Inventory and Capital Structure Decisions Under Bankruptcy Risk:

A One Period Model∗

Yasin Alan, Vishal Gaur†

November 29, 2010

Abstract

We investigate the inventory and capital structure decisions of a firm in the presence ofbankruptcy risk. The key aspect of the paper is that we model the simultaneous decisions ofan equity investor, the manager of the firm, and a bank. The investor is an expected valuemaximizer and decides how much to invest in the firm. The firm is a newsvendor who takes theinvestor’s decision as starting equity and decides its borrowing amount and inventory in order tomaximize the total return to the investor. The bank observes the firm’s equity and sets an assetbased credit limit in order to prevent the firm from over-borrowing. Our model determines thevalues of inventory, capital structure, and risk of bankruptcy of the firm that would be realizedat equilibrium in the marketplace.

Our paper builds on the literature in operations, which typically takes the firm’s equity asgiven, and the literature in corporate finance, which typically takes the firm’s earnings as given,to analyze a joint model of inventory and capital structure decisions.

∗The authors are thankful to seminar participants at Georgia Tech College of Management for valuable feedback.†Johnson Graduate School of Management, Cornell University, Sage Hall, Ithaca, NY 14853-6201, E-mail:

ya47@cornell.edu, vg77@cornell.edu

1 Introduction

Bankruptcy occurs when a firm is unable to meet its obligations to its creditors. Any firm with debt

and sufficient uncertainty of earnings faces a risk of bankruptcy. In general, the risk of bankruptcy of

a firm plays a major role in shaping its interactions with its creditors and investors. It is important

for the firm’s creditors to quantify this risk because their returns are contingent upon the firm’s

survival. Creditors take several types of actions to manage their risk, such as charging a higher

interest rate to firms that have a higher risk of bankruptcy, or setting a credit limit to reduce the

amount of debt. The success of such actions relies on the ability of the creditor to decipher the

internal operational principles of the firm (e.g., its inventory policy) and their potential financial

outcomes. The investors’ returns are also contingent upon the firm’s survival. However, contrary

to creditors, investors can benefit from the risk of bankruptcy because of limited liability, which

allows them to share this risk with creditors. Indeed, if a firm has a high risk of bankruptcy to

begin with, its investors may want the firm to take even more risky operational decisions (Brealey

and Myers 1995, section 18.3), effectively allowing them to play with other people’s money.

Classical operations management models overlook the risk of bankruptcy assuming that the

firm has sufficient working capital to finance its operations. These models can mimic operational

decisions of large corporations with sufficient cash and debt capacity. However, most small and

medium sized businesses face liquidity constraints and bankruptcy risk. For instance, according to

a survey released by the Federal Reserve Bank of New York, nearly 60% small businesses applied

for credit during the first half of 2010, and more than three quarters of applicants received only

some or none of the credit they wanted. Such financial hurdles make bankruptcy risk a severe

threat, which poses serious challenges for an operations manager.

Due to the presence of bankruptcy risk, banks commonly use asset based financing (ABF) to lend

money to firms. In ABF, the borrowing amount of a firm is linked to its current assets, including

inventory, cash, and accounts receivables. This practice is widely prevalent—for example, in 2009,

the total amount of outstanding asset based loans in the USA was $480 billion (R.S. Carmichael &

Co., Inc. 2009), which was 25% of the total amount of nonfinancial corporate loans and short term

papers (Board of Governors of the Federal Reserve System 2010). Retailing and food industries,

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which have perishable inventories, are two of the top three asset based borrowers (R.S. Carmichael

& Co., Inc. 2009). These industries are characterized by a high risk of bankruptcy. For instance,

during the 1978-1997 period roughly 17% of publicly traded retail firms filed for bankruptcy (Fisher

et al. 2002). ABF is useful to lenders because it allows them to control their exposure to the firm’s

bankruptcy risk by imposing an upper limit on the amount of debt on the firm’s balance sheet. ABF

is also useful to borrowers with insufficient liquidity because it allows them to obtain competitive

interest rates by securing their current assets. For instance, according to Paglia (2010), the median

cost of capital for a $5M asset based loan was 8% in year 2009. Although this number seems high

compared to a 5% median interest rate charged on cash flow loans, it still provides small businesses

an attractive borrowing opportunity because cash flow financing is typically available for large

companies with revenues in excess of $25 million and stable profits (Burroughs 2008). Taking these

factors into consideration, ABF provides us with a useful framework to model borrowing decisions

and bankruptcy risk in a newsvendor setting.

Our paper studies the implications of bankruptcy risk on inventory and capital structure de-

cisions of a firm in the context of ABF. We use a game theoretic model with three players - a

newsvendor firm, an investor and a bank. The investor is an expected value maximizer and allo-

cates her funds between the newsvendor and an alternative investment opportunity. Taking the

investor’s decision as starting equity, the manager of the newsvendor decides how much to borrow

from the bank and how much inventory to stock in order to maximize the total return to the in-

vestor. The bank lends money to the firm. It observes the firm’s equity and sets an asset based

credit limit in order to prevent the firm from over-borrowing. Through this model, we address

two main questions: (i) What are the inventory and capital structure outcomes of the newsvendor

at equilibrium? (ii) What is her resulting probability of bankruptcy, and how does it depend on

the stocking quantity, the interest rates, the demand distribution, and the price parameters of the

newsvendor model? Without market frictions, inventory and capital structure decisions would be

independent of each other (Modigliani and Miller 1958) and unaffected by bankruptcy risk (Stiglitz

1969). Our model includes two types of frictions - corporate taxes and a spread between the in-

terest rates applied to borrowings and deposits. We use the newsvendor model as a prototypical

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representation of single-period capacity-constrained operational decisions of a firm.

Our paper builds on Buzacott and Zhang (2004), who analyze a model of asset based financing

involving a newsvendor and a bank. The newsvendor has a given equity and decides its borrowing

amount and order quantity within a liquidity constraint imposed by the bank in order to maximize

its ending cash position. The bank sets an asset based credit limit in order to maximize its profits.

Buzacott and Zhang (2004) show that the newsvendor may over-borrow and expose the bank to

a significant risk in the absence of a credit limit. We generalize their model by incorporating two

important elements: taxation and an investor who determines the value-maximizing level of equity

to provide to the newsvendor. Taxation is important because it is a fundamental reason for a

firm to borrow money. Since interest paid on debt is tax deductible, taxation reduces the cost of

capital of a firm and makes it attractive to carry debt. The investor’s role is important because

the amounts of inventory and borrowing are functions of equity and they determine the value that

the newsvendor is able to provide to its shareholders. This begets the question as to how the

value of the newsvendor varies with its equity and what level of equity maximizes the value to

shareholders. Thus, our paper determines the bankruptcy probability and the amounts of equity,

debt, and inventory that occur at equilibrium in the marketplace.

One result of our paper is that the probability of bankruptcy takes on only two values, either

zero or a positive constant that is independent of the tax rate, order quantity, and the underlying

demand distribution. This result implies that for firms that face a positive bankruptcy probability,

the credit limit is always binding. That is, it is never optimal for the investor to give the newsvendor

an amount of equity that will be small enough to entail bankruptcy risk, but large enough to allow

borrowing less than the credit limit. Moreover, it is optimal for the bank to adjust the credit limit

in such a way that the bankruptcy probability remains constant even though inventory and capital

structure may vary. This constant probability depends on the cost parameters of the newsvendor

model and interest rates. In practice, banks set credit limits using simple rules of thumb predicated

on historical salvage values of inventory in different industries. Our result shows how banks may

set credit limits in a more sophisticated way to incorporate both the demand distribution and the

equity of the newsvendor.

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Whether the probability of bankruptcy is zero or non-zero at equilibrium depends on the rate

of return on the alternative investment available to the investor. Thus, another result of our paper

shows the existence and uniqueness of a threshold rate of return below which the bankruptcy

probability is zero. In such situations, there can be borrowing without bankruptcy risk, or in

extreme cases, the investor may choose to create a pure equity firm. Otherwise, the investor

chooses to invest a relatively small amount in the newsvendor, and the newsvendor borrows. The

remaining results of the paper show the capital structure of the newsvendor at equilibrium, and

the implications of taxes or the absence thereof on the equilibrium outcome.

The paper is organized as follows. We review the relevant literature in section 2. In section 3,

we present our model and analysis. In section 4, we discuss the implications of our results with

respect to the probability of bankruptcy, the role of corporate taxation, the implementation of the

asset-based credit limit, and the capital structure that arises at equilibrium. Finally, in section 5,

we relate our results to the implementation of secured lending in practice and discuss limitations

and extensions of our model.

2 Literature Review

We describe relevant research in the literature on capital structure decisions in corporate finance

and that on joint operational-financial decisions in operations management. We then identify the

contributions of our paper.

Modigliani and Miller (1958) show that, in a perfect market, the capital structure of a firm

is irrelevant to its optimal operational decisions. That is, the decision that maximizes the value

to shareholders is equal to the decision that maximizes the total value of the firm. Subsequent

research not only generalizes this result for milder assumptions, such as costless bankruptcy (e.g.,

Baron 1974, Stiglitz 1969), but also shows violations due to market imperfections such as inter-

est rate spread, taxation, costly bankruptcy, and liquidity constraints (e.g., Modigliani and Miller

1963, Gordon 1989). Stiglitz (1972) shows a connection between real and operational decisions

under bankruptcy risk and two investor classes, debt and equity holders. Stiglitz (1974) extends

Modigliani and Miller (1958) to a multi-period setting without bankruptcy risk, and identifies

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bankruptcy risk as a main limitation that prevents us from separating real and financial decisions.

Empirical evidence also shows that there is a relationship between capital structure and firm char-

acteristics, such as profitability, firm growth, liquidation value, return volatility, and operational

risks; see, for example, Harris and Raviv (1991).

Most papers in this stream have stylized investment functions that do not depend on the firm’s

debt-equity mix or include operational decisions. One exception is Dotan and Ravid (1985). They

consider a two-stage model with uncertain sales price, the first involving the optimal capacity

and financing decisions, and the second involving a production decision subject to the capacity

constraint. They show that joint optimization of capacity and financing decisions maximizes firm

value. See Childs et al. (2005) and references therein for the recent finance literature on the

interaction of financing and investment decisions. Our paper builds on this stream by modeling

the operational investment decision through the newsvendor problem under market imperfections,

such as taxation, interest spread, and liquidity constraints.

The operations management literature on joint operational-financial decisions addresses market

imperfections by including taxes, liquidity constraints, bankruptcy risk, costly issuance of debt

and equity, and credit limits into single- and multi-period inventory models. Among single-period

models, Xu and Birge (2004) investigate the tradeoff between bankruptcy cost and the tax benefits

of debt in a cash-constrained newsvendor model. The firm can alleviate the cash constraint by

issuing bonds to risk-neutral investors. Their analysis shows that integrating operational and

financial decisions can improve firm value. While Xu and Birge (2004) allow the borrowing interest

rate to vary with the amounts of inventory and borrowing, Buzacott and Zhang (2004) study single-

and multi-period models with asset based credit limit. The second half of their paper is relevant

to our work and we summarized it in section 1. It analyzes a single-period model in which a

newsvendor and a bank seek to maximize own profits in the presence of bankruptcy risk. Dada and

Hu (2008) use a similar framework, with the difference that the bank chooses an optimal interest

rate to charge to the newsvendor, instead of imposing a borrowing limit. They show the existence

and uniqueness of an equilibrium order quantity interest rate pair.

Among multi-period models of the firm’s inventory and financial decisions (e.g., borrowing,

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dividends, capital subscriptions), Chao et al. (2008) show the optimality of a capital dependent

base stock policy for a cash constrained pure equity firm. Li et al. (2005) study a firm that makes

production, borrowing, and dividend/equity issuance decisions in each time period. They consider

the possibility of the firm’s bankruptcy, which occurs if the firm’s retained earnings become negative.

They show the optimality of a base stock policy for both inventory and cash reserves. Extensions

of Li et al. (2005) show that, under certain assumptions, the optimal capital structure is 100% debt

or 100% equity depending only on the tax rate and interest rates (Hu and Sobel 2005), and the

joint optimization of operational and financial decisions leads to a greater exposure to risk to take

advantage of limited liability (Hu et al. 2010).

Research on the impact of financial considerations on operational decisions is not limited to

inventory models. Archibald et al. (2002) argue that a survival strategy is more sensible than a

profit maximization strategy for a cash constrained start up facing bankruptcy risk. Financial

constraints and the risk of bankruptcy also affect the choice of production technologies (Lederer

and Singhal 1994, Boyabatli and Toktay 2010), the optimal time to shut down a firm (Xu and Birge

2006), and the optimal time to offer an IPO (Babich and Sobel 2004).

All the operations management models cited above with the exception of Lederer and Singhal

(1994) ignore the investor’s equity financing decision. The single-period models take equity as

given. The multi-period models allow issuance of equity assuming that the issued amount will

always be financed in full. Our paper contributes to this literature by introducing an investor that

controls the newsvendor’s starting equity. In our model, interactions among the investor, the bank,

and the newsvendor determine the newsvendor’s debt-equity mix, inventory level, and bankruptcy

risk. We model these interactions as a Stackelberg game. We use the framework of Buzacott and

Zhang (2004) to model the interactions among the firm and the bank. Then we analyze the starting

equity as an outcome of a game played by the potential investor and the newsvendor. This game

determines the equilibrium outcome.

There has also been vast research on predicting corporate defaults. Altman and Hotchkiss

(2006), Duffie (2010), and references therein provide a comprehensive overview of this research

stream. While these studies focus on predicting bankruptcy risk using financial ratios and stock

6

market return data, we study the connection between bankruptcy risk and operational decisions.

3 Model

We study a single-period model with three players: an inventory management (newsvendor) firm,

a commercial bank, and an investor. The firm obtains equity from the investor. With the amount

of equity as given, it decides how much to borrow from the bank and how much inventory to stock

in order to maximize the expected value of its ending cash position. The bank seeks to maximize

its expected profit by taking deposits and lending to the newsvendor. The bank determines the

maximum lending amount, i.e., a credit limit, in order to prevent over-borrowing. There is a single

investor, who decides an amount to invest in the equity of the newsvendor in the presence of an

alternative investment opportunity, viz., a market portfolio. The single investor may be interpreted

as a private equity firm. Since we include an alternative investment opportunity in the investor’s

decision-space, our model is extendible to multiple investors. We simplify to a single investor for

parsimony. We assume that all parameters of the model, such as the interest rates, the demand

distribution, and the cost economics are common knowledge among all players. Thus, we focus

on the optimal decisions and the equilibrium outcome without getting into the implications of

information asymmetry. Without loss of generality, we also assume that all three players are risk

neutral decision-makers.

The sequence of events is as follows. First, the investor determines the newsvendor’s starting

capital, x. Then, the newsvendor-bank interaction takes place, in which the newsvendor borrows

w from the bank at interest rate α without exceeding a credit limit ψ set by the bank. This gives

the newsvendor a total capital of x + w available for inventory procurement. She orders q units

and pays cq to her supplier, where c denotes the per unit cost. The amount cq cannot exceed x+w

because the payment to the supplier is due when the order is placed. She puts her excess cash,

x + w− cq, into the bank to generate an interest revenue at a rate α′. A random demand ξ occurs

and the newsvendor generates a revenue of pmin{ξ, q}+s(q−ξ)+, where p denotes the selling price,

s denotes the salvage value of unsold units, and (q − ξ)+ ≡ max{q − ξ, 0}. The pre-tax operating

income of the newsvendor is thus equal to p min{ξ, q} + s(q − ξ)+ + α′(x + w − cq) − cq − αw.

7

The newsvendor pays corporate tax at rate τ if the operating income is positive, and zero tax if

she incurs an operating loss. If the income is sufficient, then the newsvendor repays the loan plus

interest, (1 + α)w, to the bank. Otherwise, she declares bankruptcy.

We apply corporate taxation to the newsvendor’s income but not to the bank or the investor for

simplicity. The bank and the investor may have other sources of income that they can offset losses

against, so we ignore the role of taxation on their decision-making. We incorporate an interest rate

spread in our model by assuming that borrowing rate α is strictly greater than deposit rate α′.

Thus, our model has two types of market frictions, taxes and interest rate spread.

We assume that the demand ξ is non-negative and follows a continuous probability distribution

with increasing failure rate (IFR). The pdf, cdf, complementary cdf (ccdf), and inverse ccdf of

the demand distribution are denoted as f , F , F , and F−1, respectively, where f is positive on

an interval and zero elsewhere. For the newsvendor problem to be non-trivial, we assume that

(1 − τ)(p − s) > (1 + α)c − s and c > s. These assumptions are necessary for the tail probability

of the demand distribution to lie between 0 and 1. We also assume that p−s(1+α)c−s ≥ 1 + α. This

assumption means that the profit margin of the newsvendor is sufficiently high so that the rate of

return of a sold unit that is purchased on credit, p−s(1+α)c−s − 1, is no less than the borrowing rate α.

Our analysis proceeds in the reverse sequence of time. We first solve the newsvendor’s and

the bank’s problems given the starting capital of the newsvendor. We then analyze the investor’s

decision, which determines the newsvendor’s capital structure and inventory at equilibrium.

3.1 The Newsvendor-Bank Subgame

This part of our model is based on Buzacott and Zhang (2004). However, our proofs and the

resulting solutions differ because of taxation. We first present our analysis and then briefly remark

on the similarities and differences with their paper.

The newsvendor is formed as a limited liability company, which, by definition, limits the liability

8

of the investor to the amount of equity, x. Hence, the newsvendor’s ending cash position is

π(q, w; x, ξ) =

((1 + α′)(x + w − cq) + pmin{ξ, q}+ smax{q − ξ, 0} − (1 + α)w

− τ[α′(x + w − cq) + pmin{ξ, q}+ smax{q − ξ, 0} − cq − αw

]+

)+

.

The newsvendor solves the following problem:

π∗(x) = max(q,w)∈C(x,ψ)

E [π(q, w; x, ξ)] (1)

where E denotes expectation with respect to the distribution of demand, and C(x, ψ) = {(q, w) ∈<2

+ : w ≤ ψ, cq ≤ x + w} is the constraint set. The first constraint is that the borrowing amount

w must be less than the credit limit ψ set by the bank. The second constraint specifies that the

cost of procurement must be less than the total cash available to the newsvendor. We solve the

newsvendor’s problem ignoring the first constraint. Then, we solve the bank’s problem to compute

ψ. Finally, we determine conditions in which the credit limit is binding.

It is important to note that the newsvendor will not simultaneously borrow and hold excess

cash, i.e.,

(x + w − cq)w = 0. (2)

Intuitively, this condition occurs because α > α′. We omit the proof, which follows by a contradic-

tion argument. This condition implies that there are three possible scenarios that can occur based

on the values of the starting capital and the order quantity of the newsvendor. We denote as (NB)

the scenario (NB) in which the newsvendor has enough cash to purchase inventory, i.e., cq ≤ x, and

does not borrow any money from the bank. In scenario (BWO), the newsvendor borrows without

bankruptcy risk. In scenario (BWR), the newsvendor borrows with bankruptcy risk. In the rest

of the paper, we use the superscripts NB, BWO and BWR to denote variables in the respective

solutions.

In (NB), let dNB(q, x) denote the level of demand above which the newsvendor has non-negative

operating income and below which it incurs an operating loss. We get dNB(q, x) = (1+α′)c−sp−s q− α′

p−sx.

Thus, the newsvendor pays tax if ξ ≥ dNB(q, x). Let x =(c + c−s

α′)q. Then, the newsvendor always

has taxable operating income if x > x for a given order quantity q because dNB(q, x) < 0 for x > x.

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Therefore, the newsvendor’s ending cash position for x > x is

πNBx>x(q, 0;x, ξ) = (1 + (1− τ)α′)x + (1− τ)

(pmin{ξ, q}+ smax{q − ξ, 0} − (1 + α′)cq

).

Taking an expectation gives

E[πNB

x>x

]= (1 + (1− τ)α′)x + (1− τ)

((p− s)

∫ q

0F (ξ)dξ − [(1 + α′)c− s]q

).

On the other hand, if cq ≤ x ≤ x, then the newsvendor’s ending cash position is

πNBcq≤x≤x(q, 0;x, ξ) = (1 + α′)(x− cq) + p min{ξ, q}+ smax{q − ξ, 0}

− τ(p− s)[min{ξ, q} − dNB(q, x)

]1{ξ ≥ dNB(q, x)}.

Here, 1{X} takes value 1 if condition X is true, and 0 otherwise. By taking an expectation, we

obtain the newsvendor’s expected ending cash position in (NB) as

E[πNB

cq≤x≤x

]= (1 + α′)x + (p− s)

(∫ q

0F (ξ)dξ − τ

∫ q

dNB(q,x)F (ξ)dξ

)− [(1 + α′)c− s]q.

In (BWO), the newsvendor does not face bankruptcy risk, so that it ends up with non-negative

cash even when the demand is zero. Therefore, we seek to determine an inventory level ¯q(x) such

that ¯q(x) > x/c and the newsvendor’s ending cash position is always non-negative. Since the lower

bound on the newsvendor’s ending cash position occurs when demand ξ = 0, we get

(1 + α)x− [(1 + α)c− s]¯q(x)− τ (αx− [(1 + α)c− s]¯q(x))+ = 0.

Observe that for any q > x/c, αx− [(1 + α)c− s]q < αx− [(1 + α)c− s]x/c = s−cc x < 0. Therefore,

(αx− ((1 + α)c− s)q)+ = 0, and the newsvendor borrows without bankruptcy risk if x/c < q ≤¯q(x) = 1+α

(1+α)c−sx. The newsvendor has taxable income if ξ ≥ dBWO(q, x) ≡ (1+α)c−sp−s q− α

p−sx. The

newsvendor’s ending cash position in (BWO) is

πBWO(q, w; x, ξ) = (1 + α)(x− cq) + p min{ξ, q}+ smax{q − ξ, 0}

− τ(p− s)[min{ξ, q} − dBWO(q, x)

]1

{ξ ≥ dBWO(q, x)

}.

Taking an expectation gives

E[πBWO

]= (1 + α)x + (p− s)

(∫ q

0F (ξ)dξ − τ

∫ q

dBWO(q,x)F (ξ)dξ

)− [(1 + α)c− s]q.

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Scenario (BWR) occurs and the newsvendor borrows with bankruptcy risk if q > ¯q(x). For each

such value of q, let dBWRL (q, x) denote the value of demand below which bankruptcy occurs. The

corresponding ending cash position is

π(q, w; x, dBWRL (q, x)) = (1 + α)x + (p− s)dBWR

L (q, x)− [(1 + α)c− s]q

− τ(αx + (p− s)dBWR

L (d, x)− [(1 + α)c− s]q)+

.

Observe that when the firm is bankrupt, its operating income cannot be positive and there are

zero taxes. Using this and setting π(q, w;x, dBWRL (q, x)) equal to zero gives the break-even point

dBWRL (q, x) = (1+α)c−s

p−s q − 1+αp−s x. For dBWR

L (q, x) < ξ ≤ dBWRU (q, x) ≡ (1+α)c−s

p−s q − αp−sx, the

newsvendor survives, but has an operating loss. She earns an operating income if demand is higher

than the threshold dBWRU (q, x). Thus, we get

πBWR(q, w; x, ξ) =

0 if ξ ≤ dBWRL (q, x),

(p− s)[min{ξ, q} − dBWR

L (q, x)]

if dBWRL (q, x) < ξ ≤ dBWR

U (q, x),

(1− τ)(p− s)[min{ξ, q} − dBWR

U (q, x)]

if ξ > dBWRU (q, x).

Taking an expectation gives

E[πBWR

]= (p− s)

[∫ q

dBWRL (q,x)

F (ξ)dξ − τ

∫ q

dBWRU (q,x)

F (ξ)dξ

].

The above analysis defines the cutoff values of inventory and demand in which the three scenarios

occur and specifies the expected ending cash position of the newsvendor for each scenario. Note that

the cutoff values of demand, dNB, dBWO, dBWRL and dBWR

U are functions of equity and inventory.

We shall drop the arguments of these functions for notational convenience. For instance, dBWRL

denotes dBWRL (q, x). Let q(x) denote the optimal order quantity for the newsvendor as a function

of its starting capital when there is no credit limit. Solving the newsvendor’s problem in the three

scenarios, we show that q(x) has the following form.

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Proposition 1 Let qBWR(x), qBWO(x), qNBL (x) and qNB

U be order quantities defined by:

qBWR(x) = F−1

((1 + α)c− s

(1− τ)(p− s)[F

(dBWR

L

)− τF(dBWR

U

)]), (3)

qBWO(x) = F−1

((1 + α)c− s

(1− τ)(p− s)[1− τF

(dBWO

)]), (4)

qNBL (x) = F−1

((1 + α′)c− s

(1− τ)(p− s)[1− τF

(dNB

)]), (5)

qNBU = F−1

((1 + α′)c− s

p− s

).

Then, the optimal order quantity q(x) for the newsvendor without a credit limit is given by qBWR(x)

if 0 ≤ x < x2, qBWO(x) if x2 ≤ x < x3, x/c if x3 ≤ x ≤ x4, qNBL (x) if x4 < x ≤ x5, and qNB

U if x >

x5. The cutoff values of the newsvendor’s equity solve x2 = (1+α)c−s1+α qBWR(x2), x3 = cqBWO(x3),

x4 = cqNBL (x4), and x5 =

(c + c−s

α′)qNBU .

In this solution, the newsvendor borrows with bankruptcy risk if its equity is less than x2,

borrows without bankruptcy risk if its equity lies between x2 and x3, and does not borrow if its

equity is greater than x3. In the last case, it is interesting to observe that the optimal inventory

varies with x because of taxes and the newsvendor’s cash constraint. When the equity is larger

than x5, the optimal order quantity becomes the same as one would obtain for an unconstrained

newsvendor without taxation.

In the next proposition, we first show that q(x) is continuous. Then we identify the slope of the

optimal order quantity with respect to the starting equity. We find that the optimal order quantity

increases (weakly) in equity when there is no bankruptcy risk.

Proposition 2 q(x) is continuous in x. Furthermore, it strictly increases in x ∈ [x2, x5] and is

constant in x > x5.

The optimal order quantity may be non-monotone when there is bankruptcy risk. Applying the

Implicit Function Theorem, we get

dqBWR(x)dx

=[(1 + α)c− s] [(1 + α)f(dL)− ταf(dU )]

((1 + α)c− s)2(f(dL)− τf(dU ))− (1− τ)(p− s)2f (qBWR). (6)

The denominator in this equation is equal to (p − s)d2E[πBWR(x)]dq2

∣∣∣q=qBWR(x)

, which is shown to be

negative in the proof of Proposition 1. However, the nominator might be positive or negative,

12

which implies that qBWR(x) may increase for some values of x and decrease for others. This result

implies that, for certain parameter values (e.g., τ = 0), the optimal order quantity increases in

decreasing equity. That is, the operations manager’s tendency to take riskier decisions increases

when she has less equity. This result provides an inventory theoretic example for the existence of

a moral hazard problem under limited liability and full information. See Easterbrook and Fischel

(1985) for a general discussion on the incentives created by limited liability to transfer the risk to

debt holders.

We now present the bank’s problem. We assume that the bank has access to sufficient cash.

This assumption is usually met in practice because firms seek loans from large individual banks or

a large consortium of banks. The bank generates income by borrowing money at a rate of α′ and

lending it at a rate of α. Then, the bank’s profit is

ω(x, ξ) = min{(1 + α)w, pmin{q, ξ}+ s(q − ξ)+

}− (1 + α′)w.

In other words, the bank’s revenue is equal to (1 + α)w if the newsvendor is able to repay the

borrowing in full. Otherwise, it is equal to the newsvendor’s sales revenue; note that the newsvendor

has no revenue from deposits when it borrows because borrowing and holding excess cash do not

occur simultaneously. Also note that the bank’s cash flow is independent of whether the newsvendor

repays the principal of the loan after or before paying its taxes. This is so because the newsvendor’s

taxes are zero in the event of bankruptcy.

Following Proposition 1, the bank does not generate any profit if the newsvendor’s equity is

greater than x3 because there is no borrowing. The bank does not need to set a borrowing limit if

the newsvendor’s equity lies between x2 and x3, i.e., she borrows without bankruptcy risk. Indeed,

in this scenario, the bank’s profit increases linearly in w. However, the bank must be cautious

when there is bankruptcy risk because the newsvendor might over-borrow and fail to repay the

loan. From (2), the borrowing amount is given by w = cq − x. Thus, the bank’s problem can be

rewritten in terms of the order quantity that is optimal from the bank’s perspective:

ω(x, ξ) = (α− α′)(cq − x)− (p− s)[dBWR

L − ξ]+

.

That is, the bank generates a profit of (α − α′)(cq − x) if the newsvendor survives. But, if the

newsvendor declares bankruptcy, the bank loses p−s for every unit of demand that is below dBWRL .

13

The following proposition describes the order quantity that maximizes the bank’s expected profit

if the newsvendor has a non-zero risk of bankruptcy.

Proposition 3 If x < x2, the bank’s expected profit by lending to the newsvendor is

E[ω(x, ξ)] = (α− α′)(cq − x)− (p− s)

[dBWR

L −∫ dBWR

L

0F (ξ)dξ

].

It is maximized when q = qB(x) ≡ βx + θ, where β = 1+α(1+α)c−s , θ = p−s

(1+α)c−s F−1

((1+α′)c−s(1+α)c−s

).

Proposition 3 shows that qB(x) is linearly increasing in x. Moreover, qB(0) = θ can be inter-

preted as the minimum credit line, expressed as an order quantity, offered to an all-debt newsvendor.

Thus, the bank is willing to lend money to the newsvendor even if its equity is zero. As a result,

an all-debt firm can be constituted. Intuitively, the value of θ arises from the tradeoff between

the marginal benefit to the bank of allowing the newsvendor to stock an extra unit, (α − α′)c,

and the marginal loss that arises if bankruptcy occurs, ((1 + α)c− s)Pr(ξ < (1+α)c−s

p−s θ). Observe

that (1 + α)c − s is the potential loss because the bank receives the salvage value s instead of

loan plus interest (1+α)c, and Pr(ξ < (1+α)c−s

p−s θ)

is the probability of bankruptcy for an all-debt

newsvendor stocking θ units.

The order quantity at equilibrium is given by q(x) if the credit limit is not binding, and by

qB(x) otherwise. In the next lemma, we prove a necessary and sufficient condition for these two

functions to intersect. We determine the intersection points between q(x) and qB(x), and show

when the credit limit is and is not binding.

Lemma 1 If qBWR(0) ≤ qB(0) then the credit limit is never binding. If qBWR(0) > qB(0), then

there exists an equity value x1 such that 0 < x1 < x2 and qBWR(x1) = qB(x1) and the credit limit

is binding for x ∈ [0, x1]. The value of x1 is given by

(1− τ)(p− s)F (βx1 + θ) + τ [(1 + α)c− s]F(

x1 + [(1 + α)c− s]θp− s

)= (1 + α′)c− s. (7)

Figure 1 illustrates the solution given by Lemma 1. In Figure 1(a), qBWR(0) > qB(0) so that

x1 exists. The newsvendor’s order quantity is restricted by the credit limit when x ≤ x1 and

unrestricted otherwise. In Figure 1(b), qBWR(0) ≤ qB(0), so that the newsvendor’s inventory level

is nowhere restricted by the credit limit. The fact that qBWR(0) can be less than qB(0) is interesting

14

because it shows that the credit line offered to an all-debt newsvendor exceeds its unconstrained

optimal borrowing amount.

When qBWR(0) > qB(0), the newsvendor’s order quantity at equilibrium in the newsvendor-

bank subgame is given by:

q∗(x) =

qB(x) if 0 ≤ x ≤ x1 (Case 1),

qBWR(x) if x1 < x < x2 (Case 2),

qBWO(x) if x2 ≤ x < x3 (Case 3),

x/c if x3 ≤ x ≤ x4 (Case 4),

qNBL (x) if x4 < x ≤ x5 (Case 5),

qNBU if x > x5 (Case 6).

(8)

Correspondingly, the newsvendor’s expected ending cash position is

π∗(x) =

(p− s)

[∫ qB

dBWRL

F (ξ)dξ − τ

∫ qB

dBWRU

F (ξ)dξ

]if 0 ≤ x ≤ x1,

(p− s)

[∫ qBWR

dBWRL

F (ξ)dξ − τ

∫ qBWR

dBWRU

F (ξ)dξ

]if x1 < x < x2,

(1 + α)x− ((1 + α)c− s)qBWO + (p− s)

[∫ qBWO

0F (ξ)dξ − τ

∫ qBWO

dBWO

F (ξ)dξ

]if x2 ≤ x < x3,

s

cx + (p− s)

[∫ x/c

0F (ξ)dξ − τ

∫ x/c

(c−s)x(p−s)c

F (ξ)dξ

]if x3 ≤ x ≤ x4,

(1 + α′)x− (1 + α′)c− s)qNBL + (p− s)

[∫ qNBL

0F (ξ)dξ − τ

∫ qNBL

dNB

F (ξ)dξ

]if x3 ≤ x ≤ x4,

(1 + (1− τ)α′)x− (1− τ)(1 + α′)c− s)qNBU + (1− τ)(p− s)

∫ qNBU

0F (ξ)dξ if x > x5.

(9)

Note that q∗(x) and π∗(x) are defined in terms of six cases. In Case 1, the newsvendor borrows

with bankruptcy risk and the bank’s credit limit is binding. In Case 2, the newsvendor borrows with

risk and the bank’s credit limit is not binding. Cases 1 and 2 are subsumed in scenario (BWR)

defined earlier. In Case 3, the newsvendor borrows without bankruptcy risk; this corresponds

to scenario (BWO). In Case 4, the newsvendor does not borrow and uses up all the equity to

procure inventory. In Case 5, the newsvendor does not borrow, is left with excess cash, and

has a positive probability of incurring an operating loss. In Case 6, the newsvendor stocks the

15

unconstrained optimal quantity, and will always have a non-negative operating income. Cases 4,

5, and 6 correspond to scenario (NB). When qBWR(0) ≤ qB(0), the equilibrium solution is similar

except that Case 1 does not arise.

Remark. A comparison of our solution for the newsvendor-bank subgame with Buzacott and

Zhang (2004) shows the differences arising from taxation. If the tax rate is zero, then our solution

reduces to the result in Buzacott and Zhang (2004). With taxes, Propositions 1 and 2 are general-

izations of the corresponding results in Buzacott and Zhang (2004) and Proposition 3 is identical

to their credit limit. Lemma 1 is a new result showing the existence of x1 to determine when the

credit limit is or is not binding. Taxes affect the solution to the newsvendor-bank subgame thus.

The equilibrium solution in our model has six cases instead of five. Taxation creates an extra sce-

nario, Case 5, in our model. Moreover, the threshold values defining each scenario, x1, . . . , x4, are

smaller because the newsvendor becomes less profitable when there is taxation. The equilibrium

order quantities are also different in all cases except 1 and 6. Most importantly, qBWO in the case

of borrowing without bankruptcy risk varies with x in our model, but is a constant in Buzacott and

Zhang (2004). In addition, qBWR decreases in x in Buzacott and Zhang (2004) due to no taxation,

but it can be non-monotone in our model.

In the next section, we will see that some of the six cases do not occur at equilibrium once the

investor’s problem is introduced. We present the solution to the investor’s problem, first under the

condition qBWR(0) > qB(0) and then under qBWR(0) ≤ qB(0).

3.2 The Investor’s Problem

We consider an investor who has capital K to invest and is given two investment options: the

equity of the newsvendor, and an exogenous asset that has a random rate of return of αI . The

investor seeks to determine the amount x to invest in the newsvendor’s equity in order to maximize

its payoff function Π(x):

Π∗ ≡ maxx∈[0,K]

Π(x) = maxx∈[0,K]

π∗(x) + E[(1 + αI )(K − x)].

16

where E denotes expectation with respect to the random rate of return αI . Let αI = EαI . We

refer to αI as the expected market return or market return. The first part of the objective function,

π∗(x), denotes the payoff from investing in the newsvendor and is given by (9). We assume that the

investor has enough cash to fund the procurement of inventory in all cases, i.e., K > cqNBU . This

assumption is made only to ease the presentation because it guarantees the feasibility of Cases 5

and 6.

We solve the investor’s problem by determining the optimal solution in each of the Cases 1-6

and then finding the highest value. We first show that Case 2 (i.e., x ∈ (x1, x2)) cannot arise in

equilibrium.

Lemma 2 Π(x) is convex in x for x ∈ (x1, x2). Thus, borrowing with risk but ordering less than

the bank’s optimal cannot arise in equilibrium.

As a consequence of this lemma, the optimal equity investment with bankruptcy risk lies in the

range x ∈ [0, x1]. Its value is given by the following proposition.

Proposition 4 Let αl = α (1−τ)(p−s)(1+α)c−s F (βx1 + θ) + (1+α′)c−s

(1+α)c−s − 1 and αh = (1 + α) (1−τ)(p−s)(1+α)c−s F (θ) +

τ (1+α′)c−s(1+α)c−s − 1. The optimal equity investment with bankruptcy risk, x∗R, is given by

x∗R =

x1 if αI < αl,

xR if αI ∈ [αl, αh],

0 if αI > αh,

where xR solves

(1− τ)(1 + α)(p− s)(1 + α)c− s

F (βxR + θ) + τF

(xR + [(1 + α)c− s]θ

p− s

)= 1 + αI . (10)

When αI is relatively high (i.e., αI > αh), the investor does not invest any amount in the

newsvendor because her opportunity cost is high. When αI is relatively low (i.e., αI < αl), the

investor invests as much as she can (i.e., x = x1). For intermediate values of αI , the investor

chooses a value of equity in order to match the return from the newsvendor, given by (10), with

αI .

17

Now consider the cases when the newsvendor does not face any risk of bankruptcy, i.e., when

x ∈ [x2,K]. Let Πj(x) denote the investor’s payoff function in Case j for j ∈ {3, . . . , 6} as a

function of the equity amount. Similarly, let Π∗j denote the investor’s optimal payoff in Case j. In

Case 3, the investor solves

Π∗3 = maxx∈[x2,x3]

Π3(x)

= maxx∈[x2,x3]

(1 + αI )(K − x) + (1 + α)x + ((1 + α)c + s)qBWO(x)

+ (p− s)

[∫ qBWO(x)

0F (ξ)dξ − τ

∫ qBWO(x)

dBWO

F (ξ)dξ

].

Taking a derivative with respect to x and using the definition of qBWO(x) from (4), we get

dΠx

dx= α− αI − ατF

(dBWO(x)

)= α

(1− τ)(p− s)(1 + α)c− s

F(qBWO(x)

)− αI . (11)

The second derivative with respect to x is −α (1−τ)(p−s)(1+α)c−s

dqBWO(x)dx f

(qBWO(x)

), which is nega-

tive because qBWO is increasing in x by Proposition 2. Therefore, the investor’s problem is

concave in Case 3. Concavity implies that (11) attains its maximum at x = x2 and its min-

imum at x = x3, which implies that the optimal solution is in (x2, x3) only if 0 is between

the values of the first derivative at x = x2 and x = x3. The first derivatives at x = x2 and

x = x3 are α (1−τ)(p−s)(1+α)c−s F

(qBWO(x2)

) − αI and α (1−τ)(p−s)(1+α)c−s F

(qBWO(x3)

) − αI , respectively. Let

α2 = α (1−τ)(p−s)(1+α)c−s F

(qBWO(x2)

)and α3 = α (1−τ)(p−s)

(1+α)c−s F(qBWO(x3)

). Then the optimal equity

investment in Case 3, x∗3, is given by

x∗3 =

x2 if αI > α2,

x3 if αI ∈ [α3, α2],

x3 if αI < α3,

where x3 is obtained by setting (11) equal to zero.

In Case 4, the investor solves

Π∗4 = maxx∈[x3,x4]

Π4(x)

= maxx∈[x3,x4]

(1 + αI )(K − x) +s

cx + (p− s)

[∫ x/c

0F (ξ)dξ − τ

∫ x/c

(c−s)x(p−s)c

F (ξ)dξ

].

18

This objective function is concave in x because the second derivative is −(1 − τ)p−sc2

f(x/c) −τ

(c−sc

)2 1p−sf

((c−s)x(p−s)c

)≤ 0. The first derivative is

−(1 + αI ) +s

c+ (1− τ)

p− s

cF

(x

c

)+ τ

c− s

cF

((c− s)x(p− s)c

), (12)

which is decreasing in x. q(x) is continuous by Proposition 2. Therefore, x3/c = qBWO(x3), and

the first derivative at x = x3 is

−(1 + αI ) +s

c+

p− s

c

((1− τ)F

(qBWO(x3)

)+ τ

c− s

p− sF

(c− s

p− sqBWO(x3)

)). (13)

Using (4), we know that τF(

c−sp−sq

BWO(x3))

= 1 − (1−τ)(p−s)(1+α)c−s F (qBWO(x3)). Substituting it into

(13) gives α (1−τ)(p−s)(1+α)c−s F (qBWO(x3)) − αI , which is equal to α3 − αI . Similar analysis shows that

the derivative at x = x4 is equal to α′ (1−τ)(p−s)(1+α′)c−s F

(qNBL (x4)

)− αI . Therefore,

x∗4 =

x3 if αI > α3,

x4 if αI ∈ [α4, α3],

x4 if αI < α4,

where α4 = α′ (1−τ)(p−s)(1+α′)c−s F

(qNBL (x4)

)and x4 is obtained by setting (12) equal to zero.

In Case 5, the investor solves

Π∗5 = maxx∈[x4,x5]

Π5(x)

= maxx∈[x4,x5]

(1 + αI )(K − x) + (1 + α′)x− ((1 + α′)c− s

)qNBL (x)

+ (p− s)

[∫ qNBL (x)

0F (ξ)dξ − τ

∫ qNBL (x)

dNB

F (ξ)dξ

].

Observe that this function is very similar to the objective function in Case 3 except that α is

replaced with α′. The first derivative is

α′(1− τ)(p− s)(1 + α′)c− s

F (qNBL (x))− αI (14)

and the second derivative is −α′ (1−τ)(p−s)(1+α′)c−s

dqNBL (x)dx f

(qNBL (x)

), which is negative since qNB

L (x) is

increasing in x by Proposition 2. Therefore, the investor’s problem is concave in Case 5. One can

show that

x∗5 =

x4 if αI > α4,

x5 if αI ∈ [α4, α5],

x5 if αI < α5,

19

where α5 = α′ (1−τ)(p−s)(1+α′)c−s F

(qNBL (x5)

)= (1−τ)α′ because qNB

L (x5) = qNBU . x5 is obtained by setting

(14) equal to zero.

In Case 6, the investor solves

Π∗6 = maxx∈[x5,K]

Π6(x)

= maxx∈[x5,K]

(1 + αI )(K − x) +(1 + (1− τ)α′

)x + (1− τ)

((p− s)

∫ qNBU

0F dξ − [(1 + α′)c− s]qNB

U

).

Therefore, the optimal equity investment in Case 6, x∗6, is given by

x∗6 =

x5 if αI > α5,

K if αI ≤ α5.

That is, the investor transfers all of his wealth to the newsvendor if the tax adjusted deposit rate,

(1− τ)α′, is higher than the expected market return.

Collecting the solutions under Cases 3-6 together, we obtain the following proposition.

Proposition 5 The optimal equity investment in [x2,K], i.e., without bankruptcy risk, is

x∗NR =

K if αI ≤ α5,

x∗5 if αI ∈ (α5, α4],

x∗4 if αI ∈ (α4, α3],

x∗3 if αI ∈ (α3, α2],

x2 if αI > α2.

Proof: The objective function Π(x) is concave in all four cases, and the first derivatives from the

right and left are equal to each other at every switching point. Therefore, collecting these four

cases together gives the optimal solution under no borrowing. 2

That is, the optimal investment amount is K when αI is very low. On the other hand, it is as

small as possible when αI is high. Thus far, we have treated the investment with and without the

bankruptcy risk as two separate scenarios. Let Π∗R(αI ) and Π∗NR(αI ) denote the investor’s payoff

functions with and without risk, respectively. In the next section, we compare Π∗R and Π∗NR to

obtain the global optimal solution for the investor.

20

3.3 When Is Investing with Bankruptcy Risk Optimal?

In comparing the investor’s payoff functions with and without bankruptcy risk, we find that Π∗R and

Π∗NR are both increasing in αI , but Π∗R is increasing at a faster rate. Moreover, Π∗NR > Π∗R when

αI is sufficiently small, and Π∗NR < Π∗R when αI is large. As a result, the two functions intersect at

a unique threshold value of αI . Let α be the threshold. If αI ≤ α, then the investor puts enough

equity into the newsvendor that there is zero probability of bankruptcy. Otherwise, the investor

finds it optimal to invest with bankruptcy risk. We formalize this result in the following proposition

and present its proof in the appendix.

Proposition 6 There exists a unique threshold return value α such that the investing without

bankruptcy risk is optimal when αI ≤ α and investing with bankruptcy risk is optimal otherwise.

Non-Binding Credit Limit. We revisit the scenario illustrated in Figure 1(b). When qB(0) ≥qBWR(0), the bank’s credit limit is greater than the newsvendor’s optimal purchase quantity for

x ∈ (0, x2). Thus, Case 1, i.e., borrowing with risk and ordering the bank’s optimal quantity,

does not arise because the credit limit is never binding. Further, from Lemma 2, the investor’s

payoff function is convex in Case 2. Therefore, the optimal solution for the borrowing with risk

scenarios is either at x = 0 or x = x2. The optimal solution for borrowing without bankruptcy risk,

Proposition 5, remains unchanged because Cases 3-6 are unaffected by the credit limit. Combining

these together, the investor’s global optimal solution when qB(0) ≥ qBWR(0) is at either x∗ = 0 or

x∗ = x∗NR. Writing the investor’s payoff functions for x = 0 and x = x∗NR and comparing them,

we obtain a threshold value such that x = 0 is optimal for αI values that exceed the threshold and

x = x∗NR is optimal for αI values that are below the threshold. We omit this step because it is

analogous to Proposition 6.

In general, the value of α can be determined by a numerical search technique to find the

intersection point between Π∗R(αI ) and Π∗NR(αI ). In the next section, we derive a formula for α for

the special case of no taxation.

21

3.4 Example: No Corporate Taxes

In this section, we characterize α for the special case of no corporate taxes. Consider the solution to

the newsvendor-bank subgame. Solving (7) with τ = 0 gives x1 = qNBU −θ

β . Thus, the expression for

the cutoff equity value between Cases 1 and 2 is simplified. Moreover, this expression shows that

qBWR(x1) = qB(x1) = qNBU . In words, the optimal inventory at the equity value of x1 is identical

to the unconstrained optimal order quantity. For the remaining cases, setting τ = 0 and solving (4)

and (5) shows that qBWO = F−1(

(1+α)c−sp−s

)and qNB

L = F−1(

(1+α′)c−sp−s

). Thus, the order quantity

in Case 3, qBWO, becomes a constant independent of the capital structure. Moreover, Cases 5

and 6 collapse into a single case because qNBU = qNB

L . In summary, the newsvendor-bank subgame

yields four or five cases depending on the condition in Lemma 1. When there are five cases, the

solution is identical to that in Buzacott and Zhang (2004).

We derive the optimal equity investment values xR and xNR by rewriting Propositions 4 and 5

under no taxation.

Proposition 7 If τ = 0, then αl = (1 + α) (1+α′)c−s(1+α)c−s − 1 and αh = (1 + α) p−s

(1+α)c−s F (θ) − 1. Let

xR = qR−θβ and qR = F−1

((1 + αI )

(1+α)c−s(1+α)(p−s)

). Then the optimal equity investment with risk, x∗R,

is equal to x1 if αI < αl, xR if αI ∈ [αl, αh], and 0 if αI > αh. Similarly, let xNR = cqNR and

qNR = F−1(

(1+αI)c−s

p−s

). Then the optimal equity investment without risk, x∗NR, is equal to K if

αI < α′, xNR if αI ∈ [α′, α], and x2 if αI > α. Furthermore, investing with risk is optimal when

αI > α and investing without risk is optimal when αI < α′.

Proposition 7 shows that the threshold return value α lies between α′ and α when τ = 0. We

know the exact form of Π∗NR for αI ∈ [α′, α]. However, we do not know the exact form of Π∗R

because αh can be anywhere between αl and ∞. Proposition 8 refines the optimal solution further

when qBWO ≥ θ. Table 1 shows the optimal order quantity, debt, and equity values for this specific

case.

22

Proposition 8 If τ = 0 and qBWO ≥ θ, then [α′, α] ⊂ [αl, αh], and the optimal investment is

x∗ =

K if αI ≤ α′,

cqNR if αI ∈ (α′, α],qR − θ

βif αI ,∈ (α, αh],

0 if αI > αh,

where α solves

(p− s)

(∫ qR(α)

η1θF (ξ)dξ −

∫ qNR(α)

0F (ξ)dξ

)− (1 + α)

qR(α)− θ

β+ ((1 + α)c− s)qNR(α) = 0. (15)

Table 1: Equilibrium values of inventory, debt, and equity when τ = 0 and qBWO > θ

Interval Order Quantity (q∗) Equity (x∗) Debt (w∗)

αI ≤ α′ qNB K 0

αI ∈ (α′, α] qNR cqNR 0

αI ∈ (α, αh] qRqR−θ

βs

1+α qR + θβ

αI > αh θ 0 cθ

4 Managerial Implications

4.1 Probability of Bankruptcy

The most important result of our paper is that the probability of bankruptcy is either 0 or (α−α′)c(1+α)c−s

at equilibrium. To see this, note that the newsvendor does not face bankruptcy risk when αI ≤ α

because the investor chooses to invest a relatively large amount. If αI > α, the investor invests a

small enough amount that forces the newsvendor to borrow with risk. Further, by Lemma 2, Case

2 cannot arise in equilibrium. Therefore, if αI > α, the credit limit is binding, and the newsvendor

declares bankruptcy if the demand is less than dBWR(qB(x), x). The probability of occurrence of

this event is Pr(ξ < dBWR(βx + θ, x)) = (α−α′)c(1+α)c−s . Therefore,

Pr(Bankruptcy) =

0 if αI ≤ α,

(α−α′)c(1+α)c−s if αI > α.

23

This formula captures the dynamics that determine bankruptcy risk at equilibrium. The factors

that directly affect the probability of bankruptcy are the interest rates α and α′ and the relative

salvage value of the product, s/c. The probability of bankruptcy increases in s/c as well as in the

interest rate spread because the bank is willing to allow the newsvendor a higher order quantity as

s/c or α− α′ increases. This increase in the order quantity increases the probability of ending up

with unsold units, so that the probability of bankruptcy increases.

Additionally, the market return αI and the threshold α also affect the bankruptcy probability

by determining whether the newsvendor will be able to raise enough equity to avoid borrowing with

risk. The market return αI is a snapshot of the financial market. Holding the underlying demand

distribution unchanged, a higher expected return in the market can increase the probability of

bankruptcy from 0 to (α−α′)c(1+α)c−s because the investor has other attractive investment alternatives.

Relationship between the bankruptcy risk and market parameters such as interests rates and S&P

500 returns has been empirically tested in corporate finance literature (e.g., Duffie et al. 2007). The

factors affecting α include the underlying demand distribution, the tax rate, and the cost/revenue

parameters.

4.2 Corporate Taxation

We find that borrowing without risk cannot arise as an equilibrium outcome when there is no

taxation (i.e., τ = 0). This result captures the tax shield of debt. We also show that the investor’s

tendency to create a pure equity newsvendor decreases as the tax rate increases.

The tradeoff theory of capital structure in corporate finance argues that the optimal capital

structure exists due to a tradeoff between the tax benefits of debt and bankruptcy costs (Kraus

and Litzenberger 1973). In order to see the tax benefits of debt, consider the borrowing without

risk scenario in our model. Setting τ = 0 gives qBWO(x) = F(

(1+α)c−sp−s

), which is a fixed quantity

independent of x. As a result, the investor’s optimization problem in Case 3 becomes:

maxx∈[x2,x3]

(1 + αI)K + (α− αI )x + (p− s)∫ qBWO

0F (ξ)dξ − [(1 + α)c− s]qBWO.

This objective function is linear in x because the capital requirement cqBWO is fixed. This implies

that an interior solution cannot arise in equilibrium when τ = 0. Thus, the optimal investment is

24

either x2 or x3. However, as we showed in Proposition 5, x∗ ∈ (x2, x3) can arise when τ > 0. This

shows that borrowing without bankruptcy risk arises due to the tax benefit of debt.

Borrowing has another benefit from the investor’s standpoint. It allows the investor to share

some of its risk with the bank due to the newsvendor’s limited liability structure. In our equilibrium,

the newsvendor has risky borrowing if αI ≥ α. Therefore, we assess how α varies in the tax rate to

determine the newsvendor’s tendency to borrow at equilibrium. Figure 2 illustrates this relationship

for a numerical example. We observe that α decreases in the tax rate implying that the range of

parameters for which the bankruptcy risk arises increases when corporate tax rate increases. In

other words, increase in taxes would lead to more firms borrowing with risk due to a decrease in

the newsvendor’s effective rate of return under taxation.

4.3 Implementing the Credit Limit

Being able to share the risk of bankruptcy with debt holders presents an attractive investment

opportunity for the investor. Section 3.3 shows that in the absence of a credit limit, the investor

can play a risk free gamble by creating a pure debt newsvendor and the newsvendor can seek to

borrow an excessively large amount. An optimally set credit limit mitigates this risk for the bank.

In this section, we show that the bank can implement an asset based credit limit by choosing an

appropriate inventory valuation factor γq. This factor allows the bank to control the newsvendor’s

sales volatility. We also briefly discuss potential drawbacks of setting the credit limit sub-optimally.

According to Proposition 3, the bank sets the maximum order quantity equal to qB(x) = βx+θ.

The bank can limit the newsvendor to a maximum of this order quantity using an asset based credit

limit, as a function of the two types of assets held by the newsvendor, cash and inventory. We define

ψ = γc(x+w−cq)+γqcq, where γc and γq denote the bank’s valuation of $1 of the newsvendor’s cash

and inventory, respectively. For instance, if γq = 0.9 then the bank values the starting inventory

at 90% of its wholesale value. The excess cash will be zero in the case in which the newsvendor

borrows with risk. This makes γc irrelevant and allows us to write ψ = γqcq. Using the expressions

for β and θ from Proposition 3, we find that the bank can limit the newsvendor to order a maximum

of qB(x) by setting γq = cqB(x)−xcqB(x)

. Writing γq as a function of x gives γq = 1−x/(

(1+α)c(1+α)c−sx + cθ

).

The most important import of this formula is that γq is not independent of x. In fact, contrary

25

to intuition, γq decreases in x. Figure 3 illustrates this relationship. This result occurs because,

when x is large, the bank tries to reduce sales volatility by lowering γq. We see that x∗ decreases

in αI . However, the bank increases γq because a decrease in x∗ implies a decrease in the order

quantity, which decreases the sales volatility. At the extreme values of αI (e.g., αI = 0.5), γq is

close to 1 because the investor invests a very small amount in the newsvendor. The newsvendor,

in turn, purchases a very small amount, which is almost assured to be sold.

In practice, secured lenders use simple rules of thumb to value inventory. According to one

commercial lender that we spoke to, banks use the historical salvage value of inventory as the only

metric for setting γq. Such inventory valuation is consistent with our formula for the probability of

bankruptcy, but fails to capture firm-specific factors such as the starting capital and the underlying

demand distribution in determining γq. As a consequence, the bank’s expected profit and the

probability of bankruptcy deviate from the equilibrium outcome. Thus, our model shows that

banks can improve their practice by tailoring inventory valuation to different firms using objective

optimization criteria.

4.4 Capital Structure at Equilibrium

In this section, we discuss how the equilibrium capital structure depends on two important pa-

rameters of the newsvendor model, demand uncertainty and profit margin. We show that the

debt-to-equity ratio at equilibrium is decreasing in demand uncertainty, and is non-monotone in

the profit margin. We illustrate these observations for the scenario in which there are no corporate

taxes and the newsvendor borrows with risk.

We choose to work with the borrowing with risk under no taxation scenario because it is

relatively straightforward to see the interactions among x∗, w∗, and q∗ and their relationship with

operational parameters. From Table 1, x∗ = q∗−θβ and w∗ = s

1+αq∗ + θβ , so that the equilibrium

debt-to-equity ratio is w∗/x∗. Here, q∗ = qR = F−1((1 + αI )

(1+α)c−s(1+α)(p−s)

)and q∗, θ and β are

functions of demand uncertainty, price and cost parameters, and interest rates.

Figure 4 shows that the equilibrium debt-to-equity ratio declines as demand uncertainty in-

creases. To wit, as demand uncertainty increases, q∗ increases as a result of an increase in the

safety stock and θ decreases. Therefore, the amount of equity, x∗, rises. Moreover, the decrease in

26

θ is sharper than the increase in q∗ implying that the debt w∗ decreases in demand uncertainty.

Both these changes, in w∗ and x∗, lead to a decline in the debt-to-equity ratio. Intuitively, the

bank reacts to the increase in demand uncertainty by reducing the credit limit, the investor ends

up increasing the equity investment in order to compensate for the decrease in debt and increase

in the order quantity, and thus, the firm becomes less leveraged.

Figure 5 shows that there is a non-monotone relationship between capital structure and the

newsvendor’s profit margin. This non-monotonicity result arises because both x∗ and w∗ increase

in profit margin, but at different rates. For numerical analysis, we define the profit margin as p−cp−s .

For a given demand distribution, we fix c and s and vary p to change the profit margin. Note that

the probability of bankruptcy remains unchanged because c and s are fixed. Both q∗ and θ increase

as p increases. Further, q∗ increases at a faster rate implying that x∗ also increases in p. For the

parameter values we chose, the rate of increase is higher for x∗ for low margin values, whereas it is

higher for w∗ for high margin values. As a result, the optimal debt-to-equity ratio first decreases

then increases.

Undoubtedly, the above numerical results depend on the chosen parameter values. Our main

takeaway is that, although the equilibrium capital structure can be written as a function of the

newsvendor parameters, its sensitivity to these parameters depends on complex interactions among

players that determine the values of q, x, and w at equilibrium. Another challenge arises due to the

fact that the equilibrium can switch from one scenario (e.g., borrowing with risk) to another (e.g.,

borrowing without risk) as parameters vary. Therefore, this exercise provides a simple explanation

for conflicting evidence observed in the empirical corporate finance literature on the impact of

various parameters on capital structure (See Harris and Raviv (1991) for mixed empirical results).

5 Conclusions

We characterize the equilibrium capital structure, inventory level, and the risk of bankruptcy of

a newsvendor in a game played between the newsvendor, its creditor bank, and its investor. The

main finding of our paper is that the probability of bankruptcy is not a function of the inventory

level of the firm at equilibrium, but depends on the cost parameters of the newsvendor model and

27

interest spread. Insights provided by our model include how the asset based credit limit can be

set by the bank, and how the expected market return, taxation, demand uncertainty, and profit

margins affect the capital structure.

We refer to αI as the expected market return. In practice, αI can also be interpreted as the

minimum return required by a potential investor. In 2009, on average, the required returns for

private equity groups and venture capital funds were 25.0% and 38.2%, respectively (Paglia 2010).

Such high numbers explain why these types of investors inject a minimal amount of equity and rely

on debt while making a new investment because these numbers are almost assured to exceed α.

Many aspects of our model can be generalized in future research. For example, players, esp. the

investor, may be modeled as expected utility maximizers, agency issues can be added, and a cost

of bankruptcy may be considered. It may also be productive to allow trading between the bank

and the investor or to replace them with two investor classes. Results similar to ours should obtain

as long as the investor classes differ in utility or endowment and vary in their preferences for debt

and equity. In this respect, our paper follows from Stiglitz (1972). Another potential direction is

to extend our model to a multi-period setting in order to capture time-varying bankruptcy risk.

We assume a fixed lending rate α for all borrowers. Modeling α as a decision variable rather

than setting an asset based credit limit is another alternative, but implementing the equilibrium

interest rate of such model is difficult in practice (Dada and Hu 2008). Difficulty of setting an

interest rate sheds some light on why asset based financing is commonly used in practice. Using a

fixed α and setting an asset based credit limit provides the decision makers with a practical solution

because β and θ are easy to compute. Future research may examine the pros and cons of different

lending strategies.

Our analysis motivates the necessity to build both theoretical and empirical models in order

to understand the link between a firm’s operational performance and its bankruptcy risk. Future

empirical research may examine how to use operational indicators to have a better understanding

of a firm’s bankruptcy risk.

28

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30

Figure 1: The newsvendor’s and the bank’s optimal order quantities for different investment levels.In Figure (a), demand is Weibull with E[ξ] = 10 and shape parameter k = 2. p = 2, c = 1,s = 0.6, τ = 0.4, α = 0.10, and α′ = 0.05. As a result, qBWR(0) = 12.16 > qB(0) = 10.26, and thecredit limit binds for x ∈ [0, x1]. x1 = 0.75, x2 = 5.04, x3 = 11.17, x4 = 11.74, x5 = 108.19, andqNBU = 12.02. In Figure (b), all the model parameters are the same as Figure (a) except s = 0.7.

As a result, qB(0) = 13.40 > qBWR(0) = 12.88, and the credit limit never binds. x2 = 4.35,x3 = 12.05, x4 = 12.71, x5 = 90.48, and qNB

U = 12.93. The graph is truncated at x = 15 to betterillustrate Cases 1 to 5 while omitting Case 6.

Equity Investment

Ord

er Q

uant

ity

(a)

Bank’s OptimalNewsvendor’s Optimal

qLNB

x c

qBWOqBWRqB

0 x1 x2 x3 x4 15

10

11

12

13

Case 1 Case 2 Case 3 Case 4 Case 5

Equity Investment

Ord

er Q

uant

ity

(b)

Bank’s OptimalNewsvendor’s Optimal

qLNB

x c

qBWO

qBWR

qB

0 x2 x3 x4 15

12

13

14

15

Case 2 Case 3 Case 4 Case 5

31

Figure 2: The threshold value α as a function of the tax rate for different interest rate spread values.ξ ∼ exp(10), p = 1, c = 0.5, s = 0.15. We set the probability of bankruptcy equal to 0.05 and changeα′ and α. (α′, α) = (0.022, 0.060) gives a 3.8% interest spread. Similarly, we use (0.041,0.080) and(0.06,0.10) to get 3.9% and 4.0% spread, respectively. The value of α decreases in the tax rateand interest rate spread. The region above each line denotes values of αI at which borrowingwith bankruptcy risk takes place, and the region below denotes values at which no borrowing orborrowing without bankruptcy risk takes place. The newsvendor is more likely to borrow withbankruptcy risk when taxes are high and interest spread is low.

0.0 0.1 0.2 0.3 0.4

0.00

0.02

0.04

0.06

0.08

Tax Rate, τ

Thr

esho

ld In

tere

st R

ate,

α~

Interest spread3.8%3.9%4%

Borrowing withbankruptcy risk

No borrowing or borrowingwithout bankruptcy risk

Figure 3: Optimal equity investment and inventory valuation factor γq as a function of αI showinghow the inventory valuation in the credit limit increases as equilibrium equity decreases. ξ ∼exp(10), p = 1, c = 0.5, s = 0.15, τ = 0.40, α = 0.1, α′ = 0.05

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.5

1.0

1.5

Expected Market Return, αI

Opt

imal

Equ

ity In

vest

men

t, x*

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.5

0.6

0.7

0.8

0.9

1.0

Expected Market Return, αI

Inve

ntor

y V

alua

tion

Fac

tor,

γq

32

Figure 4: Equilibrium order quantity and debt-to-equity ratio as a function of demand uncertainty.ξ is Weibull with E[ξ] = 10, p = 1, c = 0.3, s = 0.15, τ = 0, α = 0.1, α′ = 0.05, αI = 0.09. We varythe scale parameter between 1 and 1.75 to change the demand coefficient of variation.

0.6 0.7 0.8 0.9 1.0

0

5

10

15

20

Demand Coefficient of Variation

Ord

er Q

uant

ity

q

θ

0.6 0.7 0.8 0.9 1.0

0

5

10

15

20

Demand Coefficient of Variation

Deb

t to

Equ

ity R

atio

Figure 5: Equilibrium order quantity and debt-to-equity ratio as a function of the profit margin.ξ ∼ exp(10), c = 0.5, s = 0.15, τ = 0, α = 0.1, α′ = 0.05, and αI = 0.09. We vary p between 0.75and 2 to change the profit margin, which we define as p−c

p−s .

0.4 0.5 0.6 0.7 0.8

0

5

10

15

Profit Margin

Ord

er Q

uant

ity

q

θ

0.4 0.5 0.6 0.7 0.8

0.70

0.75

0.80

Profit Margin

Deb

t to

Equ

ity R

atio

33

Online Appendix

A. Nomenclature

p: selling price

c: purchase cost

s: salvage value

αI : rate of return of the alternative investment option with αI = EαI

α: borrowing rate

α′: deposit rate

τ : tax rate

x: newsvendor’s starting capital

w: newsvendor’s borrowing amount

ξ: random demand

F (·): complementary CDF of the demand distribution

F−1(·): inverse of complementary CDF

f(·): pdf of the demand distribution

θ =p− s

(1 + α)c− sF−1

((1 + α′)c− s

(1 + α)c− s

)

β =1 + α

(1 + α)c− s

dNB(q, x) =(1 + α′)c− s

p− sq − α′

p− sx

dBWO(q, x) = dBWRU (q, x) =

(1 + α)c− s

p− sq − α

p− sx

dBWRL (q, x) =

(1 + α)c− s

p− sq − 1 + α

p− sx

αl = α(1− τ)(p− s)(1 + α)c− s

F (βx1 + θ) +(1 + α′)c− s

(1 + α)c− s− 1

αh = (1 + α)(1− τ)(p− s)(1 + α)c− s

F (θ) + τ(1 + α′)c− s

(1 + α)c− s− 1

α2 = α(1− τ)(p− s)(1 + α)c− s

F(qBWO(x2)

)

α3 = α(1− τ)(p− s)(1 + α)c− s

F(qBWO(x3)

)

α4 = α′(1− τ)(p− s)(1 + α′)c− s

F(qNBL (x4)

)

α5 = α′(1− τ)

34

B. Proofs

Proof of Proposition 1. Setting the first derivative of E[πNB

x>x

]with respect to q equal to zero

gives the unconstrained newsvendor solution, qNBU . Replacing q with qNB

U in x =(c + c−s

α′)q gives

x5. The first derivative of E[πNB

cq≤x≤x

]with respect to q is

(1− τ)(p− s)F (q)− [(1 + α′)c− s](1− τF

(dNB

)),

whereas the second derivative is

−(1− τ)(p− s)f(q)− τ[(1 + α′)c− s]2

p− sf

(dNB

) ≤ 0.

Hence, E[πNB

cq≤x≤x

]is concave. Solving the first order condition for this subscenario gives qNB

L (x).

x4 can be obtained by solving the first order condition for the specific case in which x = cq. The

first derivative of E[πBWO] with respect to q is

(1− τ)(p− s)F (q)− [(1 + α)c− s](1− τF

(dBWO

)),

whereas the second derivative is

−(1− τ)(p− s)f(q)− τ[(1 + α)c− s]2

p− sf

(dBWO(q, x)

) ≤ 0.

Hence, E[πBWO

]is also concave. Solving the first order condition gives qBWO(x). x3 can be

obtained by setting x = cqBWO(x), and solving the first order condition. (1−τ)(p−s) > (1+α)c−s

guarantees the existence of qBWO(x), qBWR(x), and qNBL (x).

We need to show that x3 ≤ x4 to define the case in which the newsvendor does not borrow, but

uses all of her cash for procurement (i.e., q = x/c). Observe that qBWO(x3) is obtained by solving

(1− τ)(p− s)F (q)− [(1 + α)c− s](

1− τF

(c− s

p− sq

))= 0, (16)

whereas qNBL (x4) is obtained by solving the same equation in which α is replaced with α′. Implicit

differentiation of q with respect to α in (16) gives

dq

dα= −

(p− s)c(1− τF

(c−sp−sq

))

(1− τ)(p− s)2f(q) + τ [(1 + α)c− s](c− s)f(

c−sp−sq

) < 0.

Hence, qBWO(x3) < qNBL (x4) because α > α′, which implies that x3 < x4.

35

The first derivative of E[πBWR] with respect to q is

(1− τ)(p− s)F (q) + [(1 + α)c− s](τF (dBWR

U − F (dBWRL )

).

For notational convenience, let dL and dU denote dBWRL and dBWR

U , respectively. Then the second

derivative is

d2E[πBWR]dq2

= −(1− τ)(p− s)f(q) +[(1 + α)c− s]2

p− s(f(dL)− τf(dU )) . (17)

qBWR(x) must satisfy the first order condition. Therefore, at q = qBWR(x)

d2E[πBWR]dq2

= (1− τ)(p− s)F (q)

−(1− τ)(p− s)f(q) + [(1+α)c−s]2

p−s (f(dL)− τf(dU ))

(1− τ)(p− s)F (q)

= (1− τ)(p− s)F (q)(−z(q) +

(1 + α)c− s

p− s

f(dL)− τf(dU )F (dL)− τF (dU )

)

< (1− τ)(p− s)F (q)(−z(q) +

(1 + α)c− s

p− s

(f(dL)

F (dL)− τF (dU )

))

< (1− τ)(p− s)F (q)(−z(q) +

(1 + α)c− s

(1− τ)(p− s)z(dL)

)

< 0

where z is the hazard rate function. z(q) > (1+α)c−s(1−τ)(p−s)z(dL) because ξ is IFR and (1+α)c−s

(1−τ)(p−s) < 1.

x2 can be obtained by solving the first order condition of the borrowing with risk case after setting

x = (1+α)c−s1+α q.

Proof of Proposition 2. Observe that x2 is the point where the newsvendor switches from

(BWR) to (BWO). Therefore, the break-even demand point, dBWRL approaches zero as x approaches

x2. Mathematically,

limx↑x2

dBWRL

(qBWR(x), x

)= 0,

which implies that

limx↑x2

(1 + α)c− s

(1− τ)(p− s)F

(qBWR(x2)

)= lim

x↑x2

F(dBWR

L

)− τF(dBWR

U

)

= 1− τF(dBWO

).

36

The second line follows because dBWRU = dBWO. Therefore, limx↑x2 qBWR(x2) = qBWO(x2). Similar

analysis shows the continuity of q(x) at the other cutoff points.

q(x) = x/c for x ∈ [x3, x4], which implies that q(x) increases in x ∈ [x3, x4]. In addition,

∂2E[πBWO]∂q∂x

= ατ(1 + α)c− s

(p− s)2f

(dBWO

) ≥ 0.

Therefore, E[πBWO] is supermodular, which implies that qBWO(x) is increasing in x. Supermod-

ularity of E[πNBx≤x≤x] can be shown in a similar manner. Collecting these cases implies that q(x)

increases x ∈ [x2, x5] because it is a continuous function. q(x) = qNBU is constant in x > x5.

Proof of Proposition 3. Taking a derivative with respect to q and setting it equal to zero gives

qB(x) as one can show that the bank’s expected return under the newsvendor bankruptcy is concave

in q for a given x.

Proof of Lemma 1. qBWR(x) and qB(x) are both continuous functions. First, we show that

qBWR(x2) < qB(x2). To see this, note that x2 = qBWR(x2)/β, whereas qB(x2) = βx2 + θ =

qBWR(x2) + θ > qBWR(x2). Therefore, it follows that if qBWR(0) > qB(0), then there exists

0 < x1 < x2 such that qBWR(x1) = qB(x1). Substituting qBWR(x1) = qB(x1) = βx1 + θ in (3)

gives (7). x1 is unique because the left hand side of (7) increases in x. For the other direction,

suppose qB(0) ≤ qBWR(0), and the two functions intersect. The intersection point must be unique

because, as we explained above, it has to satisfy (7). If the intersection point is unique, then the

derivatives of the two functions at that point must be equal to each other (i.e., the derivative of

qB(x)− qBWR(x) must be zero). From (6) and Proposition 3, dqB(x)dx = dqBWR(x)

dx is equivalent to

β =[(1 + α)c− s] [(1 + α)f(dL)− ταf(dU )]

((1 + α)c− s)2(f(dL)− τf(dU ))− (1− τ)(p− s)2f (qBWR),

which can be written as

τ((1 + α)c− s)2f(dU ) + (1− τ)(1 + α)(p− s)2f(qBWR) = 0,

which cannot hold because ξ has an IFR distribution. Therefore, the two functions cannot intersect

if qB(0) ≤ qBWR(0).

37

Proof of Lemma 2. The order quantity in Case 2, qBWR(x), solves

(1− τ)(p− s)F (qBWR(x)) + [(1 + α)c− s](τF (dBWR

U )− F (dBWRL )

)= 0, (18)

and the investor solves

Π∗2 = maxx∈[x1,x2]

Π2(x)

= maxx∈[x1,x2]

(1 + αI )(K − x) + (p− s)

(∫ qBWR(x)

dBWRL

F (ξ)dξ − τ

∫ qBWR(x)

dBWRU

F (ξ)dξ

)

The first derivative of Π2(x) with respect to x is

dΠ2(x)dx

= −(1 + αI ) + (1 + α)F (dBWRL )− ταF (dBWR

U )

= −(1 + αI ) + (1 + α)(1− τ)(p− s)(1 + α)c− s

F (qBWR(x)) + τF (dBWRU ).

The second line follows from (18). We drop the superscript BWR for notational convenience. The

second derivative is

−(

(1− τ)(1 + α)f(q) + τ(1 + α)c− s

p− sf(dU )

)dq

dx+ τ

α

p− sf(dU ) (19)

Using (6), (19) can be written as

dΠ22(x)

dx2=(1− τ)

τα {(p− s)− (1 + α)[(1 + α)c− s]} f(dU )f(q) + (1 + α)2[(1 + α)c− s]f(q)f(dL)(1− τ)(p− s)2f(q)− [(1 + α)c− s]2(f(dL)− τf(dU ))

+τ [(1 + α)c− s]2f(dU )f(dL)

(p− s) [(1− τ)(p− s)2f(q)− [(1 + α)c− s]2(f(dL)− τf(dU ))],

which is non-negative because we assume that p−s(1+α)c−s ≥ 1 + α and the denominators of both

terms are positive from (6). Hence, we can rule out the interior, (x1, x2), as the optimal solution

will be at x = x1 or x = x2. Furthermore, the objective function is continuous, which implies that

x = x1 is taken in the account in the first case.

Proof of Proposition 4. This is Case 1 in which the investor solves

Π∗1 = maxx∈[0,x1]

Π1(x)

= maxx∈[0,x1]

(1 + αI )(K − x) + (p− s)

[∫ qB(x)

(1+α)c−sp−s

θF (ξ)dξ − τ

∫ qB(x)

dBWRU (x)

F (ξ)dξ

]

38

where qB(x) = βx + θ and dBWRU (x) = x+[(1+α)c−s]θ

p−s . The objective function is concave in [0, x1]

because the second derivative is

−(1− τ)(1 + α)(p− s)(1 + α)c− s

βf(βx + θ)− τ

p− sf

(x1 + [(1 + α)c− s]θ

p− s

)≤ 0.

Setting the first derivative equal to zero gives (10). αh is obtained by setting x = 0 in (10).

Similarly, αl is obtained by setting x = x1. That is,

αl = (1− τ)(1 + α)(p− s)(1 + α)c− s

F (βx1 + θ) + τF

(x1 + [(1 + α)c− s]θ

p− s

)− 1

= (1− τ)α(p− s)

(1 + α)c− sF (βx1 + θ) +

(1 + α′)c− s

(1 + α)c− s− 1.

The second equality follows from (7). It can be shown that xR < 0 when αI > αh and xR > x1

when αI < αl. Moreover, αh > αl because the left hand side of (10) is decreasing in x. This proves

the result.

Proof of Proposition 6. For a relatively small αI value, the investor chooses to invest x1 if

he decides to invest with bankruptcy risk. He chooses to invest K if he chooses to invest without

bankruptcy risk. Suppose αI = −1, which is less than αl and α5. Then Π∗R = maxx∈[0,x1] π∗(x) +

E[(1+αI )(K−x)] = π∗(x1) and Π∗NR = maxx∈[x2,K] π∗(x)+E[(1+αI )(K−x)] = π∗(K). Therefore,

Π∗NR ≥ Π∗R because π∗(K) has a larger feasible region by (1). On the other hand, for a large αI

value, investing with bankruptcy risk option leads to no investment in the newsvendor (i.e., x∗1 = 0).

However, the investor invests x2 if he chooses to invest without bankruptcy risk. Therefore,

Π∗NR −Π∗R = (1 + αI )(K − x2) + (1 + α)x2 − ((1 + α)c− s)qBWO(x2)

+ (p− s)

[∫ qBWO(x2)

0F (ξ)dξ − τ

∫ qBWO(x2)

dBWO(x2)F (ξ)dξ

]

−(

(1 + αI)K + (p− s)∫ θ

(1+α)c−sp−s

θF (ξ)dξ

)

= −(1 + αI )x2 + C2

where C2 is a constant. Hence, Π∗R > Π∗NR for a sufficiently large αI value. Furthermore,

dΠ∗sdαI

=∂Πs(αI , x

∗)∂αI

+∂Πs(αI , x)

∂x

∣∣x=x∗

∂x∗

∂αI

=∂Πs(αI , x

∗)∂αI

39

for s ∈ {R,NR}. This is due to ∂Πs(αI,x)

∂x

∣∣x=x∗ = 0 because x∗ solves the first order condition of

the investor’s objective function. Therefore,

dΠ∗RdαI

=

K − x1 if αI < αl,

K − xR if αI ∈ [αl, αh],

K if αI > αh,

and

dΠ∗NR

dαI

=

0 if αI ≤ α5,

K − x∗5 if αI ∈ (α5, α4],

K − x∗4 if αI ∈ (α4, α3],

K − x∗3 if αI ∈ (α3, α2],

K − x2 if αI > α2.

In addition, one can show that x∗5 > x∗4 > x∗3 > x2, x1 > xR, and x2 > x1. Also note that K−x∗5 > 0

as we assumed that the investor is large compared to the newsvendor. Therefore, dΠ∗Rdα

I>

dΠ∗NRdα

I≥ 0.

This proves the existence and uniqueness of α.

Proof of Proposition 7. Setting τ = 0 in Proposition 4 gives the first part of the proof. More

formally, x1 = qNBU −θ

β , which implies that αl = (1+α) (1+α′)c−s(1+α)c−s −1. Similarly, αh = (1+α)(p−s)

(1+α)c−s F (θ)−1. Setting τ = 0 and solving (10) gives xR = qR−θ

β and qR = F−1((1 + αI )

(1+α)c−s(1+α)(p−s)

). Setting

τ = 0 in Proposition 5 gives α2 = α3 = α and α4 = α5 = α′. As a result, we can eliminate two

out of five possible cases in which there is no bankruptcy risk. Setting (12) equal to zero gives

xNR = cqNR and qNR = F−1(

(1+αI)c−s

p−s

)when there is no taxation.

Next we show that investing with risk (i.e., x∗ ∈ [0, x1]) is optimal when αI > α. To see

this, note that the first derivative of Π(x) at x = x2 is α − αI when τ = 0. Furthermore, the

first derivative in x ∈ [x1, x2] is increasing because the investor’s profit function is convex in this

interval. Hence, if α− αI is negative at x = x2 (i.e., if αI > α) then it is negative for x ∈ [x1, x2],

which implies that Π(x1) > Π(x2). But Π(x2) is the optimal solution under no borrowing when

αI > α. Therefore, borrowing with risk is optimal when αI > α.

Investing without risk (i.e., x ∈ [x2,K]) is optimal when αI > α′. Observe that at αI = α′,

40

x∗NR = K, which implies that

Π∗NR(α′) = (1 + α′)K + (p− s)∫ qNB

U

0F (ξ)dξ − ((1 + α′)c− s)qNB

U .

On the other hand, x∗R = x1 at αI = αl, which implies that

Π∗R(αl) = (1 + αl)K + (p− s)∫ qNB

U

(1+α)c−sp−s

θF (ξ)dξ − ((1 + α′)c− s)(qNB

U − θ).

Two facts are useful to bound Π∗R(αl). First of all, α′−αl = (α−α′)c(1+α)c−s > 0. Secondly, the maximum

rate of increase of the profit functions is K. Using these results, we can write

Π∗R(α′) ≤ Π∗R(αl) + (α′ − αl)K

= (1 + α′)K + (p− s)∫ qNB

U

(1+α)c−sp−s

θF (ξ)dξ − ((1 + α′)c− s)(qNB

U − θ).

Then

Π∗R(α′)−Π∗NR(α′) ≤ ((1 + α′)c− s)θ − (p− s)∫ (1+α)c−s

p−sθ

0F (ξ)dξ.

We can write the same inequality as

Π∗R(α′)−Π∗NR(α′) ≤ (p− s)(1 + α′)c− s

(1 + α)c− sF−1

((1 + α′)c− s

(1 + α)c− s

)− (p− s)

∫ F−1(

(1+α′)c−s(1+α)c−s

)

0F (ξ)dξ

= (p− s)∫ F−1

((1+α′)c−s(1+α)c−s

)

0

((1 + α′)c− s

(1 + α)c− s− F (ξ)

)dξ,

which is negative because (1+α′)c−s(1+α)c−s ≤ F (ξ) for ξ ∈

[0, F−1

((1+α′)c−s(1+α)c−s

)]. This result is sufficient to

show that Π∗NR ≥ Π∗R for αI < α′ because both functions are increasing, but Π∗R is increasing at a

faster rate.

Proof of Proposition 8. qBWO ≥ θ implies that

(1 + α)c− s

p− s≤ F (θ)

Multiplying both sides by 1 + α and rearranging terms gives αh ≥ α. Therefore, [α′, α] ⊂ [αl, αh].

Setting the two profit functions equal to each other gives (15). Borrowing with risk scenario gives

the first two parts of the of the optimal solution. Similarly, the last two parts are obtained from

the optimal solution of Π∗NR.

41