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The Quarterly Review of Economics and Finance 54 (2014) 111– 122

Contents lists available at ScienceDirect

The Quarterly Review of Economics and Finance

jo ur nal home page: www.elsev ier .com/ locate /qre f

roduct–market flexibility and capital structure

udipto Sarkar ∗

SB 302, DeGroote School of Business, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4M4, Canada

r t i c l e i n f o

rticle history:eceived 14 March 2012eceived in revised form 6 June 2013ccepted 30 September 2013vailable online 10 October 2013

a b s t r a c t

Many companies have the ability to adjust their product’s price and/or quantity in response to changesin the marketplace. We show that this product–market flexibility or market power, hitherto ignored inthe contingent-claim modeling literature, can potentially have a significant effect on the corporate capitalstructure decision. When the firm is operating at full capacity, product–market flexibility is not important,

eywords:roduct–market flexibilityrice sensitivityapital structureontingent-claim model

hence market power has a negligible effect on optimal capital structure. However, when operating belowcapacity, product–market flexibility becomes important and market power has, in general, a positive effecton optimal debt level and optimal leverage ratio. This is consistent with available empirical evidence.Numerical results indicate that the effect of product–market flexibility on optimal debt level and optimalleverage ratio can potentially be large enough to be economically significant, hence it should not beignored as a determinant of capital structure.

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© 2013 The Board of Tru

. Introduction

Contingent-claim models are an important component of theheoretical capital structure literature in Corporate Finance. Start-ng with Brennan and Schwartz (1978), a large number of papersave examined the corporate leverage decision using contingent-laim models, e.g., Fischer, Heinkel, and Zechner (1989), Goldstein,u, and Lel (2001), Hackbarth and Mauer (2012), Leland (1994,998), Leland and Toft (1996), Mauer and Ott (2000), Mauer andarkar (2005), Sarkar and Zapatero (2003), Sundaresan and Wang2007), Titman and Tsyplakov (2007), and many more. In all of these

odels, the source of uncertainty (state variable) is the exoge-ously specified earnings/cash flows or asset value of the firm.ut this ignores an important managerial flexibility – the abilityo adjust production as market (demand) conditions fluctuate. Thenly exception, to our knowledge, is Mauer and Triantis (1994),here the firm can shut down and restart operations when mar-

et demand changes sufficiently in either direction. However, sincehey do not allow the firm to adjust price or output level, their

odel’s view of product–market flexibility is somewhat limited.This paper contributes to the literature by incorporating

roduct–market flexibility or market power in the capital struc-ure decision. We extend the standard contingent-claim model

y including the firm’s product–market power, i.e., price- anduantity-setting flexibility in the product market. This is an impor-ant issue because the available empirical evidence indicates that

∗ Tel.: +1 905 525 9140x23959; fax: +1 905 521 8995.E-mail address: sarkars@mcmaster.ca

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062-9769/$ – see front matter © 2013 The Board of Trustees of the University of Illinoisttp://dx.doi.org/10.1016/j.qref.2013.09.002

of the University of Illinois. Published by Elsevier B.V. All rights reserved.

roduct–market power does, in general, have a significant effectn corporate capital structure decisions (Guney, Li, & Fairchild,011; MacKay & Phillips, 2005; Pandey, 2004; Rathinasamy,rishnaswamy, & Mantripragada, 2000; Smith, Chen, & Anderson,008). While some papers have examined theoretically the inter-ction of product market and capital structure, they have beenimited to strategic/game-theoretic models (Brander & Lewis,986; Lyandres, 2006; Maksimovic, 1988).

Firms often have substantial flexibility in an uncertain productarket, because of their ability to adjust output price and/or quan-

ity as market conditions change. When demand for its productalls (rises), a company will reduce (increase) price and/or quan-ity, subject to its capacity limit.1 However, this product–marketexibility is ignored in the contingent-claim capital-structure mod-ls referred to above, which assume that earnings or asset values an exogenous random process that is unaffected by the firm’sroduct–market decisions such as quantity and pricing.2 But sincehe firm generally has the ability to adjust output price and/oruantity, earnings should be endogenously determined. This endo-

1 Thus, product–market flexibility will be particularly important when the firm’sapacity constraint is not binding, i.e., when it is operating below full capacity, sincet is then free to change both output quantity and price.

2 Even when the product market is considered (e.g., Titman & Tsyplakov, 2007),t is assumed that the firm’s product market is described by an exogenous pricerocess, and the firm always operates at full capacity, in which case it has no product-arket flexibility since it cannot adjust either output price or output quantity.

. Published by Elsevier B.V. All rights reserved.

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We start with a downward–sloping stochastic demand curvehat the firm faces in the output market. Thus, the firm is operatingn a differentiated-product market (monopolistic competition or

onopoly), with a downward–sloping demand curve. The firm’sroduct–market flexibility will then be determined by demandlasticity (or price sensitivity), as discussed in Section 2. We there-ore focus our attention on the effect of this parameter on optimalapital structure.

We show that, even apart from strategic or game-theoretic con-iderations, product–market characteristics can have an importantffect on corporate capital structure. When the firm is not oper-ting at full capacity, the optimal debt level and optimal leverageatio are both increasing functions of price sensitivity; the effect isnsignificant when the firm is operating at full capacity, which is noturprising since, at full capacity, the firm does not have operatingexibility. Also, the optimal debt level is a U-shaped function, andptimal leverage ratio a decreasing function, of demand volatility.

The rest of the paper follows the outline below. Section 2escribes the model and derives the firm’s earning stream and assetalue under product–market flexibility. Section 3 examines theapital structure decision, and identifies the optimal debt level andptimal leverage ratio. Section 4 presents the results, and Section

summarizes and concludes.

. The model

We use a contingent-claim model to study the corporate capi-al structure decision, as in Leland (1994), Goldstein et al. (2001),arkar and Zapatero (2003). However, since we are interested inhe effect of the company’s product’s demand characteristics, ourtate variable is the demand shock rather than earnings or assetalue (as in the existing contingent-claim models).

A firm produces q units of the output per unit time, which itells at a price of $p per unit. The firm can set the price and quantity,ubject to a couple of constraints. The first constraint is the capacityimit – we assume that the firm has a physical capacity limit of

per unit time, thus it cannot produce output at a higher rate,.e., qt ≤ Q, ∀t. The second constraint is the demand relationship –

e assume that the firm faces a downward–sloping demand curveith constant elasticity (Dobbs, 2004):

t = yt

(qt/Q )� (1)

here the state variable yt represents a stochastic shock to demand.he parameter � indicates the price sensitivity to output quantityhanges (0 ≤ � ≤ 1). The demand curve (1) implies a constant pricelasticity of demand of (1/�). A higher value of � denotes greater

arket power for the firm, and � = 0 represents no market power

perfect competition) with an exogenous price process.The company incurs a constant variable cost of $v per unit pro-

uced. Its profits are taxed at a (symmetric) tax rate of � (thats, profits and losses are treated the same).3 Also, the company is

3 Thus, if taxable income is negative, there will be a positive cash flow from cor-orate tax (i.e., the company will receive a tax rebate from the government). This

s not an exact depiction of the actual tax code, which is asymmetric; however, its a standard assumption in the contingent-claim literature for reasons of tractabil-ty (Leland, 1994; Mauer & Ott, 2000; Sundaresan & Wang, 2007). Moreover, it isot a damaging assumption because, in real life, companies can carry forward tax

osses and thereby benefit from their losses. In any case, we can take care of anyemaining asymmetry in the tax code by reducing effective tax rate � appropriatelyn our computations (as discussed in Graham & Smith, 1999).

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isk-neutral and discounts cash flows at the constant discount ratef r.4

The state variable y is the demand shock, which can be vieweds the strength of market demand, i.e., a higher y implies strongeremand. It introduces uncertainty in the model, and evolvesccording to a lognormal process (Aguerrevere, 2003; Dobbs,004):

y/y = �dt + �dz (2)

here � and � are expected growth rate and volatility, respectively,f y; and z is a standard Weiner process.

When � = 0 (perfect competition) the firm faces an exogenousrice process pt = yt, and the output policy will be as follows:roduce at the maximum rate (qt = Q) when yt > v, and nothingqt = 0) when yt ≤ v. Thus, when � = 0, the firm has no discretionver either price or quantity, i.e., no product–market flexibil-ty. Product–market flexibility is relevant only for � > 0. As � isncreased, the firm has more market power, hence greater abil-ty to benefit from the ability to adjust price and quantity. Thus,roduct–market flexibility is more valuable when � is higher.

.1. The earnings stream with product–market flexibility

We assume that the firm acts rationally, hence it adjusts pricend quantity continuously in response to changes in demand (y)o as to maximize the profit stream (Aguerrevere, 2003; Sarkar,009).5 If it produces output at a rate of q units per unit time, andells the output at a price of $p per unit, subject to Eq. (1), then thefter-tax instantaneous profit is given by:

(q) = (1 − �)(p − v)q = (1 − �)(yQ �q1−� − vq)

etting d�/dq = 0, we get the optimal quantity as a function of thetate variable: qt = Q [yt(1 − �)/v]1/�, and the optimal output price:t = v/(1 − �). The resulting profit stream is given by:

(y) = (1 − �)�y1/� (3)

here

= �Q [v/(1 − �)]1−1/� (4)

ifferentiating �(y) in Eq. (3) with respect to �, it is easily shownhat d�(y)/d� > 0. Therefore, when the capacity constraint is notinding (hence the firm has operating flexibility), the earnings levelf the firm is an increasing function of the price sensitivity �. Thishould not be surprising, since increased market power will resultn higher profits.

It is clear from the above that as the strength of demand (y) rises,he price remains unchanged but the output quantity increasesccordingly. However, there is a capacity constraint q ≤ Q; hencehe output quantity can increase only up to the limit Q. Suppose theapacity limit is reached when y rises to y; then Q (y(1 − �)/v)1/� =

. Thus, the capacity-constraint boundary (i.e., the level of y athich full capacity is reached) is y = v/(1 − �). For higher levels

f y, the output cannot rise and is therefore unchanged at Q, but

4 Risk neutrality is a standard assumption in the literature (Aguerrevere, 2003;obbs, 2004). Risk aversion can be incorporated as suggested by Aguerrevere (2003,

ootnote 5), but it will add complexity without qualitatively changing the results.5 Since the firm adjusts quantity continuously, the implicit assumption is that

djustment costs are zero. This is a common assumption in the Economics literatureHagspiel, Huisman, & Kort, 2009; He & Pindyck, 1992; Sarkar, 2009), justified by theact that reduction of output quantity is achieved simply by not using some of thenstalled capacity, and increase of output by using some more of the installed capac-ty. There are generally no significant costs attached to such increases/decreases inutput. Adjustment costs would, however, be important if the production was to beemporarily shut down and restarted (Dixit, 1989; Mauer & Triantis, 1994).

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he price rises according to the demand curve (Eq. (1)). Thus, for > y, the output level and output price are given by: q = Q, and

= y. The corresponding profit flow will be:

(y) = (1 − �)(y − v)Q (5)

rom Eq. (5), earnings level �(y) is independent of �. Therefore,hen the capacity constraint is binding (hence no operating flex-

bility), price sensitivity has no effect on earnings. When capacityonstraint is binding, increased market power does not translatento higher profit because the firm has no degrees of freedom (withapacity fixed at Q, the price just comes from the demand curvend the firm has no control over it), hence no product–marketexibility.

To summarize, there are two operating regions:

Region 1 (y ≤ y): p = v/(1 − �), q = Q [y(1 − �)/v]1/�, and�(y) = (1 − �)�y1/�.Region 2 (y > y): p = y, q = Q and �(y) = (1 − �)(yQ − vQ).

In Region 1 the firm is not forced to produce at a fixed out-ut level, but has the ability to change output depending on thetrength of demand. This represents product–market flexibility inur model. There is no product–market flexibility in Region 2.

.2. The dynamics of the earnings stream

.2.1. Region 1Differentiating �(y) from Eq. (3) using Ito’s lemma, we get

� = [(1 − �)y1/��/�][(� + 0.5�2(1/� − 1))dt + �dz]

hich simplifies to

d�

�= � + 0.5�2(1/� − 1)

�dt + �

�dz (6)

hus the earnings process is also lognormal, but both the drift andolatility are functions of the demand sensitivity �. Importantlybecause earnings volatility affects optimal capital structure), weote that earnings volatility is a decreasing function of �. A higher

denotes more market power; when the firm has greater marketower (and recall that capacity constraint is not binding in thisegion), it is better able to adjust price and/or quantity and thus itan moderate the volatility effects of market demand. This reduceshe volatility of the earnings stream when � is higher.

.2.2. Region 2Using Ito’s lemma to differentiate �(y) in Eq. (5), we get after

ome simplification:

d�

�= �[1 + (1 − �)vQ/�]dt + �[1 + (1 − �)vQ/�]dz (7)

n this region, the earnings process does not exactly inherit theemand variable’s lognormal characteristics. Note that the earn-

ngs volatility here is independent of the price sensitivity �.

.3. Unlevered firm valuation

In an infinite-horizon setting, the value of the firm’s assets (orhe unlevered firm value) will be a function of the state variable yand also the operating region). Thus, the value can be written as:

(y) ={

V1(y) in operating region1

V2(y) in operating region2(8)

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ics and Finance 54 (2014) 111– 122 113

s shown in Appendix A, V(y) must satisfy the ordinary differentialquation (ODE):

.5�2y2V ′′(y) + �yV ′(y) − rV(y) + �(y) = 0 (9)

here �(y) is the profit flow derived above, given by (1 − �)�y1/�

n region 1 and (1 − �)(yQ − vQ) in region 2. Since �(y) depends onhe operating region, the project value V(y) will also depend on theperating region, hence it is derived separately for each region.

Region 1 (y ≤ y): Solving ODE (9) with �(y) = (1 − �)�y1/�, we get:

1(y) = Zy1/� + Ay�1 (10)

here

= (1 − �)�r − �/� − 0.5�2(1/� − 1)/�

(11)

is a constant to be determined by the boundary conditions, and1 is the positive root of the quadratic

.5�2�(� − 1) + �� − r = 0 (12)

n Eq. (10), the first term is the value of operating forever without capacity constraint (i.e., permanently in operating region 1), andhe second term captures the possibility of the capacity constraintinding in the future (therefore, A < 0).

Region 2 (y > y): Solving ODE (9) with �(y) = (1 − �)(yQ − vQ),e get:

2(y) = (1 − �)[

Qy

(r − �)− vQ

r

]+ By�2 (13)

here B is a constant to be determined by the boundary conditions,nd �2 is the negative root of the quadratic Eq. (12). The two roots1 (>0) and �2 (<0) are given by:

1 = 0.5 − �/�2 +√

(0.5 − �/�2)2 + 2r/�2 (14)

2 = 0.5 − �/�2 −√

(0.5 − �/�2)2 + 2r/�2 (15)

n Eq. (13), the first term is the value of operating permanentlyn region 2, and the second term captures the possibility of theapacity constraint becoming non-binding in the future.

To complete the valuation of the firm’s assets, the two constants and B must be identified from the boundary conditions. There isne boundary here: the capacity-constraint boundary y = y. Twoonditions must be satisfied at this boundary (Dixit & Pindyck,994; Leland, 1994): (a) the value-matching or continuity con-ition: V1(y) = V2(y) and (b) the smooth-pasting or optimalityondition: V ′

1(y) = V ′2(y). The two boundary conditions give:

= (1 − �)[yQ (1 − 1/�2)/(r − �) − vQ/r] − Z(y)1/�(1 − 1/��2)(1 − �1/�2)(y)�1

(16)

= (y)1/�Z(1 − 1/��1) − (1 − �)[yQ (1 − 1/�1)/(r − �) − vQ/r](1 − �2/�1)(y)�2

(17)

e have now completed the valuation of the firm’s assets.Unlike the case of earnings (Sections 2.1 and 2.2), the effect of

on the firm’s asset value V(y) and asset value volatility cannot beerived analytically, because of the complexity of the expressions

nvolved. We therefore demonstrate the effect with numericalesults in Section 4.1. Intuitively, we expect the following. In Region, since earnings are increasing in �, value should also be an increas-

ng function of �. In Region 2, earnings are independent of �, hence

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here should be no direct effect of � on value. However, there is positive probability of moving from Region 2 to Region 1 (i.e., aositive probability that y will fall below y in the future). Since assetalue is also affected by future earnings expectations, it should alsoe an increasing function of � in Region 2. However, this is a second-rder effect, hence the effect of � on value should be much smallern Region 2 than on Region 1.

In a similar manner, the volatility of asset value should be aecreasing function of � in Region 1, since earnings volatility is

decreasing function of �. In Region 2, earnings volatility is inde-endent of �; however, since value depends on future earnings andhere is a positive probability of moving to Region 1, the volatilityf asset value in Region 2 should also be a decreasing function of �,lthough the effect will be much smaller than in Region 1.

. The leverage decision

Now suppose the firm has debt in its capital structure. Theevel of debt financing is given by the coupon obligation of $c pernit time to bondholders. We assume it is long-term debt, i.e., theoupon payments will be made in perpetuity or until bankruptcy.he bankruptcy process is modeled as in the corporate finance lit-rature (Leland, 1994; Mauer & Ott, 2000). The company will fileor bankruptcy when the state variable y falls to a sufficiently lowevel (say, y = yb). At bankruptcy, the company will incur a fractionalankruptcy cost of (1 ≥ ≥ 0), its assets will be acquired by theebt holders, and the equity holders will be left with nothing.

.1. Debt valuation

The firm’s debt is valued in a similar manner to project valua-ion, by solving the ODE (9) with �(y) replaced by c (since the cashow stream to debt holders consists of the coupon payment) andhe appropriate boundary conditions. Since the cash flow to debtolders will be the same in both operating regions, the debt val-ation formula, D(y), will be independent of the operating region.olving ODE (9) with D(y) instead of V(y), and using �(y) = c, we get:

(y) = c/r + D1y�2 (18)

here D1 is a constant to be determined from the boundary condi-ion. The term c/r represents the value of the debt if there was noankruptcy risk, and the other term represents the bankruptcy riskhence D1 < 0).

As mentioned above, at the bankruptcy boundary (y = yb) theayoff to bondholders is (1 − ˛) times the unlevered firm value.he formula for unlevered firm value at bankruptcy (i.e., V1(yb) or2(yb)) will depend on whether the firm is operating at full or par-ial capacity when it declares bankruptcy. It does not make sense toeclare bankruptcy when demand is so high that the firm is operat-

ng at full capacity (unless the debt level is extremely high, but thatould not be optimal, and this paper is concerned with optimalebt levels), hence for practical purposes the bankruptcy triggerill not be in Region 2. We therefore assume the bankruptcy trig-

er will be in Region 1, i.e., y≥yb.6 The value of the firm’s assets atankruptcy will therefore be V1(yb), giving the boundary condition:(yb) = (1 − ˛)V1(yb), from which we get the constant

1/� �1 −�2

1 = [(1 − ˛)Z(yb) + (1 − ˛)A(yb) − c/r](yb) (19)

he bankruptcy trigger yb is determined below.

6 The other case (i.e., extremely large debt level and bankruptcy in region 2) cane handled in the same manner, but is of no practical importance, since the optimalebt level (Section 4) is always such that the bankruptcy trigger is in Region 1 (y≥yb).

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ics and Finance 54 (2014) 111– 122

.2. Equity valuation

If the equity value is given by E(y), we can write:

(y) ={

E1(y) in operating region1

E2(y) in operating region2(20)

ince equity value will be state-contingent. It can be shown, as inhe earlier cases, that

1(y) = Zy1/� − (1 − �)c/r + B1y�1 + B2y�2 (21)

2(y) = (1 − �)[Qy/(r − �) − (vQ + c)/r] + B3y�2 (22)

here B1, B2 and B3 are constants to be determined from the bound-ry conditions.

In Eq. (21), the first two terms on the right hand side representquity value if the firm were to operate forever without bankruptcyr capacity constraint, and the other two terms represent the effectsf the bankruptcy and capacity constraint boundaries, y = yb and

= y respectively. In Eq. (22), the first term on the right hand sideepresents the equity value if the firm was to operate forever at fullapacity, and the other term represents the effect of the boundary

= y.Thus, four unknowns (B1, B2, B3 and yb) have to be deter-

ined from the boundary conditions. There are two boundaries: (i)ankruptcy boundary (y = yb) and (ii) capacity-constraint boundaryy = y). Once again, the value-matching and smooth-pasting condi-ions must be satisfied at each boundary. Keeping in mind that theayoff to shareholders at bankruptcy is zero, we get the followingoundary conditions:

At y = yb: Value-matching: E1(yb) = 0, Smooth-pasting: E′1(yb) = 0.

At y = y: Value-matching: E1(y) = E2(y), Smooth-pasting: E′1(y) =

E′2(y).

From the boundary conditions, we get the three constants:

1 = (1 − �)[Qy(1 − 1/�2)/(r − �) − vQ/r] − Z(1 − 1/��2)(y)1/�](1 − �1/�2)(y)�1

(23)

2 = −(yb)1/�−�2 Z/(��2) − B1(�1/�2)(yb)�1−�2 (24)

3 = B2 + B1(y)�1−�2 + Z(y)1/�−�2−(1 − �)[Qy/(r − �)−vQ/r](y)−�2

(25)

ote that B2 and B3 are functions of the optimal bankruptcy triggerb, which is determined as the solution of an implicit equation:

1 − 1/��2)Z(yb)1/� − (1 − �)c/r + B1(yb)�1 (1 − �1/�2) = 0 (26)

q. (26) has no analytical solution and has to be solved numericallyor yb. We have now completed the valuation of the firm’s debt andquity, D(y) and E(y) respectively. The total value of the firm is the

um of equity and debt: F(y) = D(y) + E(y), and is given by:

(y) ={

Zy1/� + �c/r + B1(y)�1 + (B2 + D1)(y)�2 in Region1

(1 − �)[yQ/(r − �) − vQ/r] + �c/r + (B3 + D1)(y)�2 in Region2(27)

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.3. Optimal capital structure

When issuing debt, the firm must decide on the debt level opti-ally, i.e., so as to maximize the total value of the firm (equity plus

ebt); see Leland (1994). Thus, the optimal debt level is given by:

∗ = Argmaxc

F(y, c)

here we write the total firm value as F(y,c) to make it clear thathe coupon (debt) level is a choice variable. Setting dF(y,c)/dc = 0,ives us the first order condition for optimal debt level c*:

�

r+ (y)�2

[dD1

dc+ dB2

dc

]= 0 (28)

here

dB2

dc= [Z(�2 − 1/�)(yb)1/� − �B1�1(�1 − �2)(yb)�1 ] dyb

dc

��2(yb)�2+1,

dD1

dc= [c�2 + (1 − ˛)rA(�1 − �2)(yb)�1 + r(1 − ˛)Z(1/� − �2)(yb)1/�] dyb

dc− yb

r(yb)�2+1,

dyb

dc= (1 − �)yb

rZ(1 − 1/��2)(1/� − �1)(yb)1/� + (1 − �)c�1

.

q. (28) also has to be solved numerically, as it has no analyticalolution.7

To jointly identify the optimal debt level (c*) and bankruptcyrigger (yb), we solve Eqs. (26) and (28) simultaneously. The result-ng optimal leverage ratio is then given by D(y,c*)/F(y,c*).

. Results

The model is too complex to allow for analytical solutions, hencehe results are illustrated with numerical solutions. For numer-cal results, the values of the various input parameters must bepecified. We start with a “base case” set of parameter valueshat are reasonable and based on well-known papers in the exist-ng literature.8 However, all computations are repeated numerousimes with various parameter values, to ensure that the results areobust and not dependent on the parameter values chosen.

.1. Base-case parameter values

For the “base case,” we set the plant capacity at Q = 1 and theariable operating cost per unit at v = 1; these are without anyoss of generality, since they are just scaling variables. For thether variables, we base the values on Leland (1994), which is aeminal paper in the contingent-claim capital-structure literature.hus, we set the tax rate at � = 15% (see footnote 27 of Leland,994), bankruptcy cost = 50%, interest rate r = 6%, and volatility

= 20%. The demand growth rate is set at � = 1%. We use thesearameter values to solve for the optimal capital structure withifferent levels of price sensitivity (�), the parameter associatedith product–market flexibility.9

7 Note that the first order condition (28) is the same for both operating regionsy ≤ y and y > y) since dB1/dc = 0 and dB2/dc = dB3/dc.

8 This is true for most variables, but the price sensitivity (�) has not been examinedn the existing literature, hence there is no precedent for us to rely on. For thisariable, we display the results for the reasonable range of values of the variable,.g., � = 0.05–0.95.9 In Appendix B, we present numerical results with parameter values from some

ther well-known papers; the results are found to be very similar.

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tsb

ics and Finance 54 (2014) 111– 122 115

.2. Effect of price sensitivity � on asset value V(y)

The effect of price sensitivity � on earnings �(y) has been dis-ussed in Section 2.1: in Region 1 (capacity constraint not binding)arnings level is an increasing function of �, in Region 2 (capacityonstraint binding) earnings level is independent of �. The effect of

on asset value or unlevered firm value, V(y), is a little more com-licated because the asset value in any operating region is affectedvia the boundary conditions) by the possibility of moving to thether operating region.

Fig. 1 shows V(y) as a function of �. The results are computedith the base-case parameter values specified in the previous sub-

ection (and y = 2), as well as for a range of parameter values. Since = 1, the capacity-constraint boundary is y = 1/(1 − �). The firms at the capacity-constraint boundary when � = 0.5 (since y = 2),

hich is shown in the figures by the vertical broken line. To the leftf this boundary (i.e., for � < 0.5) we have y > y, hence the firm isperating in Region 2. To the right of the vertical line (� > 0.5), weave y < y, hence the firm is operating in Region 1.

In all cases, asset value is an increasing function of �. In Region 1where the firm has operating flexibility) � has a significant effectn value, while in Region 2 (where the firm has no operating flex-bility) the effect is generally insignificant. Both relationships areonsistent with economic intuition, as discussed in Section 2.3.

Although the effect of � on firm’s asset value could not be shownnalytically, numerical results indicate the same relationship for allarameter values examined, hence we are confident of the gener-lity of the result.

esult 1. In Region 1 (capacity constraint not binding, hence firm hasperating flexibility), earnings level and asset value are both increasingunctions of price sensitivity; in Region 2 (capacity constraint binding,ence no operating flexibility) earnings level is independent of, andsset value is slightly increasing in, price sensitivity.

.3. Optimal capital structure

To illustrate a specific case, let us identify the optimal capitaltructure with the base-case parameter values, price sensitivity

= 0.5 (implying a price elasticity of 2), and y = 2. The capacity-onstraint boundary is y = v/(1 − �) = 2; thus y = y, i.e., the firm ist the boundary of the two operating regions. With these param-ter values, the solution to Eqs. (26) and (28) gives the followingutput:

ptimal debt level c∗ = 0.6599,

optimal bankruptcy trigger

yb = 0.7243, and optimal leverage ratio = 40.41%.

.4. Effect of price sensitivity �

Fig. 2 illustrates the effect of � on the optimal coupon level (c*)nd the optimal leverage ratio with the base case parameter values,or three levels of demand volatility, � = 10%, 20% and 30%. The ver-ical broken line (� = 0.5) shows the boundary between operatingegions 1 and 2, i.e., y = y. To its left (� < 0.5), the firm is operating inegion 2 (no operating flexibility), since y > y; to its right (� > 0.5),he firm is operating in Region 1 (with operating flexibility), since

< y.

There are three points worth noting. One, the sensitivity of

he leverage decision to � is greater when demand volatility � ismaller. This is not surprising, since � should have a (relatively)igger impact on asset volatility when � is smaller. Two, the effect

116 S. Sarkar / The Quarterly Review of Economics and Finance 54 (2014) 111– 122

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ig. 1. Shows firm’s asset value (or unlevered firm value) V(y) as a function of priceart (b) for different growth rates (�), part (c) for different interest rates (r), and pa

= 6%, � = 15%, = 50%, v = 1 and Q = 1. The broken vertical line shows the boundary

f � depends on whether the firm is operating at full or partialapacity when making the capital structure decision. Three, bothptimal debt level c* and optimal leverage ratio are increasing func-ions of � in operating region 1, but insensitive to � in operatingegion 2. Thus, the traditional contingent-clam models withoutroduct–market flexibility can approximate reasonably well com-anies operating at full capacity, but not companies operatingelow capacity.

Operating Region 2 (� < 0.5): For low volatility (� = 10%), both c*

nd optimal leverage ratio are initially increasing, then decreasing,nd finally increasing, in �; for moderate volatility (� = 20%), both* and optimal leverage ratio are virtually independent of �; andor higher volatility (� = 30%), both c* and optimal leverage ratiore slightly increasing in �. In all cases, the effect of � on debt level

nd capital structure is negligible.

Operating Region 1 (� > 0.5): Both c* and optimal leverage ratiore in all cases increasing functions of �, and the effect is notnsignificant.

vfrw

ivity �. Part (a) shows the relationship for different levels of volatility or sigma (�),or different tax rates (�). The base-case parameter values are used: � = 1%, � = 20%,en operating regions 1 and 2.

These relationships can be explained as follows. A higher priceensitivity � results in a lower asset volatility (the “volatility”ffect), from Section 2.3. The lower asset volatility has two effects:i) from standard option theory, lower volatility brings closer theankruptcy boundary (i.e., raises the bankruptcy trigger yb), which

ncreases bankruptcy risk and makes debt less desirable; and (ii)ower volatility also means that extreme realizations (both highnd low) are less likely, which reduces the likelihood of reaching theankruptcy trigger; this reduces bankruptcy risk and makes debtore desirable. The net effect of � on the optimal leverage ratio will

epend on which of the above two effects dominates. In general,he second effect seems to dominate, hence optimal leverage ratios generally an increasing function of �.

Recall from Section 2.3 that � has a negligible effect on asset

olatility in Region 2 but a significant effect in Region 1. There-ore, � should also have a significant effect on optimal leverageatio in Region 1, but a minor effect in Region 2. This is indeedhat we find, as illustrated in Fig. 2 – optimal leverage ratio is an

S. Sarkar / The Quarterly Review of Econom

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rice sensitivity �, for three levels of volatility, � = 10%, 20% and 30%. The base-casearameter values are used: � = 1%, r = 6%, � = 15%, = 50%, v = 1 and Q = 1. The brokenertical line shows the boundary between operating regions 1 and 2.

ncreasing function of � in Region 1, but not very sensitive to � inegion 2. However, the exception is the case of low demand volatil-

ty (� = 10%), in which case there is a region (intermediate �) overhich the optimal leverage ratio is actually a decreasing function

f �. Thus, for intermediate values of �, the first effect seems toominate. Therefore, for small demand volatility, optimal leverageatio is initially an increasing function, then a decreasing function,nd finally an increasing function again, of �.

For the optimal coupon level c*, there is an additional effect:

higher � results in higher asset value (the “value” effect), asiscussed in Sections 2.3 and 4.2.10 As illustrated in Fig. 1, thealue effect is negligible in Region 2 but significant in Region 1.

10 The “value” effect is not relevant for optimal leverage ratio because, when assetalue increases, more debt is used but equity value also rises; thus, in spite of theigher debt level, the optimal leverage ratio is independent of asset value changes.

aaoi

Ii

ics and Finance 54 (2014) 111– 122 117

herefore, � has an additional positive effect on c* that is negligiblen Region 2 but significant in Region 1. This is indeed what we find,s Fig. 2 illustrates – c* behaves very similar to optimal leverageatio in Region 2 but rises much faster in Region 1 because ofhe “value” effect. We state below the main result of our paper,ummarizing the effect of market power/price sensitivity �:

esult 2. The effect of price sensitivity on capital structure willepend on whether the capital structure decision is made when oper-ting at full or partial capacity. When the firm is operating at fullapacity, price sensitivity does not have a substantial effect on therm’s capital structure; when operating below full capacity, optimalebt level c* and optimal leverage ratio are both increasing functionsf price sensitivity.

.4.1. Magnitude of the effectFor modeling purposes, a very important question is whether

he effect of � on optimal leverage ratio is large enough to be eco-omically meaningful. If not, there is not much to be gained byaking into account product market flexibility or price sensitivity.o answer this question, we can look at how the optimal leverageatio changes when � is increased. With the base-case parameteralues, when � is increased from 0.05 to 0.95, the optimal leverageatio rises from 39.94% to 45.64%; that is, an increase in leverageatio of 5.7%. This is certainly not an insignificant increase. Theffect of � on capital structure is seen more clearly if we split it intohe two operating regions. Within Region 2 (when � is increasedrom 0.05 to 0.5), optimal leverage ratio rises by only 0.47% (from9.94% to 40.41%), but within Region 1 (when � is increased from.5 to 0.95), optimal leverage ratio rises by 5.23% (from 40.41% to5.64%). Clearly, the effect of � is concentrated in operating region. For a firm operating in region 2, the effect of � is negligible.

Table 1 summarizes the results with different parameter val-es. In this table, Lev1 is the spread in optimal leverage between

= 0.05 and 0.95, i.e., (optimal leverage ratio with � = 0.95) minusoptimal leverage ratio with � = 0.05). This measure shows how

uch the optimal leverage ratio can potentially change becausef differences in price sensitivity. Lev2 and Lev3 show how thiss split between operating region 2 and operating region 1 respec-ively, i.e., Lev2 is given by (optimal leverage ratio with � = 0.5)

inus (optimal leverage ratio with � = 0.05), and Lev3 is (opti-al leverage ratio with � = 0.95) minus (optimal leverage ratio with

= 0.5). Thus, Lev2 illustrates the effect of � in operating region 2at full capacity) and Lev3 illustrates the effect of � in operatingegion 1 (below full capacity), while Lev1 shows the overall effectf � on optimal leverage ratio.

The spread in optimal leverage ratio varies depending on thearameter values, but is generally economically significant. Whilehe range is 3.3–8.4% for the parameter values shown in the table,n most cases it is close to 5%. Thus, incorporating market powern the model can potentially result in a significantly higher optimaleverage ratio. When the firm is operating below full capacity, mar-et power can make a material difference to the capital structureecision. Clearly, the effect of � on capital structure can be eco-omically significant, and can result in substantially higher levelsf debt usage than traditional models.

In operating region 2, the firm is operating at full capacitynd has no operating flexibility, hence product–market effects

re not important and can be ignored. In this region, the effectf � on optimal leverage ratio is insignificant, hence the exist-ng contingent-claim models without product–market flexibility

t is, in fact, a general result in contingent-claim models that optimal leverage ratios independent of asset value (see, for instance, Leland, 1994).

118 S. Sarkar / The Quarterly Review of Econom

Table 1Effect of price sensitivity (�) on optimal leverage ratio.

Lev1 Lev2 Lev3

�15% 5.00% −1.59% 6.59%20% 5.70% 0.47% 5.23%25% 5.31% 1.29% 4.02%

r5% 5.55% 0.94% 4.61%6% 5.70% 0.47% 5.23%7% 5.63% −0.12% 5.75%

�0% 5.79% −0.09% 5.88%1% 5.70% 0.47% 5.23%2% 5.03% 0.65% 4.38%

�10% 8.44% 2.08% 6.36%15% 5.70% 0.47% 5.23%20% 3.29% −0.99% 4.28%

˛40% 4.04% −0.51% 4.55%50% 5.70% 0.47% 5.23%60% 6.99% 1.22% 5.77%

Base case parameter values: r = 6%, � = 1%, � = 20%, � = 15%, = 50%, Q = 1, and v = 1.Note:Lev1: optimal leverage ratio with � = 0.95 minus optimal leverage ratio with� = 0.05.Lev2: optimal leverage ratio with � = 0.5 minus optimal leverage ratio with��

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re a reasonable approximation. To summarize, the effect ofroduct–market flexibility on capital structure is (i) negligible forrms operating at full capacity, but (ii) potentially significant forrms operating below full capacity. Thus, there is an additionaleterminant of capital structure that must be taken into account inhese models.

However, we certainly do not claim that market power willlways be a significant factor in capital structure decisions. Theffect is very much a function of the circumstances facing the firmhen it makes the capital structure decision, e.g., at full capacityarket power will be insignificant in the capital structure decision.

he best we can say from the model’s outputs is that in some situ-tions, market power can have a significant effect on optimal debtevel and optimal leverage ratio. This is, in our opinion, enough rea-on to consider including this factor in models of capital structure.

From Table 1, we also note that the effect of � on optimal lever-ge ratio is particularly strong for certain parameter values (small, � and large ˛). Therefore, it is more important to include this

actor in the model for low-growth firms (e.g., old companies),ompanies with low effective tax rate (e.g., with large tax loss car-yforwards), or companies with high bankruptcy cost (e.g., with aarge percentage of intangible assets).

As briefly referred to in Section 4.1, we follow Leland (1994) inetting the base-case tax rate at � = 15%. This is based on Miller’s1977) argument that in a “trade-off” model where optimal cap-tal structure is determined by trading off the tax advantage ofebt versus the bankruptcy cost resulting from debt, the com-any’s effective tax rate is the net tax advantage of debt. Miller1977) shows that the net tax advantage is given by the expression1 − (1 − �c)(1 − �e)/(1 − �d)}, where �c, �e, and �d are the corpo-ate tax rate, tax rate on equity income, and tax rate on debt

ncome, respectively. Using the values in Leland (1994, footnote7), we get {1 − (1 − 0.35)(1 − 0.20)/(1 − 0.40)} = 0.133. For our basease, therefore, we use a slightly higher effective tax rate of

= 15%.

ipi

ics and Finance 54 (2014) 111– 122

Also, since these capital structure models are infinite-horizonr perpetual models, a forward-looking long-term estimate shouldlways be used. As mentioned by Leland (1994) and discussedn more depth by Graham and Smith (1999), the actual long-erm forward-looking effective tax rate is likely to be even smallerecause of real-life factors such as tax loss carryforwards, possibil-

ty of making losses in the future, etc. Finally, we have assumed, inommon with the contingent-claim literature, that the tax rate isymmetric (i.e., profits are taxed and losses are subsidized). How-ver, the tax rate in real life is asymmetric: while profits are taxed,osses are not subsidized. This introduces non-linearities in theffective tax schedule, which further reduces the effective tax rateGraham & Smith, 1999). Because of these various reasons, it is pos-ible (even likely) that the actual effective tax rate for many firms isower than the base-case value we use, hence our base-case resultsould have understated the importance of market power in theapital structure decision.

.4.2. Empirical implicationsThe first empirical implication of Result 2 is that the effect of �

epends on whether the company is operating at full capacity orelow capacity when it makes the capital structure decision. This is

new empirical implication, since this aspect has not been exam-ned in the literature; while there have been a number of empiricaltudies on the relationship between market power and leverageatio, the difference between firms operating at full and partialapacity has not been examined. Testing this implication might be ahallenge because the necessary information is not easily availablerom conventional data sources.

The next empirical implication is that the relationship betweenarket power and leverage ratio might be monotonic or non-onotonic (U-shaped). In the context of empirical testing, a

on-monotonic relationship would introduce non-linearity in theegression specifications. That is, if a regression is run with leverageatio as the dependent variable, and market power (appropriatelyeasured) as an explanatory variable, a positive and significant

oefficient on market power would indicate that leverage is anncreasing function of market power. However, with both marketower and market power squared as additional explanatory vari-bles, if the coefficients are negative and positive, respectively, itould indicate a U-shaped relationship.

The most important empirical implication from Result 2 ishat leverage ratio is generally an increasing function of marketower. This is consistent with a number of empirical studies.uney et al. (2011) provide evidence that leverage is positively

elated to market power, and in most cases the relationship istatistically significant. Rathinasamy et al. (2000) report that, forost of their samples, firms with higher monopoly power useore debt in their capital structure. Lovisuth (2008) reports a

ositive relationship between market power and leverage forost of the East Asian economies examined in the study, while

mith et al. (2008) report the same relationship for firms in Newealand. Istaitieh and Rodriguez (2002) find that firms in highlyoncentrated industries tend to have higher debt levels. Sincereater concentration generally implies less competition or morearket power, their evidence is consistent with our prediction.

inally, MacKay and Phillips (2005) provide empirical evidencehat firms in concentrated industries have higher leverage ratioshan firms in competitive industries, which, in the context ofur model, means that leverage ratio is an increasing function of.

A further empirical implication is that, for low demand volatil-ty, leverage ratio is a U-shaped function of market power. A fewapers have found a U-shaped relationship, although these stud-

es did not explicitly require their samples to be composed of only

conomics and Finance 54 (2014) 111– 122 119

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S. Sarkar / The Quarterly Review of E

ow-volatility firms; hence they are not necessarily direct tests ofhis implication. Guney et al. (2011) report a U-shaped relationshipetween leverage ratio and market power in certain cases, with

negative coefficient on market power and a positive coefficientn market power squared. Pandey (2004) finds that for lower andigher market power debt usage is high, and for intermediate mar-et power debt usage is low; this is also consistent with a U-shapedelationship.

Finally, the model also implies that in certain cases, mar-et power might have a negative effect on leverage ratio (theownward–sloping portion of the relationship). This is also consis-ent with some empirical findings. For instance, Rathinasamy et al.2000) carried out industry-wise regressions, and reported thator some industries, debt usage is negatively related to monopolyower. Lovisuth (2003) also found a negative relationship betweenarket power and debt usage in the U.K. retailing industry.

.5. Effect of demand volatility �

The effect of earnings volatility or asset-value volatility on cap-tal structure has been examined in existing papers, e.g., Leland1994), Fischer et al. (1989), and Sarkar and Zapatero (2003). Thisection, however, looks at the effect of demand volatility on capitaltructure.11

Demand volatility � will affect bankruptcy risk in two ways,imilar to the discussion in Section 4.4. First, from standard optionheory, a higher � will move the bankruptcy trigger further (lower),hereby reducing bankruptcy risk. On the other hand, a higher �

eans that extreme values (both high and low) are more likelyo be reached, hence the bankruptcy trigger is more likely to beeached; this increases bankruptcy risk.

In addition to the “bankruptcy risk” affect, � affects debt choiceia the “flexibility” channel. A higher � makes the embedded optionassociated with product–market flexibility) more valuable, againrom standard option theory. This increases value, hence it alsoffects the choice of debt level. Note, however, that there is noperating flexibility in Region 2; hence this effect will apply onlyo Region 1.

The overall effect will depend on which of the above effectsominates. As discussed in Section 4.4, for small � and �, the firstffect seems to dominate; thus, when � is small, bankruptcy riskill initially be a decreasing function of �, hence it will initially be

ptimal to take on more debt as � is increased. Thus, for small �, c*

hould initially be an increasing function of �.As � is increased further, the second effect starts to dominate,

ence bankruptcy risk starts rising, making debt less attractive.hus, c* becomes a decreasing function of �. Eventually, the thirdffect mentioned above becomes more important, hence a higher �esults in greater flexibility value and makes debt more attractive;hus the optimal debt level c* starts rising again as � is increased.o sum up, for small � the optimal debt level c* should be initiallyn increasing function, then a decreasing function, and finally anncreasing function again, of �. For other values of �, since therst effect is not important, only the last two segments should bebserved, hence c* should just be a U-shaped function of �.

For optimal leverage ratio, the third effect above does not apply,s discussed in Section 4.4 and footnote 10. Thus, for small �, opti-al leverage ratio will be initially increasing and subsequently

ecreasing in �, to give an inverted-U shaped relationship; forll other values of �, optimal leverage ratio will be a decreasingunction of �.

11 Earnings volatility and demand volatility may be related, but they are not theame, as shown in Section 2.2.

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ig. 3. Shows optimal coupon level (c*) and optimal leverage ratio as functions ofolatility or sigma (�), for three levels of price sensitivity, � = 0.05, 0.5 and 0.75. Thease-case parameter values are used: � = 1%, r = 6%, � = 15%, = 50%, v = 1 and Q = 1.

Fig. 3 illustrates how demand volatility � affects optimal coupon* and optimal leverage ratio. It can be seen that the results are justs expected form the discussion above. The effect of � is summa-ized below.

esult 3. The optimal coupon level is a U-shaped function of demandolatility, and the optimal leverage ratio is a decreasing function ofemand volatility, except when � is very small, in which case theptimal coupon is first increasing, then decreasing, and finally increas-ng, in �, while optimal leverage ratio is initially increasing and thenecreasing in �.

Fig. 4 shows the relationship between c* and � for differentarameter values. The same base-case parameter values are useds in Section 4.1, along with � = 0.5 (since the shape of the rela-ionship does not vary with �, with the exception of very small �,s seen above). It can be noted that the relationship is more or less

he same for all the parameter values examined. The only thing thataries is the exact point at which c* starts rising as � is increased.

When demand growth rate � is higher, the firm will more likelyperate at full capacity, hence product–market flexibility becomes

120 S. Sarkar / The Quarterly Review of Economics and Finance 54 (2014) 111– 122

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ess important. Since the flexibility is less important, c* starts risingater (i.e., when � is higher). This is exactly what we observe inig. 4(a).

The interest rate r has virtually no effect on the shape of the c*

s. � curve.When the tax rate � is higher, the after-tax cash flows from oper-

tions will be smaller, leading to a drop in the value of operatingssets. Therefore, the value of product–market flexibility will riseelative to the value of operating assets, and flexibility will becomeore important. As a result, c* will start rising earlier (i.e., when

is smaller) when � is increased, and will start rising later when is lower. This is also consistent with Fig. 4(c). In fact, for � = 10%,e note that c* is a decreasing function of � for the entire range

xamined.

Finally, when bankruptcy cost is higher, the loss in firm value

t bankruptcy is larger. Bankruptcy risk, therefore, becomes moremportant. As a result, c* will start rising later (when � is higher).nce again, this is exactly what we observe in Fig. 4(d).

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. Summary and conclusions

Contingent-claim models of corporate capital structure assumexogenous earnings or asset values. However, earnings in real lifere endogenous because they are determined by the firm’s activ-ties in the product market; for instance, when demand for theroduct is stronger (weaker), a firm will often increase (reduce)

ts output and/or price. Its ability to make these adjustments (i.e.,ts market power) could then be a factor in determining the opti-

al capital structure. Empirical studies confirm that this is indeedhe case.

We therefore extend the traditional contingent-claim literaturey incorporating the firm’s market power in the capital structureecision. This is the main contribution of our paper. We show that

arket power generally increases the optimal leverage ratio, con-

istent with the available empirical evidence. We also show that, inarious situations, this relationship might be negligible, negative,r non-monotonic (U-shaped).

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If the firm makes the leverage decision when operating at fullapacity, market power has no significant effect on the optimaleverage ratio, since the capacity constraint limits its ability to

ake adjustments. When operating below full capacity, however,e show that taking into account market power could increase the

ptimal leverage ratio by an economically significant amount. It isherefore not advisable to ignore this factor in models of corporateapital structure.

A critical question in this context is: do firms make capital struc-ure decisions before reaching full capacity? If the answer is no,hen product–market flexibility can safely be ignored in the anal-sis of optimal capital structure. However, we contend that firmsften do decide on their capital structure before reaching full capac-ty. This is because it usually takes a long time to reach full capacity,nd firms do not generally defer capital structure decisions for suchong periods. Moreover, it is often optimal to invest in a projectefore it can run at full capacity (Sarkar, 2009); in such cases, theapital structure decision will be made when operating at partialapacity. We therefore argue that the effect of product–market flex-bility should be taken into account when identifying the optimalapital structure.

We have essentially extended models such as Leland (1994),hich examines a one-time capital structure decision. In practice,rms might change their capital structures (although not fre-uently, because changing the capital structure is a costly process,aulkender, Flannery, Hankins, & Smith, 2007). This scenario haseen examined by Fischer et al. (1989) and Goldstein et al. (2001)ithout product–market flexibility. Therefore, a possible extension

f our model is to examine corporate capital structure decisionsith product–market flexibility when recapitalization is possible.

cknowledgement

Financial support from a McMaster Innovation Grant is grate-ully acknowledged. I would also like to thank two anonymouseferees for their helpful comments and suggestions, and the Editor.S. Esfahani for his recommendations during the review process.he usual disclaimer applies.

ppendix A.

We use the valuation approach of Dixit and Pindyck (1994), Bar-lan and Strange (1999). V(y) is the value of the firm’s assets (or thenlevered firm value). These assets generate a cash flow of �(y) inerpetuity.

The expected increase in value after an infinitesimally smalleriod of time (dt) is given by E(dV). From Ito’s lemma, we have:

V = Vydy + 0.5Vyy(dy)2 (A1)

We know dy = �ydt + �ydz, from Eq. (1), which also gives:dy)2 = �2y2dt. Substituting into Eq. (A1), we get:

V = Vy(�ydt + �ydz) + 0.5Vyy(�2y2dt), (A2)

Taking expectations, and given that E(dz) = 0, we get:

(dV) = Vy �ydt + 0.5Vyy(�2y2dt), or E(dV) = V ′(y)�ydt

+ 0.5V ′′(y)�2y2dt

This is the return from the project (similar to capital gain). In

ddition, there is the continuous cash flow from the project (similaro dividend), given by �(y)dt over this period. Thus, the total returnrom the project is given by: [E(dV) + �(y)dt]/V. From the localxpectations hypothesis, this should be equal to the discount rate

s

W(

ics and Finance 54 (2014) 111– 122 121

. Equating the two for the period dt, we get: E(dV) + �(y)dt = rVdt,hich gives the ordinary differential Eq. (9) of the paper.

Region 1 valuation: Here we solve ODE (9) with(y) = (1 − �)�y1/� (from Eq. (3)). The general solution to thisDE is y� . Substituting this in the ODE, we get the quadraticquation: 0.5�2�(� − 1) + �� − r = 0, which has two solutions1 (>0) and �2 (<0). Thus, the complete general solution is:1(y) = Ay�1 + A2y�2 , where A1 and A2 are constants to be deter-ined by the boundary conditions. The particular solution, verified

y direct substitution, is Zy1/�, where Z is given by Eq. (11). Thus,he complete solution is:

1(y) = Ay�1 + A2y�2 + Zy1/� (A3)

hen y = 0, it will remain at zero forever (since y = 0 is an absorbingoundary), hence the asset value will also fall to zero. This gives theondition: Lim

y→0V1(y) = 0, which implies A2 = 0 (since �2 < 0). Thus,

e get Eq. (10) of the paper.Region 2 valuation: Here we solve ODE (9) with

(y) = (1 − �)(yQ − vQ) (from Eq. (5)). The general solu-ion is the same as above, but the particular solution is1 − �)[Qy/(r − �) − vQ/r]. Thus the complete solution is:

2(y) = (1 − �)[Qy/(r − �) − vQ/r

]+ By�2 + B1y�1 (A4)

The term (1 − �)[Qy/(r − �) − vQ/r] in Eq. (A4) is the value of therm’s assets if it always operates at full capacity, with no possibilityf operating flexibility any time in the future. When y reaches veryigh levels (y → ∞), it is very unlikely that the firm will ever goack to operating at partial capacity, hence

imy→0

V2(y) = (1 − �)[Qy/(r − �) − vQ/r] (A5)

Eq. (A5) implies that B1 = 0 in Eq. (A4), since �1 > 0. Thus, we getq. (13) of the paper.

Derivation of Eq. (28): The total firm value in Region 1 is giveny Eq. (27):

1(y) = Zy1/� + �c/r + B1(y)�1 + (B2 + D1)(y)�2

Differentiating this with respect to c, and noting that B1 is inde-endent of c, we get Eq. (28) of the paper. For Region 2, total firmalue is (also from Eq. (27)):

2(y) = (1 − �)[yQ/(r − �) − vQ/r] + �c/r + (B3 + D1)(y)�2

Differentiating this with respect to c also gives Eq. (28), sinceB3/dc = dB2/dc from Eq. (25). The derivative dB2/dc is derived fromq. (24), dD1/dc from Eq. (19), and dyb/dc from Eq. (26).

ppendix B.

Here we present the model’s results with other sets of parame-er values, taken from some well-known contingent-claim modelsn the capital structure literature. We start with the model ofackbarth, Miao, and Morellec (2006), which should be represen-

ative of the corporate sector since they “. . .select parameter valueshat roughly reflect a typical S&P 500 firm” (page 531). Using theirarameter values, we get the following inputs: r = 5.5%, � = 0.5%,

= 25%, � = 15%, and = 40% (and keeping v and Q unchanged at = 1 and Q = 1). With these parameter values, the optimal lever-ge ratio with � = 0.05 is 38.43%, and the optimal leverage with

= 0.95 is 42.54%. Thus, the spread in leverage ratio over the rangef price sensitivity is a little over 4%, not very far from our base-case

cenario.

Next, using the base-case parameter values of Sundaresan andang (2007), we have: r = 6%, � = 0, � = 15%, � = 25%, and = 35%

along with v = 1 and Q = 1). The results are: optimal leverage ratio

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SStructure and Product Market: Evidence from New Zealand, SSRN Working Paper.

22 S. Sarkar / The Quarterly Review of E

s 59.69% for � = 0.05, and 65.55% for � = 0.95. The spread in optimaleverage ratio is almost 6%.

Titman and Tsyplakov (2007) focus on the gold mining indus-ry in their numerical simulations. They use parameter values thatre “. . .chosen to roughly match empirical observations for selectedrms in the gold mining industry” (page 418). Using their paper, weet the following parameter values: r = 3%, � = 1%, � = 10%, � = 35%,nd = 50% (along with v = 1 and Q = 1). With these parameter val-es, the optimal leverage ratio comes to 68.06% for � = 0.05, and1.89% for � = 0.95. The spread in optimal leverage ratio is about.8%.

Finally, from Hackbarth and Mauer (2012), we get the follow-ng parameter values: r = 6%, � = 1%, � = 25%, � = 15%, and = 25%along with v = 1 and Q = 1). The results are: optimal leverage ratios 51.41% for � = 0.05 and 52.88% for � = 0.95, giving a spread inptimal leverage ratio of only 1.5%.

It is clear that the effect of market power or price sensitivity (�)n optimal capital structure is a function of the values of the inputarameters, particularly the tax rate and bankruptcy cost (not sur-rising, in the context of a “trade-off theory” model). Therefore,he importance of market power in capital structure decisions willikely vary from company to company, and it is not possible to

ake a blanket statement about its importance. However, afterxamining the results with parameter values from a number ofell-respected models in the literature, we find that in most casesarket power does seem to be a potentially important factor in

everage decisions.Thus, while the effect of market power on capital structure

ight be insignificant in some cases, in most cases it can make substantial difference to the optimal capital structure. Hence, wergue that it is generally a good idea to include this factor in capitaltructure models.

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