due to the fundamental equilibrium requirement that investors will not
knowingly invest at an avoidable expected loss. Together with the possibility
of firm default, this means that the original model does not define a
contraction mapping, and so the standard justification for existence and
uniqueness in Stokey and Lucas Jr. (1989) does not apply to the model.
Corresponding issues may arise in other dynamic equilibrium settings in which
future payoffs depend both on the realization of a stochastic process and on
decisions of other people.

There is a large volume of the literature concerning default, particularly,
following the last financial crisis; see, for example, Glasserman and Young
(2016); Chen et al. (2021), and the references in their works. These works are
concerned with the contagion effects in a network setting. They do not model
the lending decision of the debt holder.

Closely related to our concern about establishing uniqueness in a basic risky
corporate debt model, is found in the literature on sovereign debt (Auclert
and Rognlie, 2016; Aguiar and Amador, 2019). Due to the investor break-even
condition, the natural government debt problem does not define a contraction
mapping. Our focus on the dual problem instead of the primal problem follows
Aguiar and Amador (2019). The approach will of course not apply if there are
multiple equilibria. Multiple equilibria have been found in problems with
multi-period debt (Aguiar et al., 2019; Dangl and Zechner, 2021) and in some
models with financial frictions (Hugonnier et al., 2015).

In this paper, we show that the model does actually have a unique Markov
perfect equilibrium. To prove this, we begin as in Aguiar and Amador (2019) by
obtain a dual to the original problem. But then as in Lippman (1975) and van
Nunen and Wessels (1978) we change the measure. Under the change of measure,
the dual to the original model satisfies the Blackwell (1965) conditions. So
the original model does actually have a unique equilibrium, even with firm
default and the investor break-even requirement. Firm default is a common
concern, so our approach may be useful beyond the specifics of our model.
Hotchkiss et al. (2008) provides a very good review of the default literature.

We examine two variations on the basic model. First, we replace the capital
and the production function with an exogenous cash flow in the main model.
This is quite common (Modigliani and Miller, 1958; Leland, 1994), particularly
in continuous time capital structure models. Second, we complicate the main
model by including capital adjustment costs, which is also quite common in the
literature (Summers, 1981). In both cases we maintain the assumption of single
period debt. In both cases our approach establishes existence and uniqueness
of the equilibrium. Such models are commonly solved numerically. Having a
version of the model with an equilibrium that is known to be unique, is useful
for checking code while developing more complex numerical models (Judd, 1998).

The rest of the paper is organized as follows. The model, the equilibrium
definition and the main result are presented in section 2. The main result is
proved in section 3. The proof starts with a counterexample to show that the
usual Bellman equation, namely, the primal problem, does not give a
contraction mapping. Then, we introduce a (corresponding) dual problem with
the solution to the dual uniquely related to the solution to the primal
problem. We show that with a change of measure, the Bellman equation
associated with the dual problem gives rise to a contraction mapping and hence
has a unique solution. In section 4, we provide some characterizations of the
equilibrium. In section 5, we consider two alternative models which are quite
common in the literature. One model replaces the capital and the production
with an exogenous cash flow, and the other model adds the capital adjustment
costs. We show that our approach can be used in both cases to show the
existence and the uniqueness of Markov-perfect equilibrium. Section 6
concludes with a summary and discussions.

##  2 The model and the main result

The base model assumptions are classical: the firm objective function is the
expected present value of dividends, there is a linear flow budget constraint
that imposes cash in equals cash out, there is a production function that
depends on capital and a shock, capital depreciates, there is no capital
adjustment cost, firm revenue is taxable with depreciation tax deductible. The
productivity shock is given by a Markov process. Debt is for a single period.
Interest payments are tax deductible and the firm may be in default in some
states. The investor must expect to break-even on the debt purchased from the
firm. If the firm cannot make a promised debt payment, the firm is liquidated
with the proceeds going to the debt holder.

Consider a firm with a concave production function,

$\displaystyle F(k,z)=zAk^{\alpha},$ (1)

that depends on capital $k$, a random shock $z$, and fixed parameters $A>0$
and $\alpha\in(0,1)$. The exogenous shock follows a bounded discrete-time
Markov process $\\{z_{t},t\geq 0\\}$ with a state space
$\mathbb{Z}\subseteq\Re_{+}$, where $\Re_{+}$ is the set of all non-negative
real numbers.

###### Assumption 2.1

The Markov process $\\{z_{t},t\geq 0\\}$ has a bounded state space
$\mathbb{Z}$. That is, there exists a positive constant $\bar{z}<\infty$ such
that ${\bf\sf P}(0\leq z_{t}\leq\bar{z})=1$. Transitions are homogeneous.

Unless necessary, we will usually drop the time subscript $t$ for ease of
presentation. For any (possibly vector) process $\\{x_{t},t\geq 0\\}$, when
both the subscript $t$ (the current period) and the subscript $t+1$ (the next
period) appear in the same set of equations, we write $x$ for $x_{t}$ and
$x^{\prime}$ for $x_{t+1}$.

At the start of a period the firm inherits bonds $b$, and physical capital
$k$. The value of the shock $z$, is revealed. Provided the firm is not in
default, it chooses bonds, dividends, and investment in physical capital
denoted respectively as $b,d,i$. However, if the shock is too bad, then even
with the best decisions the firm is unable to make the promised payment to
debt. The firm enters a permanent default. Then equity gets zero. Debt gets a
liquidation payoff of $L(k^{\prime})$. When the default happens, it happens
after the start of the period; so, the capital level $k^{\prime}$, instead of
$k$ is used, to account for depreciation. It is assumed that $L(\cdot)$ is a
non-decreasing function.

The price of debt is determined by market equilibrium. The debt investor must
expect to break even. There is an exogenous risk-free interest rate $\rho$,
where $0<\rho<1$. The price of debt is $q$. If the firm promises to pay
$b^{\prime}$ in the next period, it will get $qb^{\prime}$ in the current
period. Investors understand that this promise will not be kept if it is not
feasible to do so. When choosing the amount of debt to issue, the firm
recognizes that the price of debt depends on the current shock, the capital
after investment, and the amount of debt issued,
$q=q(z,b^{\prime},k^{\prime})$. These are determined as part of an
equilibrium, as shown below.

There is a corporate tax on firm revenue, $\tau$. Depreciation $\delta k$, and
interest payments $(1-q(z,b^{\prime},k^{\prime}))b^{\prime}$, are both tax
deductible. Investment in physical capital is $i=k^{\prime}-(1-\delta)k$.

The firm flow budget is,

$\underbrace{(1-\tau)F(z,k)}_{\begin{subarray}{c}\text{revenue}\\\
\text{after-tax}\end{subarray}}+\underbrace{\tau(\delta
k+(1-q)b^{\prime})}_{\text{tax deduction
revenue}}+\underbrace{qb^{\prime}-b}_{\begin{subarray}{c}\text{net debt}\\\
\text{revenue}\end{subarray}}=\underbrace{d}_{\text{dividends}}+\underbrace{i,}_{\text{investment}}$
(2)

where $q=q(z,b^{\prime},k^{\prime})$. As in Strebulaev and Whited (2012), the
interest tax deduction is received at the same time that the debt is issued.
This is a common simplification. Suppose instead that the tax deduction is
received later. Then the interest tax deduction at time $t$ would become
$\tau(1-q(z_{t-1},b_{t},k_{t}))b_{t}$. That would imply that both $z_{t-1}$
and $z_{t}$ are in the current budget, complicating the algebra.

The state of the firm is described by a three-dimensional vector $x=(z,b,k)$.
The first component is the exogenous state $z\in\mathbb{Z}$. The second
component is the (net) debt $b\in\mathbb{B}=[\underline{$b$},\bar{b}]$, where
$\underline{$b$}\leq 0\leq\bar{b}<\infty$. A negative value of $b$ means
savings. The finite upper bound $\bar{b}$ imposed on the debt is a No-Ponzi
condition. The third component is the capital level $k\in\Re_{+}$. So
$\mathbb{X}=\mathbb{Z}\times\mathbb{B}\times\Re_{+}$ is the state space. Let
$V(x)\equiv V(z,b,k)$ denote the equilibrium value of the firm equity. Note
that $V(x)=0$ if the firm is in a default state, and $V(x)>0$ if the firm is
in a non-default state.

The firm’s per-period net resources apart from debt, is
$R(z,k)=(1-\tau)F(z,k)+[1-\delta(1-\tau)]k$. Given the state $(z,b,k)$, the
net revenue of the firm is given by

$\displaystyle(1-\tau)F(z,k)+\tau\delta
k+[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-b-i$ $\displaystyle=$
$\displaystyle
R(z,k)+[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-k^{\prime}-b,$

Let

$\displaystyle S(z,k)$ $\displaystyle=$
$\displaystyle\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b}\atop
k^{\prime}\geq(1-\delta)k}\big{\\{}[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-k^{\prime}\big{\\}}.$
(3)

Then the maximum possible net revenue of the firm is given by
$R(z,k)+S(z,k)-b$. This must be non-negative for the firm to avoid default.
Notice that if the supremum $\sup$ is not achievable by any
$(b^{\prime},k^{\prime})$, then we would require $R(z,k)+S(z,k)-b$ to be
strictly positive to avoid default. We show later that this supremum is
achievable in the equilibrium. So we use the non-strict inequality for
simplicity in the definition of the feasible set.

We can now define the feasible set of non-default states.

###### Definition 2.2

In any equilibrium, the feasible set is defined as,

$\mathbb{X}_{feas}=\Big{\\{}(z,b,k)\in\mathbb{X}\ \big{|}\ R(z,k)+S(z,k)\geq
b\Big{\\}}.$

Let

$\underline{b}(z,k)=\inf\\{b\geq\underline{b}:(z,b,k)\in\mathbb{X}_{feas}\\}\quad\mbox{and}\quad\bar{b}(z,k)=\sup\\{b\leq\bar{b}:(z,b,k)\in\mathbb{X}_{feas}\\}.$

Notice that when $k$ is sufficiently large,
$\underline{$b$}(z,k)=\underline{$b$}$ and $\bar{b}(z,k)=\bar{b}$. This is due
to the fact that
$S(z,k)\leq\left[\tau+(1-\tau)/(1+\rho)\right]\bar{b}-(1-\delta)k$ (in view of
the break-even condition (7)) and then

$\displaystyle R(z,k)+S(z,k)\leq(1-\tau)F(z,k)+\tau
k+\left[\tau+(1-\tau)/(1+\rho)\right]\bar{b}\to\infty\mbox{ as }k\to\infty.$

Firm problem. Let $x=(z,b,k)\in\mathbb{X}$ and take
$q=q(z,b^{\prime},k^{\prime})$ as given.

If $x\in\mathbb{X}_{feas}$, then $\displaystyle\qquad
V(x)=\sup_{(b^{\prime},d,i)}\big{\\{}d+\beta\mathbf{\mathsf{E}}_{z}V(x^{\prime})\big{\\}},$
(4) subject to $\displaystyle\qquad\qquad(1-\tau)F(z,k)+\tau\delta
k+[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-b=d+i,$
$\displaystyle\qquad\qquad k^{\prime}=(1-\delta)k+i,$
$\displaystyle\qquad\qquad\underline{$b$}\leq b^{\prime}\leq\bar{b},d\geq
0,i\geq 0.$ $\displaystyle\mbox{Otherwise, }V(x)=0.$ (5)

If the firm is viable, the firm is maximizing the expected present value of
dividends, subject to a flow budget constraint, the investment condition, and
the bounds on debt, dividends and investment. If the firm is not viable, the
payoff is zero.

Debt investors will not intentionally invest at a loss. There are many
potential investors that compete with each other. These considerations are
commonly represented by the requirement that the investor expects to just
break even in an equilibrium.

Investor break-even condition. Given the firm value function $V$, the firm’s
exogenous state $z$, and the firm’s capital level $k^{\prime}$. If the bond
issue is $b^{\prime}\leq 0$, then $q(z,b^{\prime},k^{\prime})=1/(1+\rho)$. If
$b^{\prime}\geq 0$, then

$\displaystyle q(z,b^{\prime},k^{\prime})$
$\displaystyle=\frac{b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})=0\\}}}{b^{\prime}\left(1+\rho\right)}\qquad\mbox{
and}$ (6) $\displaystyle q(z,b^{\prime},k^{\prime})$
$\displaystyle\leq\frac{1}{1+\rho}.$ (7)

Condition (7) means that the debt investor is willing to pay a price no
greater than the price at a risk free rate. After-all, the investor has access
to that risk free rate. Combining the above two conditions, we have
$[b^{\prime}-L(k^{\prime})]\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})=0\\}}\geq
0$. This means that at the time of default, the liquidation value of the
capital $L(k^{\prime})$ must be no more than the debt payment $b^{\prime}$. We
assume that $L(k^{\prime})$ is non-decreasing in $k^{\prime}$.

Equality (6) can be rewritten as

$\displaystyle
q(z,b^{\prime},k^{\prime})=\frac{b^{\prime}-[b^{\prime}-L(k^{\prime})]\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})=0\\}}}{b^{\prime}\left(1+\rho\right)}.$

This shows that $q(z,b^{\prime},k^{\prime})$ is increasing in the value
function $V$.

With the firm problem and the investor break-even conditions in hand, we can
now define the equilibrium.

###### Definition 2.3

A Markov-perfect equilibrium is a value function $V$ and a price function $q$
such that

(1)

Given $q$, the value function $V$ satisfies the Bellman equation (4) and (5),
and the supremum in (4) is attained for some policy;

(2)

Given $V$, $q$ satisfies the break-even condition (6)-(7).

The Markov Perfect equilibrium is a popular refinement of the Nash
equilibrium. It is Markov in the sense that all equilibrium decision rules are
functions of the state variables and do not depend on the rest of the history.
The equilibrium is Perfect in the sense of being complete and consistent with
optimizing decisions in all possible future states (Levhari and Mirman, 1980;
Maskin and Tirole, 2001).

###### Theorem 2.4

Under Assumption 2.1, there exists a unique Markov-perfect equilibrium.

To justify existence and uniqueness in dynamic corporate capital structure
models it is common to cite Theorem 9.6 in Stokey and Lucas Jr. (1989). Their
theorem in turn is justified by Blackwell (1965). In section 3.1, we show by
example that due to the investor break-even condition, the usual justification
does not apply to our model. In section 3.2, we provide an alternative method
that does apply.

##  3 Proof of the main result

We start with an example to show that the obvious operator associated with the
original firm problem is not a contraction mapping (subsection 3.1). We next
introduce a dual problem in subsection 3.2 and relate the value function of
the original firm problem to the value function of the dual problem. Finally,
we show the operator associated with the dual problem has a unique fixed point
(subsection 3.3).

###  3.1 Primal approach

Here we show by example that the obvious operator associated with the original
firm problem does not define a contraction mapping. The standard approach
starts by defining an operator ${\bf\sf S}$ as follows. For any bounded
function $v$ defined on $\mathbb{X}$, define the operator by

$\displaystyle({\bf\sf S}v)(x)$ $\displaystyle=$
$\displaystyle\sup_{(b^{\prime},d,i)}\big{\\{}d+\beta\mathbf{\mathsf{E}}_{z}v(x^{\prime})\big{\\}},$
(8) subject to $\displaystyle(1-\tau)F(z,k)+\tau\delta
k+[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-b=d+i,$ (9)
$\displaystyle k^{\prime}=(1-\delta)k+i,$ $\displaystyle\underline{$b$}\leq
b^{\prime}\leq\bar{b},d\geq 0,i\geq 0,$ $\displaystyle
q(z,b^{\prime},k^{\prime})=1/(1+\rho)\mbox{ if }b^{\prime}\leq 0,$
$\displaystyle
q(z,b^{\prime},k^{\prime})=\frac{b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{v(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{v(x^{\prime})=0\\}}}{b^{\prime}\left(1+\rho\right)}\mbox{
if }b^{\prime}>0,$

if the above constraint set is feasible; and ${\bf\sf S}v=0$ otherwise. A
Markov-perfect equilibrium is a fixed point of the operator ${\bf\sf S}$. The
usual proof is to show that the operator ${\bf\sf S}$ is a contract mapping
under the uniform norm. At first it might seem that this would be a
contraction mapping due to the presence of the discounting factor $\beta<1$.
However, equality (9) causes a problem that has not been widely recognized in
the dynamic corporate finance literature.

The value function $v$ appears in equality (9) where it might not normally
appear. Equation (9) reflects the need to take into account potential losses
in default and how that affects the price of debt. The fact that investors
need to account for default losses is of course, well-known. But it creates an
extra linkage in the model that is not present in the standard Bellman
equation setup. In the standard model, the objective function takes market
prices as given. Here the market price $q$ depends on the state dependent $v$
because the investor needs to determine the value that will be received in a
default.

To provide an example, it is sufficient to find two continuous functions
$v_{1}$ and $v_{2}$ defined on $\mathbb{X}$ such that $||{\bf\sf
S}v_{2}-{\bf\sf S}v_{1}||\geq||v_{2}-v_{1}||$, where $||\cdot||$ is a uniform
norm defined on $\mathbb{X}$. Assume $v_{1}(x)=0$ for at least some $x$, so
that default may actually happen. We can find a simple example with
$v_{1}\equiv 0$.

Assume: 1) there is an exogenous and deterministic shock process, so
$\mathbb{Z}$ contains a single element $z$; 2) set $\tau=0$; 3) set $L(k)=2\nu
k^{1/2}$; 4) set $v_{1}(x)=0$ and $v_{2}(x)=\epsilon$ for all
$x\in\mathbb{X}$, where the positive constants $\nu$ and $\epsilon$ will be
chosen. Clearly $||v_{2}-v_{1}||=\epsilon$. So we need to know about ${\bf\sf
S}v_{1}$ and ${\bf\sf S}v_{2}$.

Part 1. Consider $v_{1}$, so that

$\displaystyle({\bf\sf S}v_{1})(x)$ $\displaystyle=$ $\displaystyle
F(z,k)+(1-\delta)k-b+\sup_{k^{\prime}\geq(1-\delta)k}\left\\{\frac{L(k^{\prime})}{1+\rho}-k^{\prime}\right\\}$
$\displaystyle=$ $\displaystyle
F(z,k)+(1-\delta)k-b+\left(\frac{\nu}{1+\rho}\right)^{2}.$

Set $k>0$ small enough to ensure that optimal capital is
$k^{\prime}=\left(\frac{\nu}{1+\rho}\right)^{2}\geq(1-\delta)k$. Set $b$ small
enough to ensure the above is positive. We now know that the feasible set is
non-empty.

Part 2. Consider $v_{2}$, so that

$\displaystyle({\bf\sf S}v_{2})(x)$ $\displaystyle=$ $\displaystyle
F(z,k)+(1-\delta)k-b+\beta\epsilon+\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b},k^{\prime}\geq(1-\delta)k}\left\\{\frac{b^{\prime}}{1+\rho}-k^{\prime}\right\\}$
$\displaystyle=$ $\displaystyle
F(z,k)+(1-\delta)k-b+\beta\epsilon+\left\\{\frac{\bar{b}}{1+\rho}-(1-\delta)k\right\\}.$

Combining Part 1 and Part 2 gives,

$\displaystyle({\bf\sf S}v_{2})(x)-({\bf\sf S}v_{1})(x)$ $\displaystyle=$
$\displaystyle\beta\epsilon+\left\\{\frac{\bar{b}}{1+\rho}-(1-\delta)k\right\\}-\left(\frac{\nu}{1+\rho}\right)^{2}.$

Choose $\bar{b}>0$ large enough, and $\epsilon>0$ and $\nu>0$ small enough.
Then the difference can be greater than $\epsilon$. Accordingly we have
$||{\bf\sf S}v_{2}-{\bf\sf S}v_{1}||>\epsilon=||v_{2}-v_{1}||$. This shows
that due to default, ${\bf\sf S}$ is not necessarily a contraction mapping.

This example shows that the standard approach to defining the operator
${\bf\sf S}$ may not ensure a contraction. Accordingly this direct approach
does not work. Some other method is needed.

###  3.2 Dual problem

In this section, we develop a dual problem to the original problem as done in
Aguiar and Amador (2019). We start with the following lemma whose proof is in
section A.

###### Lemma 3.1

In any equilibrium, $V(z,b,k)$ is strictly decreasing in
$b\in[\underline{$b$}(z,k),\bar{b}(z,k)]$. Moreover,
$V(z,b_{2},k)-V(z,b_{1},k)\geq b_{1}-b_{2}$ for any $b_{1}>b_{2}$ with
$b_{1},b_{2}\in[\underline{$b$}(z,k),\bar{b}(z,k)]$.

Due to the monotonicy, $V(z,b,k)$ has an inverse function with respect to $b$.
Let $B(z,k,v)$ be the inverse of the equilibrium value function $V$ in $b$,
with $v\in[\underline{$v$}(z,k),\bar{v}(z,k)]$, where
$\underline{$v$}(z,k)=V(z,\underline{$b$}(z,k),k)$ and
$\bar{v}(z,k)=V(z,\bar{b}(z,k),k)$. In other words, $V(z,B(z,k,v),k)\equiv v$
for $v\in[\underline{$v$}(z,k),\bar{v}(z,k)]$ and $B(z,k,V(z,b,k)\equiv b$ for
$b\in(\underline{$b$}(z,k),\bar{b}(z,k))$.

Finding the value function $V$ is the same as finding its inverse, the bond
function $B(z,k,v)$. Recall the primal problem that is satisfied by the value
function $V$ is a maximization problem (of the value $v$) with given
$(z,b,k)$. Next, we construct a dual problem that we will show is satisfied by
the bond function $B$. It is a maximization problem (of the bond $b$) with
given $(z,k,v)$.

To specify the dual problem, we introduce a three-dimensional dual space
$\mathbb{Y}=\mathbb{Z}\times\Re_{+}^{2}$. For $y=(z,k,v)\in\mathbb{Y}$, the
first component $z$ is the exogenous state, the second component $k$ is the
capital level, and the third component $v$ is the value of the firm.

The dual. Let $x^{\prime}=(z^{\prime},b^{\prime},k^{\prime})$. Given an
equilibrium condition, we define the dual problem for (8) as follows:

$\displaystyle\hat{B}(z,k,v)=\sup_{(b^{\prime},d,i)}\Big{\\{}(1-\tau)F(z,k)+\tau\delta
k-d-i+\tau b^{\prime}$
$\displaystyle\phantom{\hat{B}(z,k,v)=}\quad+\frac{1-\tau}{1+\rho}\big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+\big{[}b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})=0\\}}\big{]}1_{\\{b^{\prime}>0\\}}\big{\\}}\Big{\\}}$
(10) subject to $\displaystyle\qquad(1-\tau)F(z,k)+\tau\delta k-d-i+\tau
b^{\prime}$
$\displaystyle\qquad\qquad\qquad+\frac{1-\tau}{1+\rho}\big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+\big{[}b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})=0\\}}\big{]}1_{\\{b^{\prime}>0\\}}\big{\\}}\Big{\\}}\geq
0,$ $\displaystyle\qquad v=d+\beta\mathbf{\mathsf{E}}_{z}V(x^{\prime}),$
$\displaystyle\qquad k^{\prime}=(1-\delta)k+i,$
$\displaystyle\qquad\underline{$b$}\leq b^{\prime}\leq\bar{b},d\geq 0,i\geq
0.$

The following lemma confirms that $\hat{B}(z,k,v)$ in the dual problem (10)
indeed gives the inverse function $B(z,k,v)$.

###### Lemma 3.2

$B(z,k,v)=\hat{B}(z,k,v)$.

For the proof see section A.

The next step is to define an operator and show that $B$ is a fixed point of
this operator. Define an operator ${\bf\sf T}$ on all continuous function $f$
of $y=(z,k,v)\in\mathbb{Y}$ as follows,

$\displaystyle({\bf\sf
T}f)(z,k,v)=\sup_{(b^{\prime},d,i)}\Big{\\{}(1-\tau)F(z,k)+\tau\delta
k-d-i+\tau b^{\prime}$ $\displaystyle\phantom{{\bf\sf
T}f(z,k,v)=}\quad+\frac{1-\tau}{1+\rho}\big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+\big{[}b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})=0\\}}\big{]}1_{\\{b^{\prime}>0\\}}\big{\\}}\Big{\\}}$
(11) subject to $\displaystyle\qquad(1-\tau)F(z,k)+\tau\delta k-d-i+\tau
b^{\prime}$
$\displaystyle\qquad\qquad\qquad+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+\big{[}b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})=0\\}}\big{]}1_{\\{b^{\prime}>0\\}}\Big{\\}}\geq
0,$ $\displaystyle\qquad v=d+\beta\mathbf{\mathsf{E}}_{z}w(x^{\prime}),$
$\displaystyle\qquad b^{\prime}\leq
f(z^{\prime},k^{\prime},w(x^{\prime}))\mbox{ for all
}x^{\prime}\in\mathbb{X}\mbox{ such that }w(x^{\prime})\geq 0\mbox{ for all
}x^{\prime}\in\mathbb{X},$ $\displaystyle\qquad k^{\prime}=(1-\delta)k+i,$
$\displaystyle\qquad\underline{$b$}\leq b^{\prime}\leq\bar{b},d\geq 0,i\geq
0,$

where $q^{*}=1/(1+\rho)$.

The maximization problem (10) is to maximize the bond $b$. So we expect any
feasible bond $b\leq\hat{B}(z,k,v)=B(z,k,v)$. This implies that
$b^{\prime}\leq B(z^{\prime},k^{\prime},v^{\prime})$. By constructing the
operator above, the goal is to show that $B$ is a fixed point. Then for any
continuous function $f$, we require $b^{\prime}\leq
f(z^{\prime},k^{\prime},v^{\prime})$, with the next period value function
constructed by $v^{\prime}=w(x^{\prime})$. The next lemma confirms that $B$ is
in fact a fixed point of the constructed operator.

###### Lemma 3.3

Any equilibrium $B(z,k,v)$ is a fixed point of ${\bf\sf T}$.

For the proof see section A.

With this lemma it is sufficient for us to prove that the operator ${\bf\sf
T}$ has a unique fixed point.

###  3.3 Unique fixed point for the dual operator

Typically, the operator takes the form ${\bf\sf T}v=sup\\{r+\beta E(v)\\}$
subject to some constraints. We need to be sure that $r$ is bounded, otherwise
it is not clear that the operator ${\bf\sf T}$ is bounded and discounting. So
we cannot just directly apply Blackwell (1965).

Some papers directly add an extra assumption saying that $r$ is bounded.
Another approach is to use other features of the model to identify a bound.
For example, Harris (1987) studies an optimal growth model. He obtains a bound
inductively primarily by using the flow resource constraints. Either of these
could potentially be applied to our problem.

We have instead chosen to follow Lippman (1975) and van Nunen and Wessels
(1978). To do that we identify a function $\phi$ in Lemma 3.5. This function
ensures that for any bounded continuous function $f$: 1) ${\bf\sf T}$
satisfies the monotonicity property (Lemma 3.4), 2) ${\bf\sf T}f$ is bounded
under $\phi$-norm (Lemma 3.6), and, 3) it satisfies the discounting property
(Lemma 3.7). Then the existence and uniqueness of a fixed point for ${\bf\sf
T}$ is concluded by applying Blackwell (1965) as in van Nunen and Wessels
(1978).

The first step is monotonicity, which clearly holds.

###### Lemma 3.4

(Monotonicity) For any bounded continuous functions $f$ and $g$ on
$\mathbb{Y}$, if $f\leq g$, then ${\bf\sf T}f\leq{\bf\sf T}g$.

As a preparation for the next two steps, we define the norm. For boundedness
and discounting we need to identify a positive function $\phi$. Then we need
to show that the operator ${\bf\sf T}$ is bounded and satisfies discounting,
under the $\phi$-norm. For any positive function $\phi(y)$ and for any
$y=(z,k,v)\in\mathbb{Y}$, we define $\phi$-norm of function $f$ on
$\mathbb{Y}$ as

$\displaystyle||f||_{\phi}=\sup_{y}\left\\{\frac{|f(y)|}{\phi(y)}:y\in\mathbb{Y}\right\\}.$
(12)

We choose the function $\phi$ for the operator ${\bf\sf T}$ as follows. Let
$\mathbb{A}(z,k,v)$ denote the set of the feasible policy for the operator
problem (11); that is, the set of all $(b^{\prime},d,i)$ that satisfy the
constraints of the optimization problem (11).

The second step is to establish boundedness. To do this we write the single
period objective function as follows. For any $y=(z,k,v)\in\mathbb{Y}$ and
$a=(b^{\prime},d,i)\in\mathbb{A}(z,k,v)$, the objective function in (11) can
be written as

$\displaystyle r(y,a)$ $\displaystyle=$
$\displaystyle(1-\tau)F(z,k)+\tau\delta k-d-i+\tau b^{\prime}$
$\displaystyle+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+\big{[}b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})=0\\}}\big{]}1_{\\{b^{\prime}>0\\}}\Big{\\}},$

where the state $x^{\prime}=(z^{\prime},b^{\prime},k^{\prime})$. We call
$r(y,a)$ the single-period reward function.

###### Lemma 3.5

For any $\epsilon>0$, there exists an $\eta>0$ such that the function,

$\displaystyle\phi(y)$ $\displaystyle\equiv$
$\displaystyle\phi(z,k,v)\equiv\phi_{0}(k)$ $\displaystyle=$
$\displaystyle\left[(1-\tau)\bar{z}A+\tau\delta\right]k+\eta,$

ensures that for all $y\in\mathbb{Y}$,

$\displaystyle r(y,a)\leq\phi(y)\mbox{ for all
}a\in\mathbb{A}(y)\qquad\mbox{and}\qquad\phi(y^{\prime})\leq(1+\epsilon)\phi(y)\mbox{
for all }a\in\mathbb{A}(y),$

where $\phi(y^{\prime})\equiv\phi_{0}(k^{\prime})$ with
$k^{\prime}=(1-\delta)k+i$.

For the proof see section A.

With this choice of $\phi$ function, we have

###### Lemma 3.6

(Boundedness) Let $f$ be a bounded continuous function on $\mathbb{Y}$. Then
${\bf\sf T}f$ is bounded under $\phi$-norm.

This lemma follows immediately from Lemma 3.5. Note that even if $f$ is
bounded, ${\bf\sf T}f$ may not be bounded under the uniform norm, which is
needed in order to apply Blackwell (1965).

The third step is to establish discounting.

###### Lemma 3.7

(Discounting) For any bounded continuous function $f$ on $\mathbb{Y}$ and
constant $a>0$, ${\bf\sf T}(f+a\phi)\leq{\bf\sf T}f+\theta a\phi$ for some
$\theta<1$.

For the proof see section A.

With monotonicity, boundedness, discounting all established, we can now apply
the Blackwell (1965) sufficient conditions for a contraction to the dual.
Accordingly, there exists a unique Markov perfect equilibrium.

Because the dual problem satisfies the conditions for a contraction mapping,
using value function iteration on the dual to find the unique solution is
justified. So there may be an advantage for people working on numerical
solutions to such models, to work with the dual problem.

To summarize: the model has a financial problem and a technical problem. The
financial problem reflects the fact that investors must expect to break-even.
The technical problem reflects the need to establish the existence of a unique
equilibrium. Both of these issues can be addressed in several ways, and the
best method may depend to some degree on the specifics of a model. We believe
that the approach taken here may be useful for a variety of applications since
break-even requirements for outside participants are common to many models of
firms.

##  4 Equilibrium capital, debt and dividends

From Theorem 2.4 we know that there exists a unique equilibrium consisting of
a value function $V$ and price function $q$. What can be said about the
properties of the equilibrium? In this section, we consider a number of
properties regarding equilibrium capital, debt, and dividends. Throughout the
rest of this subsection, we assume $x=(z,b,k)\in\mathbb{X}_{feas}$. Otherwise,
the firm would default immediately which is not so interesting.

We start with some properties of the value function, the price of the debt,
and the debt level. Specifically, the value function is shown to strictly
decrease with the debt level. The price of the debt is shown to decrease in
the amount of debt issued. The debt level is shown to be nonnegative; that is,
the firm does not save. The proof of this proposition is in section B.

###### Proposition 4.1

In the equilibrium,

(a)

$V(z,b,k)$ is strictly decreasing in
$b\in[\underline{$b$}(z,k),\bar{b}(z,k)]$. Moreover,
$V(z,b_{2},k)-V(z,b_{1},k)\geq b_{1}-b_{2}$ for any $b_{1}>b_{2}$ with
$b_{1},b_{2}\in[\underline{$b$}(z,k),\bar{b}(z,k)]$.

(b)

$q(z,b^{\prime},k^{\prime})$ is decreasing in $b^{\prime}$.

(c)

$b^{\prime}\geq 0$, so there is no saving by the firm.

To go further it is useful to distinguish three cases depending on the current
state $x=(z,b,k)$. We refer to these as high capital, moderate capital, and
low capital, and they are defined by relative to the ability to reach the
capital target $k^{*}(z)$, which depends on the current exogenous state as
determined by the equality (15).

In the first case (“high capital”), the firm already has a great deal of
capital. The current capital level is high enough that the firm does not want
to invest in more. The firm will optimize the debt level under the current
state $z$ and the future capital level $(1-\delta)k$. The excess is paid out
as dividends. Whether the state is high enough for this to happen depends on
whether depreciated capital $(1-\delta)k$ is larger than a target capital
level $k^{*}(z)$.

In the second case (“moderate capital”), the capital target is attainable.
Current debt is not too high and the capital level is not too low. Their
combination allows the firm to raise enough cash through production and
optimal debt. The firm’s capital is raised to the optimal target for next
period. Again, the firm will pay the excess funds as dividends. Proposition
4.2 provides a formal characterization.

In the third case (“low capital”), the capital target is not attainable. Given
current debt and capital, even with the best feasible capital and debt
choices, the firm cannot reach the optimal target level. The firm will not pay
current dividends due to the great need for capital investment.

We have not been able to completely prove the high capital and low capital
cases formally. We provide a more detailed description as conjectured.

To describe the target, define

$\displaystyle N(z,k^{\prime})$ $\displaystyle=$ $\displaystyle\max_{0\leq
b^{\prime}\leq\bar{b}}\big{[}\tau+(1-\tau)q(z,b^{\prime},k^{\prime})-\beta\big{]}b^{\prime},$
(13) $\displaystyle b^{*}(z,k^{\prime})$ $\displaystyle=$
$\displaystyle\arg\max_{0\leq
b^{\prime}\leq\bar{b}}\big{[}\tau+(1-\tau)q(z,b^{\prime},k^{\prime})-\beta\big{]}b^{\prime}.$
(14)

$N(z,k^{\prime})$ is the maximum revenue from the bond issuing, and
$b^{*}(z,k^{\prime})$ is the optimal bond level from issuing under the current
exogenous state $z$ and the future capital level $k^{\prime}$. Notice that we
have replaced lower bound $\underline{$b$}$ by $0$ in the above due to
Proposition 4.1 (c).

The optimal capital target is determined by

$\displaystyle k^{*}(z)=\arg\max_{k^{\prime}}\Big{\\{}\beta(1-\tau){\bf\sf
E}_{z}F(z^{\prime},k^{\prime})-\big{[}1-\beta+\beta\delta(1-\tau)\big{]}k^{\prime}+N(z,k^{\prime})\Big{\\}}.$
(15)

Corresponding to this capital target, there is a debt target $b^{*}(z)\equiv
b^{*}(z,k^{*}(z))$ (where $b^{*}(z,k)$ is determined by (14)) and a dividend
target

$\displaystyle d^{*}(z)=R(z,k)+N\left(z,k^{*}(z)\right)+\beta
b^{*}(z)-k^{*}(z)-b.$ (16)

###### Proposition 4.2

Suppose that $x=(z,b,k)\in\mathbb{X}_{feas}$. If $k^{*}(z)\geq(1-\delta)k$ and
$d^{*}(z)\geq 0$, then the equilibrium optimal investment
$i=k^{*}(z)-(1-\delta)k$ (or equivalently capital level is raised to
$k^{*}(z)$), optimal debt $b^{\prime}=b^{*}(z)$, and optimal dividend
$d=d^{*}(z)$.

For the proof see section B.

The conditions $k^{*}(z)\geq(1-\delta)k$ and $d^{*}(z)\geq 0$ in the above
proposition ensure that the constraints related to $i\geq 0$ and $d\geq 0$ in
the firm problem (4) are satisfied automatically. When the condition
$k^{*}(z)\geq(1-\delta)k$ fails, the constraint $i\geq 0$ must be explicitly
imposed in the firm’s problem. This corresponds the high capital case. When
the condition $k^{*}(z)\geq(1-\delta)k$ holds but the condition $d^{*}(z)<0$,
the constraint $d\geq 0$ must be explicitly imposed in the firm’s problem.
This corresponds the low capital level case.

Now we provide a more detailed description of the high capital level case,
with $(1-\delta)k>k^{*}(z)$. The current capital level after depreciation is
already larger than the optimal target. So in order to achieve this target,
the firm would need to make a negative investment $i=k^{*}(z)-(1-\delta)k<0$.
But negative investment is not allowed by assumption. The closest solution
would be a zero investment. So we conjecture that the equilibrium optimal
investment is $i=0$. This means that the next period capital is
$k^{\prime}=(1-\delta)k$. Mathematically, this conjecture says that
$k^{\prime}=(1-\delta)k$ is the optimal solution to the maximization problem
(15) subject to the constraint $k^{\prime}\geq(1-\delta)k$. A sufficient
condition is that the maximand in (15) is quasi-concave in $k^{\prime}$. The
challenge to establish this lies in the difficulty in characterizing the
function $N(z,k^{\prime})$, which is defined by (13) and which involves the
equilibrium price function $q(z,b^{\prime},k^{\prime})$.

Corresponding to this optimal capital level, the optimal debt
$b^{\prime}=b^{*}(z,(1-\delta)k)$ and the optimal dividend (which maximizes
(13))

$\displaystyle d^{*}_{1}(z)$ $\displaystyle=$
$\displaystyle(1-\tau)F(z,k)+\tau\delta k+N\left(z,(1-\delta)k\right)+\beta
b^{*}(z,(1-\delta)k)-b.$ (17)

The feasibility assumption $x=(z,b,k)\in\mathbb{X}_{feas}$ guarantees
$d^{*}_{1}(z)\geq 0$.

Finally consider the low capital case. So, $k^{*}(z)\geq(1-\delta)k$ and
$d^{*}(z)<0$. Dividends cannot be negative, so the constraint $d\geq 0$ in the
firm problem (4) may be binding; and we conjecture that it is binding. It is
optimal not to pay any dividends if the firm has inadequate capital. The firm
currently has a small amount of capital and a high debt. It cannot raise
enough total revenue from production and new debt to payoff the existing debt
and at the same time raise the capital level to the target $k^{*}(z)$.
However, recall that we are assuming $x\in\mathbb{X}_{feas}$, so the firm can
raise enough to avoid default. Under the conjecture, the equilibrium optimal
debt and capital pair $(b^{\prime},k^{\prime})$ is the solution to the
following optimization problem,

$\displaystyle\max_{0\leq b^{\prime}\leq\bar{b}\atop
k^{\prime}\geq(1-\delta)k}$ $\displaystyle{\bf\sf
E}_{z}V(z^{\prime},b^{\prime},k^{\prime})$ s.t.
$\displaystyle(1-\tau)F(z,k)+[1-\delta(1-\tau)]k+\left[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})\right]b^{\prime}-k^{\prime}=b,$

where the last equality is the budget flow equation with $d=0$.

Financial targets play a prominent role in the capital structure literature at
least since Myers (1984); see DeAngelo and Roll (2015), Faulkender et al.
(2012), Frank and Shen (2019) and Ai et al. (2021). Some papers study time
invariant targets. It is important to stress that here the target is state
dependent.

We close this section with a comparison to the optimal target of a financial
autarky model which is a natural benchmark. In this model, the firm can choose
capital, but there is no debt market. Otherwise, the problem has the same
structure as above. Any capital purchase must be paid for using revenue
generated from operations. Of course, if the firm has no capital to start with
it can never purchase any. So we assume that the firm has some capital to
start with. It is shown in Chen and Frank (2021) that the optimal capital
target is

$\displaystyle k^{*}_{n}(z)=\arg\max_{k^{\prime}}\Big{\\{}\beta(1-\tau){\bf\sf
E}_{z}F(z^{\prime},k^{\prime})-\big{[}1-\beta+\beta\delta(1-\tau)\big{]}k^{\prime}\Big{\\}}.$
(18)

This target has an explicit solution,

$k^{*}_{n}(z)=\left(\frac{\alpha\beta A(1-\tau){\bf\sf
E}_{z}\left(z^{\prime}\right)}{1-\beta+\delta\beta(1-\tau)}\right)^{1/(1-\alpha)}.$

The difference between the target $k^{*}_{n}(z)$ in equation (18) and the
target $k^{*}(z)$ in equation (15) is the extra term $N(z,k^{\prime})$ in
(15). Since the maximand in equation (18) is concave in $k^{\prime}$ and
$N(z,k^{\prime})$ is non-decreasing in $k^{\prime}$, we know that
$k^{*}(z)\geq k^{*}_{n}(z)$. In other words, the target capital with a debt
market is greater than the target capital when there is no debt market.

##  5 Model modifications

We have provided a method to establish existence and uniqueness for a very
standard trade-off model. Our approach seems applicable to many of the cases
studied in the literature. To help show this, in this section we consider two
popular alternative assumptions to those in our basic model.

###  5.1 Exogenous cash flow

Starting with Modigliani and Miller (1958), many capital structure papers have
taken the firm’s cash flows to be exogenous, see Leland (1994); Goldstein et
al. (2001); Abel (2018); DeMarzo and He (2021). The original motivation was to
remove the impact of the firm’s real investment decisions on the firm’s choice
of debt. Of course, as shown in our basic model these are fundamentally
connected due to the terms on which investors will provide funds. Here we show
that removing that link simplifies the model and our approach still works.

Assume that the cash flows follow an exogenous process $\\{z_{t},t\geq 0\\}$.
The state space $\mathbb{X}=\mathbb{Z}\times[\underline{$b$},\bar{b}]$. The
firm flow budget (2) is modified to

$z_{t}+\tau[1-q(z_{t},b_{t+1})]b_{t+1}+q(z_{t},b_{t+1})b_{t+1}-b_{t}=d_{t}+i_{t}.$
(19)

Let

$\displaystyle S(z)$ $\displaystyle=$ $\displaystyle\sup_{\underline{$b$}\leq
b^{\prime}\leq\bar{b}}[\tau+(1-\tau)q(z,b^{\prime})]b^{\prime}.$ (20)

The maximum possible net revenue of the firm is given by $z+S(z)-b$, which
must be non-negative for the firm to avoid default. The feasible set in any
equilibrium is defined as, $\mathbb{X}_{feas}=\Big{\\{}(z,b)\in\mathbb{X}\
\big{|}\ z+S(z)\geq b\Big{\\}}$. Let
$\underline{$b$}(z)=\inf\\{b\geq\underline{$b$}:(z,b)\in\mathbb{X}_{feas}\\}$
and $\bar{b}(z)=\sup\\{b\leq\bar{b}:(z,b)\in\mathbb{X}_{feas}\\}$.

Firm problem. Let $x=(z,b)\in\mathbb{X}$ and take $q=q(z,b^{\prime})$ as
given. If $x\in\mathbb{X}_{feas}$, then

$\displaystyle V(x)$ $\displaystyle=$
$\displaystyle\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b},d\geq 0}\quad
d+\beta\mathbf{\mathsf{E}}_{z}V(x^{\prime}),$ $\displaystyle\mbox{subject
to}\quad z+[\tau+(1-\tau)q(z,b^{\prime})]b^{\prime}-b=d,$

otherwise, $V(x)=0$.

Investor break-even condition. Given the firm value function $V$ and the
firm’s exogenous state $z$. If the bond issue is $b^{\prime}\leq 0$, then
$q(z,b^{\prime})=1/(1+\rho)$. If $b^{\prime}\geq 0$, then

$\displaystyle q(z,b^{\prime})$
$\displaystyle=\frac{\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}}{1+\rho}.$
(22)

Similar to the main model (Definition 2.3), a Markov-perfect equilibrium is
defined to be a value function $V$ and a price function $q$ such that

(1)

Given $q$, the value function $V$ satisfies the Bellman equation (5.1), and
the supremum in (5.1) is attained for some policy;

(2)

Given $V$, $q$ satisfies the break-even condition (22).

###### Proposition 5.1

Under Assumption 2.1, there exists a unique Markov-perfect equilibrium.

The proof is provided in Section C.

Exogenous cash flow capital structure models and investment based capital
structure models are often treated as if they are entirely distinct
approaches, see Strebulaev and Whited (2012) and Ai et al. (2021). In
practice, it is common to use continuous time for models with exogenous cash
flows and discrete time for models with an investment choice. However, the
models are not as different as they sometimes appear. The fact that the same
approach works in a simpler model should not be surprising. The two approaches
really are closely connected.

###  5.2 Capital adjustment costs

Both the investment literature (Hayashi, 1982; Cooper and Haltiwanger, 2006)
and the dynamic capital structure literature (Strebulaev and Whited, 2012)
commonly assume that investment is subject to adjustment costs. A quadratic
functional form is particularly popular, as in Nikolov et al. (2021). With
such well behaved adjustment costs our approach still applies. To see this
assume that capital adjustment costs are,

$\Psi(k_{t},k_{t+1})=\frac{1}{2}\psi\left(\frac{k_{t+1}-(1-\delta)k_{t}}{k_{t}}\right)^{2}k_{t}\equiv\frac{1}{2}\psi\left(\frac{i_{t}}{k_{t}}\right)^{2}k_{t}.$
(23)

The firm flow budget condition (2) becomes

$\underbrace{(1-\tau)F(z_{t},k_{t})}_{\begin{subarray}{c}\text{revenue}\\\
\text{after-tax}\end{subarray}}+\underbrace{\tau(\delta
k_{t}+(1-q_{t})b_{t+1})}_{\text{tax deduction
revenue}}+\underbrace{q_{t}b_{t+1}-b_{t}}_{\begin{subarray}{c}\text{net
debt}\\\
\text{revenue}\end{subarray}}=\underbrace{d_{t}}_{\text{dividends}}+\underbrace{i_{t}}_{\text{investment}}+\underbrace{\Psi(k_{t},k_{t+1})}_{\begin{subarray}{c}\text{adjustment}\\\
\text{cost}\end{subarray}}.$

The definition of the function $S$ in (3) becomes,

$\displaystyle S(z,k)$ $\displaystyle=$
$\displaystyle\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b}\atop
k^{\prime}\geq(1-\delta)k}\big{\\{}[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-k^{\prime}-\Psi(k,k^{\prime})\big{\\}}.$
(24)

The definition 2.2 for the feasible set remains the same. The firm problem
remains almost the same. The change is that the first constraint in the
Bellman optimization (4) is replaced by

$\displaystyle
R(z,k)+[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-b=d+k^{\prime}+\Psi(k,k^{\prime}).$

The existence and uniqueness of a Markov perfect equilibrium still applies.
The proof only requires minor adjustments to the proof of Theorem 2.4.
Quadratic capital adjustment costs do not fundamentally disrupt the proof.

##  6 Conclusion

If outside investors are going to fund a firm, they must expect to break-even.
Otherwise, they will not participate. The price of debt adjusts accordingly.
This basic finance requirement means that even a very basic trade-off theory
model with single period risky debt may not directly satisfy the usual Stokey
and Lucas Jr. (1989) method of proving existence and uniqueness.

We first illustrated the difficulty. Then we showed how to establish existence
and uniqueness. This was done by turning to the dual problem, and then
changing the measure. We provide two further closely related versions of the
model where the approach also applies. So this approach may be useful for a
range of such models.

A noteworthy feature of the model is that in the equilibrium, the firm has a
state dependent capital target and a debt target. The firm’s debt and
dividends decisions adapt to the needs of productive efficiency. While this
description sounds reminiscent of the pecking order (Myers, 1984), the
underlying mechanism is of course quite different. There is no adverse
selection in our model. Productive efficiency subject to feasibility
constraints, is the key driving force. We also show that when the firm has
access to a debt market the capital target is higher than when the firm is
required to self-finance.

##  Appendix A Proofs of lemmas in section 3

  * Proof

Proof of Lemma 3.1 In this proof, we assume that all the states are in
$\mathbb{X}_{feas}$. Fix $(z,k)$ with $z\in\mathbb{Z}$ and $k\geq 0$. For any
given $b_{1},b_{2}\in[\underline{$b$}(z,k),\bar{b}(z,k)]$ with $b_{1}>b_{2}$,
choose any $0<\epsilon<b_{1}-b_{2}$. Following the definition of the firm’s
problem (4), there exists a policy $(b^{\prime}_{1},d_{1},i_{1})$ such that

$\displaystyle V(z,b_{1},k)<d_{1}+\beta{\bf\sf
E}_{z}V(z^{\prime},b^{\prime}_{1},k^{\prime}_{1})-\epsilon,\mbox{and}$
$\displaystyle(1-\tau)F(z,k)+\tau\delta
k+[\tau+(1-\tau)q(z,b^{\prime}_{1},k^{\prime}_{1})]b^{\prime}_{1}-b_{1}=d_{1}+i_{1},$

where $k^{\prime}_{1}=(1-\delta)k+i^{\prime}_{1}$.

If we replace $b_{1}$ by $b_{2}$ and $d_{1}$ by $d_{2}=d_{1}+b_{1}-b_{2}$,
then the policy $(b^{\prime}_{1},d_{2},i_{1})$ is feasible with the given
state $(z,b_{2},k)$. In particular, the budget constraint still holds and the
future state remains the same ($b^{\prime}_{2}=b^{\prime}_{1}$ and
$k^{\prime}_{2}=k^{\prime}_{1}$). So the above inequality can be written as

$\displaystyle V(z,b_{1},k)$ $\displaystyle<$ $\displaystyle
d_{1}+\beta{\bf\sf E}_{z}V(z^{\prime},b^{\prime}_{1},k^{\prime}_{1})-\epsilon$
$\displaystyle=$ $\displaystyle d_{2}+\beta{\bf\sf
E}_{z}V(z^{\prime},b^{\prime}_{1},k^{\prime}_{1})-(b_{1}-b_{2})-\epsilon\leq
V(z,b_{2},k)-(b_{1}-b_{2})-\epsilon.$

Because $\epsilon>0$ can be arbitrarily small, the above inequality implies

$\displaystyle V(z,b_{1},k)\leq V(z,b_{2},k)-(b_{1}-b_{2}).$

  * Proof

Proof of Lemma 3.2. Fix $x_{0}=(z_{0},b_{0},k_{0})\in\mathbb{X}_{feas}$.
Suppose that $(b^{\prime}_{0},d_{0},i_{0})$ is a feasible solution to the
equilibrium problem (8). Let $k^{\prime}_{0}=(1-\delta)k_{0}+i_{0}$,
$v_{0}=V(x_{0})$,
$x^{\prime}_{0}=(z^{\prime}_{0},b^{\prime}_{0},k^{\prime}_{0})$, and

$\displaystyle d_{0}$ $\displaystyle=$
$\displaystyle(1-\tau)F(z_{0},k_{0})+\tau\delta k_{0}-b_{0}-i_{0}+\tau
b^{\prime}_{0}$
$\displaystyle+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}_{0}1_{\\{b^{\prime}_{0}\leq
0\\}}+\big{[}b^{\prime}_{0}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})>0\\}}+L(k^{\prime}_{0})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})=0\\}}\big{]}1_{\\{b^{\prime}_{0}>0\\}}\Big{\\}}.$

Now it is immediate to verify that with $(z_{0},k_{0},v_{0})$, the policy
$(b^{\prime}_{0},d_{0},i_{0})$ is feasible for the dual problem (10); hence,

$\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle(1-\tau)F(z_{0},k_{0})+\tau\delta k_{0}-d_{0}-i_{0}+\tau
b^{\prime}_{0}$
$\displaystyle+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}_{0}1_{\\{b^{\prime}_{0}\leq
0\\}}+\big{[}b^{\prime}_{0}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})>0\\}}+L(k^{\prime}_{0})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})=0\\}}\big{]}1_{\\{b^{\prime}_{0}>0\\}}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle\hat{B}(z_{0},k_{0},v_{0}).$

Since $B(z,k,v)$ is the inverse function of $V(z,\cdot,k)$, we have
$b_{0}=B(z_{0},k_{0},v_{0})$. Hence, we have shown
$B(z_{0},k_{0},v_{0})\leq\hat{B}(z_{0},k_{0},v_{0})$.

Suppose that $B(z_{0},k_{0},v_{0})<\hat{B}(z_{0},k_{0},v_{0})$. For any
$\epsilon>0$, there exists a feasible policy
$(\hat{b}^{\prime}_{\epsilon},\hat{d}_{\epsilon},\hat{i}_{\epsilon})$ for the
dual problem (10) such that

$\displaystyle\hat{B}(z_{0},k_{0},v_{0})-\epsilon$ $\displaystyle<$
$\displaystyle(1-\tau)F(z_{0},k_{0})+\tau\delta
k_{0}-\hat{d}_{\epsilon}-\hat{i}_{\epsilon}+\tau\hat{b}^{\prime}_{\epsilon}$
$\displaystyle+\frac{1-\tau}{1+\rho}\Big{\\{}\hat{b}^{\prime}_{\epsilon}1_{\\{\hat{b}^{\prime}_{\epsilon}\leq
0\\}}+\big{[}\hat{b}^{\prime}_{\epsilon}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(\hat{x}^{\prime}_{\epsilon})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(\hat{x}^{\prime}_{\epsilon})=0\\}}\big{]}1_{\\{\hat{b}^{\prime}_{\epsilon}>0\\}}\Big{\\}},$

where $\hat{k}^{\prime}_{\epsilon}=(1-\delta)k_{0}+\hat{i}_{\epsilon}$ and
$\hat{x}^{\prime}_{\epsilon}=(z^{\prime}_{0},\hat{b}^{\prime}_{\epsilon},\hat{k}^{\prime}_{\epsilon})$.
Let $\hat{b}_{0}=\hat{B}(z_{0},k_{0},v_{0})$ and note
$v_{0}=\hat{d}_{\epsilon}+\beta\mathbf{\mathsf{E}}_{z_{0}}V(\hat{s}^{\prime}_{\epsilon})$.
We can rewrite the above inequality as

$\displaystyle v_{0}-\epsilon$ $\displaystyle<$
$\displaystyle(1-\tau)F(z_{0},k_{0})+\tau\delta
k_{0}-\hat{i}_{\epsilon}+\tau\hat{b}^{\prime}_{\epsilon}$
$\displaystyle+\frac{1-\tau}{1+\rho}\Big{\\{}\hat{b}^{\prime}_{\epsilon}1_{\\{\hat{b}^{\prime}_{\epsilon}\leq
0\\}}+\big{[}\hat{b}^{\prime}_{\epsilon}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(\hat{x}^{\prime}_{\epsilon})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(\hat{x}^{\prime}_{\epsilon})=0\\}}\big{]}1_{\\{\hat{b}^{\prime}_{\epsilon}>0\\}}\Big{\\}}+\beta\mathbf{\mathsf{E}}_{z_{0}}V(\hat{x}^{\prime}_{\epsilon})$
$\displaystyle\leq$ $\displaystyle V(z_{0},\hat{b}_{0},k_{0}),$

where the last inequality is due to the fact that the policy
$(\hat{b}^{\prime}_{\epsilon},\hat{d}_{\epsilon},\hat{i}_{\epsilon})$ is a
feasible policy for the equilibrium problem (8) with
$(z_{0},\hat{b}_{0},k_{0})$. Hence, we show that
$v_{0}-\epsilon=V(z_{0},b_{0},k_{0})-\epsilon<V(z_{0},\hat{b}_{0},k_{0})$ for
all $\epsilon>0$. Let $\epsilon\to 0$; we have $V(z_{0},b_{0},k_{0})\leq
V(z_{0},\hat{b}_{0},k_{0})$. Since $V(z,b,k)$ is monotone decreasing in $b$,
we have $b_{0}\geq\hat{b}_{0}$ or
$B(z_{0},k_{0},v_{0})\geq\hat{B}(z_{0},k_{0},v_{0})$, a contradiction. $\Box$

  * Proof

Proof of Lemma 3.3. Fix any $x_{0}=(z_{0},k_{0},v_{0})\in\mathbb{Y}$. We first
show that $B(z_{0},k_{0},v_{0})\leq({\bf\sf T}B)(z_{0},k_{0},v_{0})$. For any
$\epsilon>0$, there exists a feasible policy $(b^{\prime}_{0},d_{0},i_{0})$ to
the dual problem (10) such that

$\displaystyle B(z_{0},k_{0},v_{0})$ $\displaystyle<$
$\displaystyle(1-\tau)F(z_{0},k_{0})+\tau\delta k_{0}-d_{0}-i_{0}+\tau
b^{\prime}_{0}$
$\displaystyle+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}_{0}1_{\\{b^{\prime}_{0}\leq
0\\}}+\big{[}b^{\prime}_{0}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})>0\\}}+L(k^{\prime}_{0})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})=0\\}}\big{]}1_{\\{b^{\prime}_{0}>0\\}}\Big{\\}}+\epsilon.$

Let $w(x^{\prime}_{0})=V(x^{\prime}_{0})$. Then,
$b^{\prime}_{0}=B(z^{\prime}_{0},k^{\prime}_{0},V(x^{\prime}_{0}))=B(z^{\prime}_{0},k^{\prime}_{0},w(x^{\prime}_{0}))$
as $B$ is an inverse function of $V$ in terms of its second argument. This
shows that $(b^{\prime}_{0},d_{0},i_{0})$ and $w(x^{\prime}_{0})$ is feasible
to the $\sup$ problem in the definition of the operator with given
$(z_{0},k_{0},v_{0})$. Hence, the above inequality becomes

$\displaystyle B(z_{0},k_{0},v_{0})<({\bf\sf
T}B)(z_{0},k_{0},v_{0})+\epsilon.$

Letting $\epsilon\to 0$ in the above yields $B(z_{0},k_{0},v_{0})\leq({\bf\sf
T}B)(z_{0},k_{0},v_{0})$.

Now we show $({\bf\sf T}B)(z_{0},k_{0},v_{0})\leq B(z_{0},k_{0},v_{0})$. For
any $\epsilon>0$, there exists a feasible policy
$(b^{\prime}_{0},d_{0},i_{0})$ together with $w(x^{\prime}_{0})$ to the
operator problem (11) such that

$\displaystyle({\bf\sf T}B)(z_{0},k_{0},v_{0})\ <\
(1-\tau)F(z_{0},k_{0})+\tau\delta k_{0}-d_{0}-i_{0}+\tau b^{\prime}_{0}$
$\displaystyle\qquad\quad+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}_{0}1_{\\{b^{\prime}_{0}\leq
0\\}}+\big{[}b^{\prime}_{0}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{w(x^{\prime}_{0})>0\\}}+L(k^{\prime}_{0})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{w(x^{\prime}_{0})=0\\}}\big{]}1_{\\{b^{\prime}_{0}>0\\}}\Big{\\}}+\epsilon,$
(25) $\displaystyle b^{\prime}_{0}\leq
B(z^{\prime}_{0},k^{\prime}_{0},w(x^{\prime}_{0})).$

The last inequality implies

$\displaystyle V(z^{\prime}_{0},b^{\prime}_{0},k^{\prime}_{0})\geq
V(z^{\prime}_{0},B(z^{\prime}_{0},k^{\prime}_{0},w(x^{\prime}_{0})),k^{\prime}_{0})=w(x^{\prime}_{0}),$

where the first inequality is due to $V(z^{\prime}_{0},b,k^{\prime}_{0})$
being decreasing in $b$ and the second equality is due to $V(z,b,k)$ being the
inverse function of $B(z,k,\cdot)$. In view of the condition (7) and the above
inequality, we can rewrite the inequality (25) as

$\displaystyle({\bf\sf T}B)(z_{0},k_{0},v_{0})\ <\
(1-\tau)F(z_{0},k_{0})+\tau\delta k_{0}-d_{0}-i_{0}+\tau b^{\prime}_{0}$
$\displaystyle\qquad\quad+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}_{0}1_{\\{b^{\prime}_{0}\leq
0\\}}+\big{[}b^{\prime}_{0}\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})>0\\}}+L(k^{\prime}_{0})\mathbf{\mathsf{E}}_{z_{0}}1_{\\{V(x^{\prime}_{0})=0\\}}\big{]}1_{\\{b^{\prime}_{0}>0\\}}\Big{\\}}+\epsilon.$

Thus, we have established $(b^{\prime}_{0},d_{0},i_{0})$ is a feasible policy
for the dual problem (10). Then, the above inequality implies

$\displaystyle({\bf\sf T}B)(z_{0},k_{0},v_{0})<B(z_{0},k_{0},v_{0})+\epsilon.$

Letting $\epsilon\to 0$ in the above concludes the proof.

  * Proof

Proof of Lemma 3.5. We start with establishing some useful inequalities. In
view of the bound (7) given by the break-even condition, we have for any
$y\in\mathbb{Y}$ and $a\in\mathbb{A}(y)$,

$\displaystyle r(y,a)$ $\displaystyle\leq$
$\displaystyle(1-\tau)F(z,k)+\tau\delta
k-d-i+\left(\tau+\frac{1-\tau}{1+\rho}\right)b^{\prime}$ $\displaystyle\leq$
$\displaystyle(1-\tau)F(z,k)+\tau\delta k+\eta_{1}$ $\displaystyle\leq$
$\displaystyle\left[(1-\tau)\bar{z}A+\tau\delta\right]k+\eta_{1}+(1-\tau)\bar{z}A,$
(27)

where

$\displaystyle\eta_{1}=\frac{1+\rho\tau}{1+\rho}\bar{b},$

and we used $k^{\alpha}\leq 1+k$ for all $k\geq 0$ in obtaining the last
inequality. In view of the first constraint for the operator ${\bf\sf T}$
definition (which is equivalent to $r(y,a)\geq 0$) and the inequality (Proof),
we have

$\displaystyle i$ $\displaystyle\leq$
$\displaystyle(1-\tau)\bar{z}Ak^{\alpha}+\tau\delta k+\eta_{1}.$ (28)

Now, let

$\displaystyle\phi(z,k,v)\equiv\phi_{0}(k)=\left[(1-\tau)\bar{z}A+\tau\delta\right]k+\eta.$

For any given $\epsilon>0$, we will choose $\eta$ such that the inequalities
in the lemma holds.

The first inequality in the lemma clearly follows from the inequality (27) if
we choose $\eta\geq\eta_{1}+(1-\tau)\bar{z}A$.

Consider the second inequality in the lemma. In view of
$k^{\prime}=(1-\delta)k+i$, it follows from the inequality (28),

$\displaystyle\phi_{0}(k^{\prime})$ $\displaystyle=$
$\displaystyle\left[(1-\tau)\bar{z}A+\tau\delta\right]k^{\prime}+\eta$
$\displaystyle\leq$
$\displaystyle\left[(1-\tau)\bar{z}A+\tau\delta\right]\big{\\{}(1-\tau)\bar{z}Ak^{\alpha}+[1-\delta(1-\tau)]k+\eta_{1}\big{\\}}+\eta.$

With the above inequality, in order for
$\phi(x^{\prime})\leq(1+\epsilon)\phi(x)$, it suffices to have

$\displaystyle\left[(1-\tau)\bar{z}A+\tau\delta\right]\big{\\{}[\epsilon+\delta(1-\tau)]k-(1-\tau)\bar{z}Ak^{\alpha}-\eta_{1}\big{\\}}+\epsilon\eta\geq
0.$

It is elementary to show that

$\displaystyle m(\epsilon)=\min_{k\geq
0}\big{\\{}[\epsilon+\delta(1-\tau)]k-(1-\tau)\bar{z}Ak^{\alpha}-\eta_{1}\big{\\}},$

is a finite negative number. Hence, the above inequality holds if
$\eta\geq-m(\epsilon)\left[(1-\tau)\bar{z}A+\tau\delta\right]/\epsilon$. The
lemma is proved by choosing

$\displaystyle\eta=\max\left\\{\eta_{1}+(1-\tau)\bar{z}A,-\frac{m(\epsilon)\left[(1-\tau)\bar{z}A+\tau\delta\right]}{\epsilon}\right\\}.$

  * Proof

Proof of Lemma 3.7. We note that

$\displaystyle{\bf\sf
T}(f+a\phi)(z,k,v)=\sup_{(b^{\prime},d,i)}\Big{\\{}(1-\tau)F(z,k)+\tau\delta
k-d-i+\tau b^{\prime}+$ $\displaystyle\phantom{{\bf\sf
T}(f+a\phi)(z,k,v)=\sup_{(b^{\prime},d,i)}\Big{\\{}}+\frac{1-\tau}{1+\rho}\big{[}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+h(z,b^{\prime},k^{\prime})b^{\prime}1_{\\{b^{\prime}>0\\}}\big{]}\Big{\\}},$
subject to $\displaystyle\qquad(1-\tau)F(z,k)+\tau\delta k-d-i+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\big{[}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+h(z,b^{\prime},k^{\prime})b^{\prime}1_{\\{b^{\prime}>0\\}}\big{]}\geq
0,$ $\displaystyle\qquad v=d+\beta\mathbf{\mathsf{E}}_{z}w(x^{\prime}),$
$\displaystyle\qquad b^{\prime}\leq
f(z^{\prime},k^{\prime},w(x^{\prime}))+a\phi(z^{\prime},k^{\prime},w(x^{\prime}))\mbox{
for all }x^{\prime}\in\mathbb{X}\mbox{ such that }w(x^{\prime})\geq 0\mbox{
for all }x^{\prime}\in\mathbb{X},$ $\displaystyle\qquad
k^{\prime}=(1-\delta)k+i,$ $\displaystyle\qquad\underline{$b$}\leq
b^{\prime}\leq\bar{b},d\geq 0,i\geq 0,$

where

$\displaystyle
h(z,b^{\prime},k^{\prime})=\frac{b^{\prime}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}+L(k^{\prime})\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})=0\\}}}{b^{\prime}}.$

We note that $h(z,b^{\prime},k^{\prime})$ is decreasing in $b^{\prime}$ and
$h(z,b^{\prime},k^{\prime})\leq 1$ for $b^{\prime}\geq 0$ due to the break-
even condition (7).

Next, we note $b^{\prime}1_{\\{b^{\prime}\leq 0\\}}=\min\\{0,b^{\prime}\\}$
and $b^{\prime}1_{\\{b^{\prime}>0\\}}=\max\\{0,b^{\prime}\\}$. The distinction
between ${\bf\sf T}(f+a\phi)$ and ${\bf\sf T}f$ is in the third constraint,
the former having $b^{\prime}\leq
f(z^{\prime},k^{\prime},w(x^{\prime}))+a\phi(z^{\prime},k^{\prime},w(x^{\prime}))$
or $b^{\prime}-a\phi(z^{\prime},k^{\prime},w(x^{\prime}))\leq
f(z^{\prime},k^{\prime},w(x^{\prime}))$, and the latter having $b^{\prime}\leq
f(z^{\prime},k^{\prime},w(x^{\prime}))$.

The last two terms in the above objective function can be written as (with
$\phi^{\prime}$ short for $\phi(z^{\prime},k^{\prime},w(x^{\prime}))$),

$\displaystyle\tau
b^{\prime}+\frac{1-\tau}{1+\rho}[\min\\{0,b^{\prime}\\}+h(z,b^{\prime},k^{\prime})\max\\{0,b^{\prime}\\}]$
(29) $\displaystyle=$
$\displaystyle\tau(b^{\prime}-a\phi^{\prime})+\frac{1-\tau}{1+\rho}[\min\\{0,b^{\prime}-a\phi^{\prime}\\}+h(z,b^{\prime}-a\phi^{\prime},k^{\prime})\max\\{0,b^{\prime}-a\phi^{\prime}\\}]$
$\displaystyle+\tau
a\phi^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}\big{[}\min\\{0,b^{\prime}\\}-\min\\{0,b^{\prime}-a\phi^{\prime}\\}\big{]}$
$\displaystyle+\big{[}h(z,b^{\prime},k^{\prime})\max\\{0,b^{\prime}\\}-h(z,b^{\prime}-a\phi^{\prime},k^{\prime})\max\\{0,b^{\prime}-a\phi^{\prime}\\}\big{]}\Big{\\}}$
$\displaystyle\leq$
$\displaystyle\tau(b^{\prime}-a\phi^{\prime})+\frac{1-\tau}{1+\rho}\big{(}\min\\{0,b^{\prime}-a\phi^{\prime}\\}+h(z,b^{\prime}-a\phi^{\prime},k^{\prime})\max\\{0,b^{\prime}-a\phi^{\prime}\\}\big{)}$
$\displaystyle+\tau a\phi^{\prime}+\frac{1-\tau}{1+\rho}a\phi^{\prime},$

where the last inequality follows from considering each of the three cases
$b^{\prime}\leq 0$, $0\leq b^{\prime}\leq a\phi^{\prime}$ and $b^{\prime}\geq
a\phi^{\prime}$ to establish

$\displaystyle\Big{\\{}\big{[}\min\\{0,b^{\prime}\\}-\min\\{0,b^{\prime}-a\phi^{\prime}\\}\big{]}$
$\displaystyle\qquad+\big{[}h(z,b^{\prime},k^{\prime})\max\\{0,b^{\prime}\\}-h(z,b^{\prime}-a\phi^{\prime},k^{\prime})\max\\{0,b^{\prime}-a\phi^{\prime}\\}\big{]}\Big{\\}}\leq
a\phi^{\prime}.$

Let

$\displaystyle\theta=(1+\epsilon)\left[\tau+\frac{1-\tau}{1+\rho}\right];$

we choose $\epsilon>0$ small enough to ensure $\theta<1$. Then, combining the
inequality (29) and the second inequality in Lemma 3.5 and , we have

$\displaystyle\tau
b^{\prime}+\frac{1-\tau}{1+\rho}[\min\\{0,b^{\prime}\\}+h(z,b^{\prime},k^{\prime})\max\\{0,b^{\prime}\\}]$
$\displaystyle\leq$
$\displaystyle\tau(b^{\prime}-a\phi^{\prime})+\frac{1-\tau}{1+\rho}[\min\\{0,b^{\prime}-a\phi^{\prime}\\}+h(z,b^{\prime}-a\phi^{\prime},k^{\prime})\max\\{0,b^{\prime}-a\phi^{\prime}\\}]+\theta
a\phi.$

In view of the definition for the operator, the above inequality establishes
the lemma.

##  Appendix B Proofs of propositions in section 4

  * Proof

Proof of Proposition 4.1. (a) This follows immediately from Theorem 2.4 and
Lemma 3.1.

(b) The monotonicity result clearly holds for $b^{\prime}\leq 0$. For
$b^{\prime}>0$, we note that the comment following the break-even condition
(6) that $q(z,b^{\prime},k^{\prime})$ is increasing in the value function.
Then the monotonicity result follow from part (a) (with respect to
$b^{\prime}$).

(c) When $b^{\prime}\leq 0$, we have
$q(z,b^{\prime},k^{\prime})=1/(1+\rho)\equiv q$. If there exists an optimal
policy with $b^{\prime}<0$ for a given state $(z,b,k)\in\mathbb{X}_{feas}$,
then for any $\epsilon>0$, there exists a feasible policy
$(b^{\prime},d_{\epsilon},i_{\epsilon})$ such that

$V(z,b,k)<d_{\epsilon}+\beta{\bf\sf
E}_{z}V(z^{\prime},b^{\prime},k^{\prime}_{\epsilon})+\epsilon,$ (30)

where $k^{\prime}_{\epsilon}=(1-\delta)k+i_{\epsilon}$. It is immediate to
verify that
$(b^{\prime}-\Delta,d^{\prime}_{\epsilon}+[\tau+(1-\tau)q]\Delta,i_{\epsilon})$
for any $\Delta>0$ is a feasible policy for the state $(z,b,k)$; hence,

$\displaystyle V(z,b,k)$ $\displaystyle\geq$ $\displaystyle
d^{\prime}_{\epsilon}+[\tau+(1-\tau)q]\Delta+\beta{\bf\sf
E}_{z}V(z^{\prime},b^{\prime}-\Delta,k^{\prime}_{\epsilon}),$ (31)

where the inequality follows from the Bellman equation (4).

Now consider $(z^{\prime},b^{\prime},k^{\prime}_{\epsilon})$. It follows from
the Bellman equation (4) that there exists a feasible policy
$(b^{\prime\prime}_{\epsilon},d^{\prime}_{\epsilon},i^{\prime}_{\epsilon})$
with $d^{\prime}_{\epsilon}>0$ such that

$V(z^{\prime},b^{\prime},k^{\prime}_{\epsilon})<d^{\prime}_{\epsilon}+\beta{\bf\sf
E}_{z^{\prime}}V(z^{\prime\prime},b^{\prime\prime}_{\epsilon},k^{\prime\prime}_{\epsilon})+\epsilon,$

where
$k^{\prime\prime}_{\epsilon}=(1-\delta)k^{\prime}_{\epsilon}+i^{\prime}_{\epsilon}$.
It is immediate to check that
$(b^{\prime\prime}_{\epsilon},d^{\prime}_{\epsilon}-\Delta,i^{\prime}_{\epsilon})$
is a feasible solution for the state
$(z^{\prime},b^{\prime}-\Delta,k^{\prime}_{\epsilon})$ when
$0<\Delta<d^{\prime}_{\epsilon}$; hence,

$V(z^{\prime},b^{\prime}-\Delta,k^{\prime}_{\epsilon})\geq
d^{\prime}_{\epsilon}-\Delta+\beta{\bf\sf
E}_{z^{\prime}}V(z^{\prime\prime},b^{\prime\prime}_{\epsilon},k^{\prime\prime}_{\epsilon}).$

Combining the above two inequalities yields,

$V(z^{\prime},b^{\prime},k^{\prime}_{\epsilon})<V(z^{\prime},b^{\prime}-\Delta,k^{\prime}_{\epsilon})+\Delta+\epsilon;$

this, together with the inequality (31), yields,

$\displaystyle V(z,b,k)$ $\displaystyle>$ $\displaystyle
d^{\prime}_{\epsilon}+[\tau+(1-\tau)q]\Delta+\beta\left[{\bf\sf
E}_{z}V(z^{\prime},b^{\prime},k^{\prime}_{\epsilon})-\Delta-\epsilon\right].$

The latter inequality together with the inequality (30) gives

$\displaystyle V(z,b,k)$ $\displaystyle>$
$\displaystyle[\tau+(1-\tau)q-\beta]\Delta+V(z,s,k)-(1+\beta)\epsilon,$

The above inequality implies that
$[\tau+(1-\tau)q-\beta]\Delta<(1+\beta)\epsilon$ for all $\epsilon>0$, clearly
contradicting $\tau+(1-\tau)q\geq\beta$ by noting that $q=\beta$.

  * Proof

Proof of Proposition 4.2. The existence of a finite optimal solution (15) is
because the first two terms in the maximand are concave and have a a finite
optimal solution and the third in the maximand is bounded.

The optimal target case corresponds the case when the Bellman equation (4) is
achieved with the constraints $d\geq 0$ and $i\geq 0$ not strictly binding. In
this case, we can write

$\displaystyle V(x)$ $\displaystyle=$ $\displaystyle R(z,k)-b+h(z),$ (32)

where

$\displaystyle h(z)$ $\displaystyle=$ $\displaystyle\max_{0\leq
b^{\prime}\leq\bar{b},k^{\prime}}\big{\\{}[\tau+(1-\tau)q(z,b^{\prime},k^{\prime})]b^{\prime}-k^{\prime}+\beta{\bf\sf
E}_{z}V(x^{\prime})\big{\\}}.$

(Given the existence of the equilibrium, the $\sup$ is clearly achieved, so we
will use $\max$.) Substituting the value function (32) into the Bellman
equation (4) yields,

$\displaystyle h(z)$ $\displaystyle=$ $\displaystyle\beta{\bf\sf
E}_{z}h(z^{\prime})$ (33)
$\displaystyle+\max_{k^{\prime}}\Big{\\{}\beta(1-\tau){\bf\sf
E}_{z}F(z^{\prime},k^{\prime})-\big{[}1-\beta+\beta\delta(1-\tau)\big{]}k^{\prime}+N(z,k^{\prime})\Big{\\}}.$

The optimal solution $k^{\prime}$ in the equality (33) is given by
$k^{\prime}=k^{*}_{d}(z)$. Under the assumption of the proposition, clearly,
$i=k^{*}_{d}(z)-(1-\delta)k$ is the optimal investment for the firm, and the
remainder of the proof is straightforward.

##  Appendix C Proof of proposition in section 5

This subsection is mainly on the proof of Proposition 5.1 (Exogenous Cash Flow
Model). The proof is similar to the proof of Theorem 2.4, so we outline
without getting into the details.

First, the prime problem is given by (c.f. (8)),

$\displaystyle V(x)$ $\displaystyle=$
$\displaystyle\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b},d\geq
0}\quad\big{\\{}d+\beta\mathbf{\mathsf{E}}_{z}V(s^{\prime})\big{\\}},$
$\displaystyle\mbox{subject to}\quad d=z-b+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+b^{\prime}1_{\\{b^{\prime}>0\\}}\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}\Big{\\}}$

for $x\in\mathbb{X}_{feas}$, and $V(x)=0$ for $x\not\in\mathbb{X}$.

Now we consider a two-dimensional dual space
$\mathbb{Y}=\mathbb{Z}\times\Re_{+}$. For $y=(z,v)\in\mathbb{Y}$; its first
component $z$ denotes the exogenous state, and its second component $v$
denotes the value of the firm. Given an equilibrium condition, we define the
dual problem for (C) as follows:

$\displaystyle\hat{B}(z,v)$ $\displaystyle=$
$\displaystyle\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b},d\geq 0}\left\\{z-d+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+b^{\prime}1_{\\{b^{\prime}>0\\}}\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}\Big{\\}}\right\\}$
(35) subject to $\displaystyle z-d+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+b^{\prime}1_{\\{b^{\prime}>0\\}}\mathbf{\mathsf{E}}_{z}1_{\\{V(x^{\prime})>0\\}}\Big{\\}}\geq
0,$ $\displaystyle v=d+\beta\mathbf{\mathsf{E}}_{z}V(x^{\prime}),$

where we note $x^{\prime}=(z^{\prime},b^{\prime})$.

Following almost the same line of the proof, we can show that a version of
part (a) of Proposition 4.1 holds in this case; that is, $V(z,b)$ is strictly
increasing in $b\in(\underline{$b$}(z),\bar{b}(z))$ for all
$x=(z,b)\in\mathbb{X}_{feas}$. Let $B(z,v)$ be the inverse of the equilibrium
value function $V$ in $b$; i.e., $V(z,B(z,v))\equiv v$ for all
$y\in\mathbb{Y}$ and $B(z,V(z,b))\equiv b$ for all
$b\in(\underline{$b$}(z),\bar{b}(z))$. We have a version of Lemma 3.2.

###### Lemma C.1

$B(z,v)=\hat{B}(z,v)$.

Next, we similarly define an equilibrium operator ${\bf\sf T}$ on all
continuous function $f$ of $y=(z,v)\in\mathbb{Y}$ as follows,

$\displaystyle({\bf\sf T}f)(z,v)$ $\displaystyle=$
$\displaystyle\sup_{\underline{${\scriptstyle b}$}\leq
b^{\prime}\leq\bar{b},d\geq 0}\left\\{z-d+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+b^{\prime}1_{\\{b^{\prime}>0\\}}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}\Big{\\}}\right\\}$
(36) subject to $\displaystyle z-d+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+b^{\prime}1_{\\{b^{\prime}>0\\}}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}\Big{\\}}\geq
0,$ $\displaystyle\qquad v=d+\beta\mathbf{\mathsf{E}}_{z}w(x^{\prime}),$
$\displaystyle\qquad b^{\prime}\leq f(z^{\prime},w(x^{\prime}))\mbox{ for all
}x^{\prime}\in\mathbb{X}\mbox{ such that }w(x^{\prime})\geq 0\mbox{ for all
}x^{\prime}\in\mathbb{X}.$

A version of Lemma 3.3 holds here as well (whose proof is also similar).

###### Lemma C.2

Any equilibrium $B(z,k,v)$ is a fixed point of ${\bf\sf T}$.

What remains is to show that the operator ${\bf\sf T}$ has a unique fixed
point. The key component for this proof is a version of Lemma 3.5, which gives
an upper bound for the objective function in (36).

Let $\mathbb{A}(z,v)$ denote the set of the feasible policy for the operator
problem (36); that is, the set of all $(b^{\prime},d)$ that satisfies the
constraints of the optimization problem (36). For any $y=(z,v)\in\mathbb{Y}$
and $a=(b^{\prime},d)\in\mathbb{A}(z,v)$, we the objective function in (36)
(which we also call the (single-period) reward function) by

$\displaystyle r(y,a)$ $\displaystyle=$ $\displaystyle z-d+\tau
b^{\prime}+\frac{1-\tau}{1+\rho}\Big{\\{}b^{\prime}1_{\\{b^{\prime}\leq
0\\}}+b^{\prime}1_{\\{b^{\prime}>0\\}}\mathbf{\mathsf{E}}_{z}1_{\\{w(x^{\prime})>0\\}}\Big{\\}},$

where the state $x^{\prime}=(z^{\prime},b^{\prime})$. The version of Lemma 3.5
follows (whose proof is obvious).

###### Lemma C.3

For any $\epsilon>0$, there exists an $\eta>0$ such that the function,

$\displaystyle\phi(y)\equiv\phi(z,v)\equiv\phi_{0}=\bar{z}+\eta,$

ensures that for all $y\in\mathbb{Y}$,

$\displaystyle r(y,a)\leq\phi(y)\mbox{ for all
}a\in\mathbb{A}(y)\qquad\mbox{and}\qquad\phi(y^{\prime})\leq(1+\epsilon)\phi(y)\mbox{
for all }a\in\mathbb{A}(y).$

The remaining proof follows the same line as in the proof of Theorem 2.4.

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