GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 257

The effects of external financing costs on investment
timing and sizing decisions

Michi Nishihara and Takashi Shibata

April 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

The eﬀects of external ﬁnancing costs on investment
timing and sizing decisions∗
Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
We develop a dynamic model in which a ﬁrm exercises an option to expand production
on either a small or large scale with cash reserves and costly external funds. We show thatthe ﬁnancing costs greatly distort the ﬁrm’s ﬁnancing and investment behavior and resultin a policy contingent on the dynamics of the cash ﬂow and reserves. Most notably, we
prove that an intermediate level of cash reserves is likely to accelerate investment in thesmall-scale project by interactions among ﬁnancing costs, investment timing, and investment sizing. Our results ﬁll the gap between two types of results: (i) empirical ﬁndings ina U-shaped relation between the investment volume and internal funds, and (ii) empiricalpredictions of a U-shaped relation between the investment timing and internal funds.
JEL Classiﬁcations Code: G13; G31; G32.
Keywords: Investment timing; Investment size; Costly external ﬁnancing; Optimal stopping.

∗†This version: 12 April, 2012.
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,

Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277‡
Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo
192-0397, Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298

1

1

Introduction

Subsequent to the departures from Modigliani and Miller (1958)’s irrelevance proposition
in a frictionless market, there has been a long tradition in corporate ﬁnance to investigate the eﬀects of various frictions on ﬁnancing and investment decisions. Recently,
an increasing number of papers have analyzed not only the static but also the dynamic
behaviors of corporate ﬁnancing and investment in the presence of frictions.1 Amongthese, a real options approach plays an important role in unveiling investment timing
decisions in the presence of such frictions as liquidity constraints (Boyle and Guthrie(2003)), shareholders-debtholders conﬂicts (Mauer and Sarkar (2005), Sundaresan andWang (2007)), and asymmetric information (Grenadier and Wang (2005), Shibata andNishihara (2010) Morellec and Schürhoﬀ (2011), and Grenadier and Malenko (2011)).
We extend this line of research by revealing the interactions of costs of external ﬁnancing, investment timing, and investment size. The model is as follows: A ﬁrm owns
an option to expand production on either a small or large scale, where the price of theoutput follows a geometric Brownian motion. The investment project is ﬁnanced withcash reserves and costly external funds. The ﬁrm’s cash reserves gradually increase as theﬁrm’s existent production generates cash ﬂows. If the ﬁrm waits for a suﬃcient level ofcash reserves for each project, the investment project can be ﬁnanced entirely with cash
reserves. Otherwise, the ﬁrm must rely partially on costly external ﬁnancing. For thesmall-scale expansion, the suﬃcient cash level is lower than that of for the large-scale expansion. Considering the trade-oﬀ, the ﬁrm determines its ﬁnancing, investment timing,and investment sizing policy.
As in the standard real options literature (e.g., McDonald and Siegel (1986), Dixit andPindyck (1994)), our model assumes the irreversibility of investing as a friction. A key
diﬀerence from most of the related papers is that we incorporate the investment sizing
decision in addition to the investment timing decision. The assumption of either a smallor large-scale choice builds on Dixit (1993) and Décamps, Mariotti, and Villeneuve (2006).
Indeed, our model generalizes their models to a case with costs of external ﬁnancing. The
ﬁnancing costs are known as one of the most inﬂuential frictions in the corporate ﬁnanceliterature (e.g., Altinkilic and Hansen (2000), Hennessy and Whited (2007)). According to
the pecking order hypothesis, asymmetric information problems associated with externalfunding generate higher costs; therefore, managers prefer internal over external ﬁnancing(Myers (1984), Myers and Majluf (1984)). As a proportional cost accounts for the largest
part of external ﬁnancing costs, we focus primarily on the case with a proportional cost.
Before describing the results, we emphasize a contribution of this paper from thetheoretical viewpoint. Most of the related papers demonstrate their results only by numerical examples because the model’s complexity precludes analytic results (e.g., Boyle
1

An incomplete list includes Hennessy and Whited (2005), Hennessy and Whited (2007), Hennessy, Levy,

and Whited (2007), Tsyplakov (2008), Tserlukevich (2008), and Morellec and Schürhoﬀ (2011).

2and Guthrie (2003), Hirth and Uhrig-Homburg (2010a), Shibata and Nishihara (2012)).
Contrasted to the stream of literature, this paper analytically proves the interesting properties of the dynamic corporate ﬁnancing and investment policy by developing techniquesin the mathematical ﬁnance literature (e.g., Broadie and Detemple (1997), Detemple(2006), Bobtcheﬀ and Villeneuve (2010)) into a case involving a non-convex payoﬀ and a
non-geometric Brownian motion.
The results are summarized as follows. The presence of ﬁnancing costs, unlike previous
results with no ﬁnancing cost in Décamps, Mariotti, and Villeneuve (2006), leads the ﬁrm
to take a ﬁnancing and investment policy contingent not only on the cash ﬂow dynamics
but also on the cash reserves dynamics. Speciﬁcally, higher ﬁnancing costs enhance theﬁrm’s incentive to wait for a suﬃcient level of cash reserves and use entirely internalﬁnancing, especially for the small-scale project.
The investment threshold for the large-scale project monotonically decreases with cash
reserves. This monotonic relation is straightforwardly consistent with conventional viewsof underinvestment due to ﬁnancing constraints. On the other hand, the investment region for the small-scale project is not monotonic with cash reserves. The small-scale
investment is encouraged with cash reserves until cash reserves reach the investment costand, after that, the investment is discouraged with cash reserves. The rationale behindthe non-monotonic relation is that the ﬁrm optimizes not only investment timing but also
investment size. Consider the ratio of the total cost associated with the large-scale expansion to that of the small-scale expansion. The ratio, which changes with cash reserves, ismaximized when cash reserves are equal to the amount of the small-scale investment cost.
Indeed, at that moment the small-scale project requires no external funds while the largescale project requires a great amount of external funds. The greatest advantage of thesmall-scale project over the large-scale project plays a role in speeding up the small-scale
investment at the intermediate level of cash reserves.
Most notably, our results can link two signiﬁcant results in corporate ﬁnance. The
ﬁrst one is a U-shaped relation between the investment volume and internal funds. Since
arguments among Fazzari, Hubbard, and Petersen (1988), Kaplan and Zingales (1997),
and Hubbard (1998), investment-cash ﬂow sensitivities have been the center of attention incorporate ﬁnance. In particular, recent empirical evidence regarding this issue documented
that the investment volume does not necessarily decrease with internal funds but canhave a U-shaped relation with internal funds (Cleary, Povel, and Raith (2007), Guariglia(2008)).
The second result is an empirical prediction that the investment threshold has a Ushaped relation with a degree of ﬁnancial constraints. The prediction has been seen in the
recent real option literature. Boyle and Guthrie (2003) examined the eﬀects of a liquidity
constraint to the investment timing decision and predicted that the investment thresholdhas a U-shaped relation with a degree of the liquidity constraint. Shibata and Nishihara(2012), who examined the eﬀects of a debt capacity constraint in a dynamic ﬁnancing and3

capital structure model, showed that the investment threshold has a U-shaped relation
with a degree of the debt capacity constraint.2
If one identiﬁes “earlier” investment as “increased” investment, the two results areinconsistent with each other. However, this argument pays no attention to the point thatthe investment timing studies consider ﬁxed-scale investment models. Our results can
explain both types of results in terms of the interactions of investment timing and sizing
decisions with costly external ﬁnancing. In the presence of ﬁnancing costs, cash reservesinﬂuence the trade-oﬀ between the two choices: small- or large-scale expansion. Whencash reserves are close to the amount of the small-scale investment cost, the ﬁrm hasa great incentive to invest in the small-scale project for which the investment thresholdis relatively low. When cash reserves are much higher or lower than that level, theﬁrm is likely to undertake the large-scale expansion for which the investment thresholdis relatively high. This mechanism can explain U-shaped relations regarding both theinvestment volume and timing in the previous studies.
In summary, the main contributions of this paper are threefold. First and most importantly, this paper ﬁlls the gap between two types of results in the corporate ﬁnanceliterature: (i) empirical evidence regarding a U-shaped relation between the investment
volume and internal funds and (ii) predictions of a U-shaped relation between the investment timing and internal funds. Second, this paper complements the investment timing
and sizing literature by proving that costs of external ﬁnancing greatly distort the decision, leading especially to a policy dependent the dynamics of both cash ﬂow and cash
reserves. Third, this paper contributes the mathematical ﬁnance literature by proving
the properties of exercise regions of an optimal stopping problem involving a non-convex
payoﬀ function and a non-geometric Brownian motion.
The remainder of this paper is organized as follows. Section 2 presents the setup
and the results in the case without ﬁnancing costs. Section 3 presents the results in thecase with a proportional cost and explains empirical implications. Section 4 examines
the comparative statics with respect to the price volatility and a case with ﬁxed andproportional costs in numerical examples. Section 5 concludes the paper. All proofs
appear in the appendix.

2

Regarding the relation between the investment timing and cash holdings, Hirth and Uhrig-Homburg (2010b),

who extended Boyle and Guthrie (2003) to a case with ﬁnancing costs, pointed out the possibility of various
non-monotonic relations, and Nishihara and Shibata (2011) proved that a ﬁxed cost of external ﬁnancing leads
to a non-monotonic relation.

4

2

Preliminaries

2.1

Setup

Consider a risk-neutral ﬁrm that produces a commodity at a constant rate. The output
is sold at the market price X(t), which follows a geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,(1)

where B(t) denotes the standard Brownian motion deﬁned in a probability space (Ω, F, P)
and µ, σ(> 0) and x(> 0) are constants. For convergence, we assume that r > µ, where r
is a positive and constant interest rate. Assume that the ﬁrm owns an option to expand
production on either a small or large scale, A1 (> 1) or A2 (> A1 ), respectively, at any
time. If the small-scale (large-scale) growth option is exercised at time τ , the ﬁrm pays aﬁxed investment cost at time τ and receives an instantaneous cash ﬂow A1 X(t) (A2 X(t))
after time τ . Assume that the investment cost is I1 (> 0) (I2 (> I1 )) for the small-scale(large-scale) expansion if the whole amount of the cost is internally ﬁnanced. If partof the investment cost is externally ﬁnanced, the ﬁrm pays a proportional cost C(≥0) of external ﬁnancing. The total cost associated with the investment is expressed as
Ii + C max(Ii − Y (τ ), 0) (i = 1, 2), where Y (τ ) denotes the ﬁrm’s cash reserves at time τ .
Until the investment time τ , cash reserves Y (t) follow
dY (t) = rY (t)dt + X(t)dt, (0 < t < τ ) Y (0) = y,(2)where y(≥ 0) is a constant representing the initial cash reserves. Note that Y (t) is anincreasing process.
Boyle and Guthrie (2003) assume the dynamics of cash reserves exogenously and consider an option to initiate a new project. In contrast, we relate cash reserves Y (t) to
operating cash ﬂows X(t) more directly and consider the option to expand production.
In the case of C = 0, the setup corresponds to an alternative investment model studied
in Dixit (1993) and Décamps, Mariotti, and Villeneuve (2006). For a comprehensive listof typical situations ﬁtting the standard model, refer to Dixit (1993), Dixit and Pindyck(1994), and Décamps, Mariotti, and Villeneuve (2006). We extend the standard model,
which presumes that the ﬁrm needs no costs of external ﬁnancing (otherwise, it has sufﬁcient internal funds), to a model involving costs of external ﬁnancing. Unlike Boyle andGuthrie (2003), who focused on a liquidity constraint in ﬁxed-size investment, we examine
the interactions of investment timing, size, and ﬁnancing costs.3
Our assumption of costly external ﬁnancing is justiﬁed as follows. In the pecking ordertheory, agency and asymmetric information problems cause costs of external ﬁnancing,
3

Hirth and Uhrig-Homburg (2010b) extended Boyle and Guthrie (2003) to a case involving both a liquidity

constraint and ﬁnancing costs. Nishihara and Shibata (2011) developed a model involving ﬁnancing costs,
whereas Shibata and Nishihara (2012) focused on a debt issuance constraint. However, these papers assumeﬁxed-scale investment.

5which leads to a preference for internal over external ﬁnance (Myers (1984), Myers and
Majluf (1984)). Practically, ﬁnancing costs consist of a ﬁxed cost (which is independent of
the issue size) and a variable cost (which depends on the issue size). A ﬁxed cost includes
taxes, fees, and setup expenses. A variable cost increases with issue size primarily because
more underwriting services are required for more funds raised. In the standard view ofthe literature, a variable cost is convex with respect to the issue size (e.g., Altinkilic
and Hansen (2000)). Hennessy and Whited (2007) estimated that proportional costs ofequity ﬁnancing are approximately 5% (10%) for large (small) ﬁrms. They showed thata proportional cost can almost completely account for costs of equity ﬁnancing for large
ﬁrms, although the ﬁxed cost eﬀect may be not negligible for small ﬁrms. In taking
account of their results, as well as preserving tractability of the model, we examine thecase with only a proportional cost in full detail in Sections 3 and 4.1, and succinctly
explain how the results change in the case involving both ﬁxed and proportional costs as
a supplement in Section 4.2.

2.2

Case with no financing costs

As a benchmark, this section explains results in the case of C = 0. In this case, the ﬁrm
solves the following problem:
sup E [max E
x

τ ∈T

i=1,2

τ,X(τ )

∫ τ
∫ ∞
−rt
[
e X(t)dt +e−rt Ai X(t)dt − e−rτ Ii ]],
0

(3)

τ

where T denotes the set of all stopping times and Ex [·] (Eτ,X(τ ) [·]) denotes the expectation
conditional on t = 0, X(0) = x (t = τ, X(t) = X(τ )). In (3), τ represents the time toexpand the scale of production, whereas maxi=1,2 Eτ,X(τ ) [·] represents the sizing choice attime τ . By the strong Markov property of X(t), (3) can be reduced to()
Ai − 1xx −rτ
+ sup E [emaxX(τ ) − Ii ] .
i=1,2r − µ τ ∈T
r−µ
|
{z
}
=:V0 (x) the growth option value

The second term, denoted by V0 (x), represents the growth option value. Décamps, Mariotti, and Villeneuve (2006) derived a closed-form solution for this type of problem.4
Indeed, we have V0 (x) depending on the relation of A1 , A2 , I1 and I2 as follows:4

For this type of problem, refer also to Nishihara and Ohyama (2008).

6

If {(A2 − 1)/(A1 − 1)}β/(β−1) < I2 /I1 , () ( )βx(A1 − 1)x∗1


− I1



r−µ
x∗1






 (A1 − 1)x − I


1

r−µ





(A1 − 1)x∗2 γ−1 − (A2 − 1)x∗1 γ−1V0 (x) =xβ
∗ β−1 x∗ γ−1 − x∗ γ−1 x∗ β−1 )

β(r
−
µ)(x

1
2
1
2



β−1
∗

(A1 − 1)x2
− (A2 − 1)x∗1 β−1



+
xγ

γ−1 ∗ β−1
β−1 ∗ γ−1∗∗γ(r − µ)(x1
x2
− x1
x2
)







 (A2 − 1)x − I2r−µwhere β := 1/2−µ/σ 2 +√(0 < x < x∗1 ) waiting(x∗1 ≤ x ≤ x∗21 ) small-scale expansion(x∗21 ≤ x ≤ x∗22 ) waiting(x ≥ x∗22 ) large-scale expansion,(µ/σ 2 − 1/2)2 + 2r/σ 2 (> 1) and γ := 1/2−µ/σ 2 −√(4)(µ/σ 2 − 1/2)2 + 2r/σ 2(< 0) is positive and negative characteristic roots, respectively. Threshold x∗1 is deﬁned
by x∗1 := β(r − µ)I1 /{(A1 − 1)(β − 1)}, which is equal to the threshold for an option
to invest only in the small-scale project. On the other hand, thresholds x∗21 and x∗22 aredetermined by the value matching (continuous ﬁt) conditions at the boundaries.
If {(A2 − 1)/(A1 − 1)}β/(β−1) ≥ I2 /I1 , () ( )β(A2 − 1)x∗2
x


− I2
(0 < x < x∗2 ) waiting


r−µ
x∗2
V0 (x) =
 (A2 − 1)x − I2(x ≥ x∗2 ) large-scale expansion,
r−µ

(5)

where x∗2 := β(r − µ)I2 /{(A2 − 1)(β − 1)}. In this case, the problem is reduced to a
problem of investing only in the large-scale project because the value (5) dominates thevalue of the small-scale investment, (A1 − 1)x/(r − µ) − I1 , for all x > 0.
Note that β/(β − 1) monotonically increases with σ because of ∂β/∂σ < 0. A higher
σ is more likely to lead to the case of {(A2 − 1)/(A1 − 1)}β/(β−1) ≥ I2 /I1 . The intuition isthat a higher σ increases the value of the large-scale growth option so that the value (5)can dominate (A1 − 1)x/(r − µ) − I1 for all x > 0. For details, refer to Décamps, Mariotti,
and Villeneuve (2006). We also note that)}{(
Ai − 1x −rτ
X(τ ) − Ii ] ,V0 (x) ≥ max sup E [ei=1,2 τ ∈T
r−µ
|
{z
}

(6)

the value by the initial sizing choice

where in the left-hand side the sizing decision i ∈ {1, 2} is Fτ -measurable, while in theright-hand side i ∈ {1, 2} is F0 -measurable. Dixit (1993) focused only on the right-handproblem, and Décamps, Mariotti, and Villeneuve (2006) complemented his argument bysolving the left-hand problem. Because of the diﬀerence, in general we have the inequality(6). However, the equality{
V0 (x) = max sup E [e
x

i=1,2

−rτ

τ ∈T

7

(

)}
Ai − 1
X(τ ) − Ii ] ,
r−µ

(7)

holds for x ≤ x∗1 . This means that, if the initial value X(0) = x is suﬃciently low, thisproblem is unchanged from the case in which the ﬁrm chooses the investment size notdynamically but at the initial time.

3

Analytic Results

3.1

Model solutions

This section provides analytic results in the case of C > 0. In this case, the growth option
value, denoted by VC (x, y), is expressed as)(
Ai − 1x,y −rτVC (x, y) = sup E [emaxX(τ ) − Ii − C max(Ii − Y (τ ), 0) ],
i=1,2
r−µ
τ ∈T

(8)

where where Ex,y [·] denotes the expectation conditional on t = 0, X(0) = x, Y (0) = y.
The term C max(Ii − Y (τ ), 0) represents that a proportional cost is required when theﬁrm is short of cash reserves, whereas maxi=1,2 (·) means that the ﬁrm chooses the optimal
size at the investment time τ .
First, we prove several properties of the option value VC (x, y). Consider the following
problems as approximations of VC (x, y):()
Ai − 1x −rτ
VL (x, y) := sup E [emaxX(τ ) − Ii − C max(Ii − y, 0) ],
i=1,2
r−µ
τ ∈T
)
(
Ai − 1
X(τ ) − Ii − C(Ii − Y (τ )) ].
VU (x, y) := sup Ex,y [e−rτ max
i=1,2
r−µ
τ ∈T
VL (x, y) is the same type of problem as V0 (x) so that it allows an explicit solution like (4)or (5). By the strong Markov property of X(t), we can easily show
(
)
(
)
x
Ai − C − 1
x,y −rτVU (x, y) = C y ++ sup E [emaxX(τ ) − (1 + C)Ii ].
i=1,2
r−µ
r−µ
τ ∈T
If A2 − C − 1 > 0, we have an explicit solution like (4) or (5). Otherwise, VU (x, y) =C(y + x/(r − µ)) holds, which means that the growth option will be never exercised.
Note that in both VL (x, y) and VU (x, y) the exercise policies are independent of Y (t).
The following proposition indicates that VL (x, y) and VU (x, y) are closed-form bounds of
VC (x, y).
Proposition 1 (Option value)
If y < I2 , VL (x, y) ≤ VC (x, y) ≤ VU (x, y) is satisﬁed.
Otherwise, VC (x, y) = V0 (x) holds.
Proposition 1 states that, once Y (t) reaches the large-scale investment cost I2 , the problem
with ﬁnancing costs is reduced to that of no ﬁnancing cost. We will also use the following
lemma to show the properties of the optimal ﬁnancing and investment policy.

8Lemma 1
If A2 − C − 1 > 0,0 ≤ VC (x + ∆, y) − VC (x, y) ≤(A2 − 1)∆r−µ(9)holds for any positive constant ∆.
0 ≤ VC (x, y + ∆) − VC (x, y) ≤ C∆

(10)

holds for any positive constant ∆.
Next, we concentrate on the exercise regions for the problem (8). The standard argument proves that the exercise region of the option is expressed as
SC := {(x, y) ∈ R2+ | V (x, y) = max{(Ai − 1)x/(r − µ) − Ii − C max(Ii − y, 0)}}. (11)i=1,2Furthermore, the exercise region SC can be decomposed into disjoint sets (some may be
empty) deﬁned by
SC,1,E := {(x, y) ∈ R2+ | V (x, y) = (A1 − 1)x/(r − µ) − I1 − C(I1 − y), y < I1 },SC,1,I := {(x, y) ∈ R2+ | V (x, y) = (A1 − 1)x/(r − µ) − I1 , y ≥ I1 },SC,2,E := {(x, y) ∈ R2+ | V (x, y) = (A2 − 1)x/(r − µ) − I2 − C(I2 − y), y < I2 },SC,2,I := {(x, y) ∈ R2+ | V (x, y) = (A2 − 1)x/(r − µ) − I2 , y ≥ I2 }.
The regions SC,1,E and SC,1,I represent small-scale investment regions partially with external ﬁnancing and entirely with internal ﬁnancing, respectively, whereas SC,2,E and SC,2,I
represent large-scale investment regions partially with external ﬁnancing and entirely withinternal ﬁnancing, respectively. Below, we analytically prove the interesting properties of
SC,1,E , SC,1,I , SC,2,E , and SC,2,I . It immediately follows from Proposition 1 that the exercise regions coincide with those of V0 (x) for y ≥ I2 . Note that thresholds x∗1 , x∗21 , x∗22 ,and x∗2 were explained in Section 2.2.
Proposition 2 (Case of suﬃcient cash reserves)
If {(A2 −1)/(A1 −1)}β/(β−1) < I2 /I1 , SC,1,I ∩R+ ×[I2 , ∞) = [x∗1 , x∗21 ]×[I2 , ∞) and SC,2,I =[x∗22 , ∞) × [I2 , ∞) hold. Otherwise, SC,1,I ∩ R+ × [I2 , ∞) = ∅ and SC,2,I = [x∗2 , ∞) × [I2 , ∞)hold.
Now, we examine the properties of the exercise regions in the region y < I2 . In thisregion, the optimal policy is contingent on a combination of X(t) and Y (t), and, hence, it
is quite diﬀerent from the threshold policy depending only on X(t) in the region y ≥ I2 .
The following proposition reveals the properties of the large-scale investment partiallywith external ﬁnancing, SC,2,E . Note that the region of the large-scale investment entirely
with internal ﬁnancing, SC,2,I , is explicitly derived in Proposition 2.

9

Proposition 3 (Large-scale expansion)
If A2 − C − 1 > 0 is satisﬁed,
SC,2,E = {(x, y) ∈ R2+ | x ≥ x∗C (y), y < I2 },(12)

where x∗C (·) is a continuous and monotonically decreasing function. x∗C (y) is between
the corresponding thresholds in VL (x, y) and VU (x, y), and limy↑I2 x∗C (y) is equal to the
maximum of x∗2 (or x∗22 ) and (C + 1)rI2 /(A2 − C − 1).
Otherwise, SC,2,E = ∅.
Here we limit our attention to a case in which a proportional cost is relatively low, i.e.,
A2 −C −1 > 0. We will explain the case of A2 −C −1 ≤ 0 below Proposition 6. In the caseof A2 − C − 1 > 0, the ﬁrm invests in the large-scale investment partially with externalﬁnancing when the output price X(t) exceeds the threshold x∗C (Y (t)). Proposition 3
ensures monotonicity in the lager-scale investment threshold x∗C (Y (t)) with respect tocash reserves Y (t). This monotonicity can be straightforwardly explained as follows. An
increase in Y (t) decreases a ﬁnancing cost C(I − Y (t)), which accelerates the large-scale
investment.
We now examine the properties of the small-scale investment regions, SC,1,E and SC,1,I .
Proposition 4 suggests the possibility that SC,1,E and SC,1,I are generated by costs ofexternal ﬁnancing. Proposition 5 describes the properties of SC,1,E and SC,1,I on thepresumption that they exist.
Proposition 4 (Possibility of the small-scale expansion)
If {(A2 − 1)/(A1 − 1)}β/(β−1) < I2 /I1 , SC,1,E may not be empty and SC,1,I ̸= ∅.
If I2 /I1 ≤ {(A2 − 1)/(A1 − 1)}β/(β−1) < {I2 + C(I2 − I1 )}/I1 , SC,1,E and SC,1,I may not
be empty.
Otherwise, SC,1,E = SC,1,I = ∅.
Proposition 5 (Small-scale expansion)
If (x, y) ∈ SC,1,E , (x, I1 ) ∈ SC,1,I and (x, y ′ ) ∈ SC,1,E holds for any y ′ ∈ [y, I1 ).
If (x, y) ∈ SC,1,I , (x, y ′ ) ∈ SC,1,I holds for any y ′ ∈ [I1 , y].
If {(A2 − 1)/(A1 − 1)}β/(β−1) < I2 /I1 , min{x ≥ 0 | (x, y) ∈ SC,1,I } = x∗1 holds for any
ﬁxed y ≥ I1 .
Proposition 5 shows that the small-scale investment region enhances with cash reserves
Y (t) until Y (t) = I1 and from the point it decreases with Y (t). This non-monotonic
property is in sharp contrast with the monotonicity of the large-scale investment region
proved in Proposition 3. Below, we explain the interesting result in terms of the investment
sizing choice changing with Y (t).
[Insert Figure 1 about here.]10Figure 1 illustrates how the ratio of the total cost associated with the large-scale expansion to that of the small-scale expansion, {I2 + C max(I2 − Y (t), 0)}/{I1 + C max(I1 −Y (t), 0)}, changes with Y (t). Note that the ratio regarding the proﬁt expansion, (A2 −1)/(A1 − 1), is independent of Y (t). The ratio is equal to I2 /I1 for Y (t) = 0 because external funds cover entire investment costs in both projects. For Y (t) ∈ (0, I1 ], the internalfunds cover Y (t) out of I1 and I2 in the small- and large-scale projects, respectively. The
coverage ratio in the small-scale project, Y (t)/I1 , increases more with Y (t) than that of thelarge-scale project. Then, the ratio {I2 + C max(I2 − Y (t), 0)}/{I1 + C max(I1 − Y (t), 0)}monotonically increases up to the maximum level {I2 + C(I2 − I1 )}/I1 for Y (t) = I1 .
This increase leads to the result that a higher Y (t) encourages the ﬁrm to invest in thesmall-scale project until Y (t) = I1 . For Y (t) ∈ [I1 , I2 ), the small-scale project requires
no external ﬁnancing, while the large-scale project requires the external funds I2 − Y (t).
An increase in Y (t) decreases the total cost associated with the large-scale investment,
preserving the small-scale investment cost unchanged. Then, the ratio {I2 + C max(I2 −Y (t), 0)}/{I1 + C max(I1 − Y (t), 0)} monotonically falls to the minimum level I2 /I1 for
Y (t) = I2 . This decrease leads to the novel result that a higher Y (t) discourages the ﬁrm
to invest in the small-scale project within the region Y (t) ∈ [I1 , I2 ].
As explained by the mechanism above, the possibility of the small-scale expansion
is maximized at the point Y (t) = I1 . Proposition 4 states that, if the maximum ratio{I2 + C(I2 − I1 )}/I1 is smaller than the critical level {(A2 − 1)/(A1 − 1)}β/(β−1) , the ﬁrm
never undertakes the small-scale investment. Now, compare this result with that of thecase with no ﬁnancing costs. In the absence of ﬁnancing costs, the ﬁrm never undertakes
the small-scale investment if I2 /I1 < {(A2 − 1)/(A1 − 1)}β/(β−1) is satisﬁed (see Section
2.2). In the presence of ﬁnancing costs, on the other hand, the ﬁrm has a possibility ofinvesting in the small-scale project in the region I2 /I1 ≤ {(A2 − 1)/(A1 − 1)}β/(β−1) <{I2 + C(I2 − I1 )}/I1 . This demonstrates that costs of external ﬁnancing can trigger thesmall-scale investment which is never undertaken in the case with no ﬁnancing costs. This
also suggests the counter-intuitive eﬀect that ﬁnancing costs may speed up investment (butit is small-scale).
We can prove the following proposition regarding the possibility of external ﬁnancing.
Proposition 6 (Possibility of external ﬁnancing)
If C < A1 − 1, SC,1,E may not be empty and SC,2,E ̸= ∅.
If A1 − 1 ≤ C < A2 − 1, SC,1,E = ∅ and SC,2,E ̸= ∅.
Otherwise, SC,1,E = SC,2,E = ∅.
It is clear that high ﬁnancing costs prevent a ﬁrm from ﬁnancing the project with external funds. Further, and more interestingly, this proposition suggests that the ﬁrm ismore likely to rely on external funds in the large-scale expansion than in the small-scale
expansion. To see this, suppose that the cost of external ﬁnancing is intermediate, i.e.,
A1 − 1 ≤ C < A2 − 1. In this case, the small-scale project is always deferred until the11project can be ﬁnanced entirely with internal funds. On the other hand, the large-scale
project is undertaken partially with external ﬁnancing when X(t) reaches a suﬃciently
high level. The reasoning is as follows. In the region Y (t) < I1 , the ﬁrm receives costsavings of CX(t)dt + r(1 + C)I1 dt and loses (A1 − 1)X(t)dt by deferring the small-scale
investment by an inﬁnitesimally short period dt. Then, the ﬁrm never ﬁnances the smallscale project with external funds for any X(t) if C ≥ A1 − 1 is satisﬁed. In the case of thelarge-scale project, the positive eﬀect CX(t)dt remains unchanged,5 while the negativeeﬀect is enlarged to (A2 − 1)X(t)dt. This increases the ﬁrm’s incentive to access externalﬁnancing in the large-scale expansion.

3.2

Empirical implications

In this subsection, we provide empirical implications obtained from the propositions in
Section 3.1 and clarify our contributions to the literature.
[Insert Figure 2 about here.]
Figure 2 summarizes the properties of the ﬁnancing and investment policy in Propositions 3–6.6 The ﬁrst, second, and third rows correspond to the cases with high, intermediate, and low cost of external ﬁnancing, respectively. The ﬁrst, second, and thirdcolumns correspond to the cases with high, intermediate, and low values of I2 /I1 (or(A1 − 1)/(A2 − 1)). In the hatched regions the ﬁrm invests entirely with internal ﬁnancing, while in the other regions it relies partially on external ﬁnancing. The ﬁgure also
indicates the investment size the ﬁrm chooses in the exercise region.
When I2 /I1 (or (A1 −1)/(A2 −1)) is suﬃciently low, i.e., in the ﬁrst column, the ﬁnancing and investment policy is the same as that of the case with only the large-scale project(ﬁxed-scale investment model). This case is studied in details by Nishihara and Shibata(2011), and, hence, we omit the explanation. The second and third columns present moreinteresting panels, in which the ﬁrm optimizes the investment size as well as investment
timing. The panels show that costs of external ﬁnancing bring many diﬀerences from thecase with no ﬁnancing constraint in Section 2.2 (refer also to Décamps, Mariotti, andVilleneuve (2006)). Among all, a key diﬀerence is regarding the necessity of dynamicallydeciding the investment size. Except for Panel (ix) the ﬁrm cannot determine the investment size at the initial time even if the initial value X(0) = x is low. The ﬁrm invests ineither the small- or large-scale project depending on the dynamics of (X(t), Y (t)). Recall
5
6

The term r(1 + C)I2 dt is negligible when X(t) is high.
In all ﬁgures in this paper, we set the axes in the same way as Boyle and Guthrie (2003), Hirth and Uhrig-

Homburg (2010b), and Nishihara and Shibata (2011) for comparison. This ﬁgure illustrates the cases in whichx∗2 (or x∗22 ) in problem V0 (x) is larger than (C + 1)rI2 /(A2 − C − 1) in Proposition 3. Otherwise, there is a gap
between limy↑I2 x∗C (y) and x∗2 (or x∗22 ) for Y (t) = I2 . The characteristics remain unchanged except for the gap
at the point Y (t) = I2 .

12that in Section 2.2 the dynamic choice is not necessary for a suﬃciently low X(0) = x.
Our results complement Dixit (1993) and Décamps, Mariotti, and Villeneuve (2006) byrevealing the signiﬁcance of the ﬁrm’s dynamic sizing choice in the presence of costs ofexternal ﬁnancing.
Next, we clarify a signiﬁcant contribution to the corporate ﬁnance literature. In corporate ﬁnance, there have been long-term arguments about sensitivities of investment tocash ﬂow since seminal works by Fazzari, Hubbard, and Petersen (1988), Kaplan andZingales (1997), and Hubbard (1998). Recently, several empirical studies regarding thisissue have demonstrated a counter-intuitive result that the investment volume does notnecessarily decrease with a degree of ﬁnancing constraints. Speciﬁcally, Cleary, Povel, andRaith (2007) and Guariglia (2008) documented a U-shaped relation between the investment volume and internal funds. This suggests that an intermediate level of ﬁnancing
constraint can lead to underinvestment.
On the other hand, a recent stream of real options literature has provided another prediction: an intermediate level of ﬁnancing constraint can lead a ﬁrm to hasten investment.
Boyle and Guthrie (2003) showed that the investment threshold has a U-shaped relation
with a degree of a liquidity constraint because of a ﬁrm’s incentive to avoid the risk ofa cash shortfall. Shibata and Nishihara (2012) showed that a ﬁrm’s consideration of the
optimal capital structure leads to a U-shaped relation between the investment threshold
and a debt capacity constraint. If “hastened” investment is equal to “increased” investment, the prediction from the real options literature is contrary to the empirical evidence.
However, this argument is not precise because the investment timing studies consider
ﬁxed-scale investment models. Indeed, by integrating both investment timing and sizing
problems, we can demonstrate that the two results do not conﬂict but complement eachother.
[Insert Figure 3 about here.]
To see this clearly, we concentrate on the most interesting case, Panel (viii) (or (v))
in Figure 2.7 Figure 3 illustrates the minimum threshold price in which the ﬁrm invests
and the project that is chosen at the threshold in this case. This ﬁgure indicates thatthe ﬁrm with low or high cash reserves undertakes the large-scale project on the later
timing, while, with intermediate cash reserves (Y (t) ≈ I1 ), it invests in the small-scale
project on the earlier timing. An intermediate level of cash holdings can play both rolesin accelerating the investment time and decreasing the investment size. In the presenceof the cost of external ﬁnancing, as explained with Figure 1, an intermediate level ofcash reserves provides the greatest incentive for the ﬁrm to undertake the small-scale
expansion rather than the large-scale expansion. Then, from the viewpoint of investment
7

In other cases, our model can also explain the conventional results, i.e., the monotonic relations regarding

the investment volume and timing. In the real options literature, Milne and Robertson (1996), Nishihara andShibata (2010), and Hirth and Uhrig-Homburg (2010a) are in line with the monotonicity.

13

size, the investment is scaled down from the ﬁrst-best level of a case with no ﬁnancing
cost. At the same time, from the viewpoint of investment timing, the investment takes
place earlier than the ﬁrst-best timing because the small-scale investment has a lower
value of the deferring option than that of the large-scale investment. Unlike Boyle andGuthrie (2003) and Shibata and Nishihara (2012), by the interaction of ﬁnancing costs,
investment timing, and investment size, we can explain both non-monotonic relations
regarding the investment volume and timing. This mechanism ﬁlls the gap between theinvestment volume and timing literature.
Lastly, we explain a contribution to the literature from a technical viewpoint. This
paper, unlike most of the related papers relying on numerical simulations, attains analytic
results. Although some of the techniques used in the proof are inspired by the mathematical ﬁnance literature (e.g., Broadie and Detemple (1997), Detemple (2006), Bobtcheﬀ andVilleneuve (2010)), several developments are attained by this paper.8 While the mathematical ﬁnance studies investigated the exercise regions of American options that involvea multi-dimensional geometric Brownian motion, the stochastic process Y (t) in our model
is not a geometric Brownian motion; instead, it is deﬁned by (2). Furthermore, the payoﬀ function of problem (8) is not convex, which makes the proofs more diﬃcult. These
technical developments can be potentially applied to a variety of investment timing andsizing problems.

4

Numerical examples

As mentioned in Section 1, the main contribution of this paper is to attain analytic results
regarding the ﬁrm’s optimal ﬁnancing and investment policy in Section 3. We supplement
the results from two aspects in numerical examples. Section 4.1 presents the comparative
statics results with respect to the output price volatility σ. Section 4.2 we present theresults in a case with a ﬁxed cost in addition to a proportional cost of external ﬁnancing.

4.1

Base case

We explore the eﬀects of output price uncertainty to the ﬁnancing and investment policy.
The base parameter values are set as follows:r = 0.07, µ = 0.03, σ = 0.2, I1 = 50, I2 = 100, A1 = 1.385, A2 = 1.5, C = 0.1.

(13)

For comparison, we set the parameter values for the large-scale expansion at the samevalues as Nishihara and Shibata (2011). These parameter values are also similar to thoseof the standard real options literature.
8

Nishihara and Shibata (2011) developed similar techniques to prove the properties of the optimal ﬁnancing

and investment policy in the presence of a proportional cost of external ﬁnancing, but they assumed a ﬁxed
investment size.

14

For the base parameter values (13), we have {(A2 − 1)/(A1 − 1)}β/(β−1) = 1.9569 <I2 /I1 = 2 regarding the possibility of the small-scale expansion (Proposition 4). Thismeans that the ﬁrm may undertake the small-scale investment. We also have C = 0.1 <A1 − 1 = 0.285 regarding the possibility of external ﬁnancing (Proposition 6). This meansthat the projects may be ﬁnanced partially with external ﬁnancing. In the computation,
we make a tri-nomial lattice model that approximates to a geometric Brownian motion(1), and we use a value function iteration algorithm.
[Insert Figure 4 about here.]
Figure 4 plots the exercise regions with varying levels of σ. We can see from Figure 4that a higher σ decreases the exercise regions, which implies that the investment threshold
and the option value enhance with σ. This volatility eﬀect is straightforwardly consistent
with the standard results with neither ﬁnancing costs nor sizing choice.
Further and more notably, Figure 4 demonstrates that the volatility eﬀect is stronger
for the small-scale investment than for the large-scale investment. As σ increases up
to 0.2, the region of the small-scale investment partially with external ﬁnancing, SC,1,E ,disappears ﬁrst. As σ increases up to 0.25, the region of the small-scale investment entirely
with internal ﬁnancing, SC,1,I , disappears either. Then, for σ = 0.25 and 0.3, the ﬁnancing
and investment policy is the same as that of a case in which the investment size is ﬁxed
at the large scale. The rationale is based on Proposition 4. Indeed, we have {(A2 −1)/(A1 − 1)}β/(β−1) < I2 /I1 for σ = 0.1, 0.15, 0.2, I2 /I1 ≤ {(A2 − 1)/(A1 − 1)}β/(β−1) <{I2 + C(I2 − I1 )}/I1 for σ = 0.213, and {I2 + C(I2 − I1 )}/I1 ≤ {(A2 − 1)/(A1 − 1)}β/(β−1)
for σ = 0.25, 0.3. Note that β/(β − 1) monotonically increases with σ. As a result, theﬁnancing and investment policy becomes like panel (ix) → panel (vi) → panel (v) → panel(iv) in Figure 2 with an increase in σ. A higher σ, like a higher I2 /I1 or (A1 − 1)/(A2 − 1),increases the advantage of the large-scale expansion over the small-scale expansion. Itshould be also noted that the panel of σ = 0.213 corresponds to the most interesting case,panel (v) in Figure 2 (or Figure 3).

4.2

Extension to a case with fixed and proportional costs

So far we have concentrated on a proportional cost of external ﬁnancing because a proportional cost accounts for the greatest part of ﬁnancing (e.g., Hennessy and Whited (2007)).
However, according to Hennessy and Whited (2007), the eﬀects of a ﬁxed cost may beobservable for small ﬁrms. To check the robustness, we present a case with both ﬁxed
and proportional costs of external ﬁnancing.
[Insert Figure 5 about here.]
Figure 5 illustrates the exercise regions in the extended case with varying levels of σ. In
the numerical examples, the ﬁxed cost is set at 1. This means that the total ﬁnancing cost15

is equal to 0.1 max(Ii − Y (t), 0) + 1{Ii −Y (t)>0} , where 1{·} denotes the indicator function.
For Y (t) ↑ Ii , a ﬁxed cost provides the greatest incentive for the ﬁrm to wait and invest
entirely with internal funds. Then, the ﬁrm never invests when Y (t) reach nearly I1 =50 or I2 = 100, which leads to disconnected exercise regions. This ﬁxed cost eﬀect isdocumented in Nishihara and Shibata (2011) who examined the ﬁxed cost eﬀects in aﬁxed-scale investment model. Furthermore, we can see from Figure 5 that the ﬁxed costeﬀect is strong particularly in the small-scale project. Indeed, compared to Figure 4,
Figure 5 has no small-scale investment region partially with external ﬁnancing, SC,1,E ,at all. The presence of a ﬁxed cost removes the possibility of investing in the smallscale project partially with external ﬁnancing. Figure 5 inherits the other characteristics
from Figure 4. Most notably, the panel of σ = 0.213 still holds an island-like small-scale
investment region even if a ﬁxed cost is considered.

5

Conclusion

This paper examined the interactions of ﬁnancing costs, investment timing, and investment sizing in a dynamic model. We assumed that a ﬁrm can invest in either a smallor large-scale project with cash reserves, which are increasing with time, and externalfunds that require a proportional cost. We, unlike most of the related papers, analytically
proved the properties of the corporate ﬁnancing and investment policy. The results are
summarized as follows.
Financing costs lead to the ﬁnancing and investment policy strongly contingent on the
dynamics of both the cash ﬂow and reserves. In particular, in the presence of ﬁnancing
costs, the ﬁrm is more likely to invest in the small-scale project entirely using internalfunds. The investment threshold for the large-scale project monotonically decreases withcash reserves, while the investment region for the small-scale project has a non-monotonic
relation with cash reserves. Most notably, an intermediate level of cash reserves can playboth roles in speeding up the investment and in decreasing the investment size. Our
results can explain two types of results in corporate ﬁnance: (i) empirical ﬁndings thatthe investment volume has a U-shaped relation with internal funds, and (ii) empiricalpredictions that the investment threshold has a U-shaped relation with internal funds.

Acknowledgment
This work was supported by KAKENHI (22710146, 22710142, 23310103, 21241040).

16

A

Proof of Proposition 1

Since Y (t) monotonically increases from the initial value y, we haveIi − Y (t) ≤ max(Ii − Y (t), 0) ≤ max(Ii − y, 0)(i = 1, 2)

at any time t. This implies that VL (x, y) ≤ VC (x, y) ≤ VU (x, y) holds. The problem (8)can be reduced to the problem with no ﬁnancing cost when y ≥ I2 .

B

Proof of Lemma 1

First, we prove (9). Note that E1 [·] represents the expectation with X(t) starting from
X(0) = 1. For any positive constant ∆, we have
VC (x + ∆, y)()∫ τ(Ai − 1)(x + ∆)= sup E1 [e−rτ max
X(τ ) − Ii − C max(Ii − erτ y −er(τ −s) (x + ∆)X(s)ds, 0) ]
i=1,2
r−µ
τ ∈T
0
(
(
)
∫ τ
(A
−
1)x
i
1 −rτ
rτ
r(τ −s)≤ sup E [emaxX(τ ) − Ii − C max(Ii − e y −exX(s)ds, 0)
i=1,2
r−µ
τ ∈T
0
)
∫ τ
(A2 − 1)∆r(τ −s)+X(τ ) + C∆e
X(s)ds ]
(14)
r−µ
0
)
(
∫ τ
(Ai − 1)xX(τ ) − Ii − C max(I − erτ y −er(τ −s) xX(s)ds, 0) ]≤ sup E1 [e−rτ max
i=1,2
r−µ
τ ∈T
0
|
{z
}
=VC (x,y)∫ τr(τ −s))(A2 − 1)∆e
X(s)ds ]+ sup E [eX(τ ) + C∆
r−µ
τ ∈T
0
(
)
(∫ ∞
)
∫ ∞
(A2 − 1)∆
1 −rτ
−rs
−rs
=VC (x, y) + sup E [eX(τ ) + C∆
e X(s)ds −e X(s)ds ]
r−µ
τ ∈T
0
τ
(A2 − 1 − C)∆C∆=VC (x, y) ++sup E1 [e−rτ X(τ )]
(15)
r−µ
r−µ
τ ∈T
|
{z
}
1

−rτ

(

=1

=VC (x, y) +(A2 − 1)∆,r−µwhere (14) follows from A1 < A2 , and in (15) supτ ∈T E1 [e−rτ X(τ )] = 1 follows fromµ < r.

17

Next, we prove (10). For any positive constant ∆, we have
VC (x, y + ∆))∫ τ
(Ai − 1)rτr(τ −s)= sup E [emaxX(τ ) − Ii − C max(Ii − e (y + ∆ −eX(s)ds, 0) ]
i=1,2
r−µ
τ ∈T
0
(
(
)
)
∫ τ
(Ai − 1)x −rτrτr(τ −s)rτ
≤ sup E [emaxX(τ ) − Ii − C max(Ii − e y −eX(s)ds, 0) + e C∆ ]
i=1,2
r−µ
τ ∈T
0
(
)
∫ τ
(Ai − 1)x −rτrτr(τ −s)= sup E [emaxX(τ ) − Ii − C max(Ii − e y −eX(s)ds, 0) ] +C∆
i=1,2
r−µ
τ ∈T
0
|
{z
}
x

−rτ

(

=VC (x,y)=VC (x, y) + C∆.

C

Proof of Proposition 2

Note that Y (t) monotonically increases from the initial value y. Once Y (t) reached I2 ,VC (x, y) can be reduced to V0 (x, y). Then, the exercise region agrees with the exercise
region for the benchmark problem V0 (x, y) which is explicitly obtained in Section 2.2.

D

Proof of Proposition 3

First, consider the case of A2 − C − 1 > 0. Fix (x, y) ∈ SC,2,E and (x′ , y ′ ) satisfying x ≤ x′
and y ≤ y ′ < I2 . Using Lemma 1, we have
VC (x′ , y ′ ) = VC (x′ , y ′ ) − VC (x, y ′ ) + VC (x, y ′ ) − VC (x, y) + VC (x, y)(A2 − 1)(x′ − x)
+ C(y ′ − y) + VC (x, y)r−µ(A2 − 1)(x′ − x)
(A2 − 1)x=
+ C(y ′ − y) +− I2 − C(I2 − y)r−µr−µ(A2 − 1)x′=− I2 − C(I2 − y ′ ),r−µ≤where the last inequality implies (x′ , y ′ ) ∈ SC,2,E . This proves that SC,2,E is expressed as(12) with the decreasing function x∗C (·). By Proposition 1, we can immediately show thatx∗C (y) is between the corresponding thresholds in VL (x, y) and VU (x, y).
Next, we derive limy↑I2 x∗C (y). Clearly we have limy↑I2 x∗C (y) ≥ x∗2 . Here we explain
the case of {(A2 − 1)/(A1 − 1)}β/(β−1) ≥ I2 /I1 . Replace x∗2 with x∗22 in the case of{(A2 − 1)/(A1 − 1)}β/(β−1) < I2 /I1 . Denote
f2 (x, y) :=(A2 − 1)x− I2 − C(I2 − y).
r−µ

18

(16)

We have
Lf2 (x, y) − rf2 (x, y) ≤ 0(A2 − 1)µx⇔
+ C(x + ry) − r
r−µ(C + 1)rI2⇔x ≥,A2 − C − 1((A2 − 1)x− I2 − C(I2 − y)
r−µ

)
≤0
(17)

where L denotes the generating operator of (X(t), Y (t)), i.e.,
L := µx∂1∂2∂+ σ 2 x2 2 + (x + ry) .
∂x 2
∂x
∂y

(18)

Since the general theory of optimal stopping ensures LVC (x, y) − rVC (x, y) ≤ 0 (refer
to Peskir and Shiryaev (2006)), (17) implies that f2 (x, y) is not equal to VC (x, y) forx < (C + 1)rI2 /(A2 − C − 1) and y < I2 . In other words, the option is not exercisedin the region {(x, y) ∈ R2+ | x < (C + 1)rI2 /(A2 − C − 1), y < I2 }. This proves thatlimy↑I2 x∗C (y) ≥ (C + 1)rI2 /(A2 − C − 1).
[Insert Figure 6 about here.]Now, suppose that (C + 1)rI2 /(A2 − C − 1) ≤ x∗2 < limy↑I2 x∗C (y). See Figure 6. We
can lead to contradiction as follows. Consider problem
supτ ∈T ,τ ≤T

Ex,y [e−rτ f2 (X(τ ), Y (τ ))](19)

with a ﬁnite maturity T . Generally, the exercise region of an American option converges
to the region Lf2 − rf2 ≤ 0, when the remaining life of the option goes to zero (refer
to Detemple (2006)). Then, because of (17), the exercise region of problem (19) with aﬁnite maturity T converges to {(x, y) ∈ R2+ | x ≥ (C + 1)rI2 /(A2 − C − 1)} when T ↓ 0.
Consider the exercise region of problem (8) for a ﬁxed x satisfying x∗2 < x < limy↑I2 x∗C (y)and y ↑ I2 . Note that inf{t ≥ 0 | X(t) ≥ x∗2 , Y (t) ≥ I2 } converges to 0 as y ↑ I2 .
Accordingly, the exercise region of the problem (8) for the ﬁxed x and y ↑ I2 converges tothat of problem (19) with T ↓ 0. This implies that limy↑I2 x∗C (y) = (C +1)rI2 /(A2 −C −1),
which contradicts the assumption of (C + 1)rI2 /(A2 − C − 1) < limy↑I2 x∗C (y). Similarly
we can lead to contradiction if x∗2 < (C + 1)rI2 /(A2 − C − 1) < limy↑I2 x∗C (y) is supposed.
Thus, we have limy↑I2 x∗C (y) = max(x∗2 , (C + 1)rI2 /(A2 − C − 1)).
We can show the continuity of x∗C (·) as follows. By Lemma 1 we have the continuity ofVC (x, y). Since VC (x, y) and (A2 − 1)x/r − µ − I2 − C max(I2 − y, 0) are both continuous,
SC,2,E ∪ SC,2,I is a closed set. Consider any ﬁxed y(< I2 ). Then, we have limϵ↓0 (x∗C (y +ϵ), y + ϵ) ∈ SC,2,E , which leads to limϵ↓0 x∗C (y + ϵ) ≥ x∗C (y). We have limϵ↓0 x∗C (y + ϵ) ≤x∗C (y) because x∗C (·) is a decreasing function. Thus, we obtain the right-continuity ofx∗C (·). Now, suppose that there exists y(< I2 ) satisfying x∗C (y) < limϵ↓0 x∗C (y − ϵ). We canlead to contradiction as the same method as the proof of limy↑I2 x∗C (y) = max(x∗2 , (C +1)rI2 /(A2 − C − 1)). Consider the exercise region of problem (19) for a ﬁxed x satisfying

19x∗C (y) < x < limϵ↓0 x∗C (y − ϵ) and y − ϵ. Note that inf{t ≥ 0 | X(t) ≥ x∗C (Y (t))}
converges to 0 as ϵ ↓ 0. Then, the exercise region converges to that of problem (19)with T ↓ 0. This implies limϵ↓0 x∗C (y − ϵ) = (C + 1)rI2 /(A2 − C − 1), which contradicts(C + 1)rI2 /(A2 − C − 1) ≤ x∗C (y) < limϵ↓0 x∗C (y − ϵ). Thus, we obtain the left-continuityof x∗C (·).
Lastly, consider the case of A2 − C − 1 ≤ 0. In this case, we have for any (x, y) ∈ R2+
Lf2 (x, y) − rf2 (x, y) = −(A2 − C − 1)x + (C + 1)rI2 > 0,
where L is the generating operator deﬁned by (18) and f2 (x, y) is deﬁned by (16). Since
LVC (x, y) − rVC (x, y) ≤ 0 follows from the general theory of optimal stopping, f2 (x, y)does not agree with VC (x, y). This implies SC,2,E = ∅. The proof is completed.

E

Proof of Proposition 4Note that VC (x, y) can be reduced to V0 (x, y) for y ≥ I2 . In the case of {(A2 − 1)/(A1 −1)}β/(β−1) < I2 /I1 , the problem V0 (x, y) has the small-scale expansion region [x∗1 , x∗21 ](recall Section 2.2). Then, we have [x∗1 , x∗21 ] × [I2 , ∞) ⊂ SC,1,I , which implies that SC,1,I ̸=∅.
Next, consider the case of {(A2 − 1)/(A1 − 1)}β/(β−1) ≥ (I2 + C(I2 − I1 ))/I1 . By
Proposition 1, we have VC (x, y) ≥ VL (x, y). Since we have() β
A2 − 1 β−1I2 + C(I2 − I1 )≥A1 − 1I1
I2 + C max(I2 − y, 0)≥
I1 + C max(I1 − y, 0)
for any y, the problem VL (x, y) has no small-scale expansion region (cf. Section 2.2). Thisimplies that

(A1 − 1)x− I1 − C max(I1 − y, 0).
r−µBy (20) and VC (x, y) ≥ VL (x, y), we have SC,1,E = SC,1,I = ∅.
VL (x, y) >(20)

Note that, in the case of I2 /I1 ≤ {(A2 − 1)/(A1 − 1)}β/(β−1) < (I2 + C(I2 − I1 ))/I1 ,We have no clear results, but, as seen in the argument above, SC,1,E and SC,1,I may exist
within the region 0 ≤ y < I2 .

F

Proof of Proposition 5

Take any (x, y) ∈ SC,1,E and y ′ ∈ [y, I1 ). By (10) in Lemma 1, we have
VC (x, y ′ ) ≤ VC (x, y) + C(y ′ − y)
(A1 − 1)x− I1 − C(I1 − y) + C(y ′ − y)r−µ(A1 − 1)x=− I1 − C(I1 − y ′ ),
r−µ

=

20

where the last inequality implies (x, y) ∈ SC,1,E . Similarly, we have
VC (x, I1 ) ≤(A1 − 1)x− I1 ,
r−µwhich means (x, I1 ) ∈ SC,1,I .
Next, consider any (x, y) ∈ SC,1,I and y ′ ∈ [I1 , y). We have
VC (x, y ′ ) ≤ VC (x, y)=(A1 − 1)x− I1 ,
r−µwhere the last inequality implies (x, y) ∈ SC,1,I .
Lastly, we focus on the case of {(A2 − 1)/(A1 − 1)}β/(β−1) < I2 /I1 . In this case, itfollows from (4) thatVC (x, y) ≤ V0 (x)() ( )β(A1 − 1)x∗1
x
=
− I1
,
r−µ
x∗1

(21)

for x < x∗1 . Consider the problem VC (x, y) with a initial point (X(0), Y (0)) = (x, y)satisfying x < x∗1 and y ≥ I1 . For the problem, we can realize the right-hand side of (21)by the stopping time inf{t ≥ 0 | X(t) ≥ x∗1 }. Then, this threshold policy inf{t ≥ 0 |X(t) ≥ x∗1 } is optimal. In other words, min{x ≥ 0 | (x, y) ∈ SC,1,I } = x∗1 holds for any
ﬁxed y ≥ I1 .

G

Proof of Proposition 6

We have already proved the properties of SC,2,E in Proposition 3, and, hence, we focus
on the properties of SC,1,E below. In the same manner as the proof of Proposition 3, we
can show the properties. Deﬁne
f1 (x, y) :=(A1 − 1)x− I1 − C(I1 − y),r−µand consider the case of C ≥ A1 − 1. We have for any (x, y) ∈ R2+
Lf1 (x, y) − rf1 (x, y) = −(A1 − C − 1)x + (C + 1)rI1 > 0,
where L is the generating operator deﬁned by (18). Since LVC (x, y) − rVC (x, y) ≤ 0
follows from the general theory of optimal stopping, f1 (x, y) does not agree with VC (x, y)
for any (x, y) ∈ R2+ . This implies that SC,1,E = ∅. The proof is completed.

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23{‫ܫ‬ଶ+C(‫ܫ‬ଶ െ‫ܫ‬ଵ)}/‫ܫ‬ଵ‫ܫ‬ଶ /‫ܫ‬ଵ

‫ܫ‬ଵ

0

‫ܫ‬ଶ

Y(t)

Figure 1: The ratio {I2 +C max(I2 −Y (t), 0)}/{I1 +C max(I1 −Y (t), 0)}. The ratio is unimodaland has the maximum value {I2 + C(I2 − I1 )}/I1 for Y (t) = I1 .

24‫ܫ‬ଶ Ȁ‫ܫ‬ଵ or ሺ‫ܣ‬ଵ െͳሻȀሺ‫ܣ‬ଶെͳሻ
Low

IntermediateHighlarge-scale‫ݔ‬ଶ‫כ‬large-scale‫ݔ‬ଶ‫כ‬‫ܫ‬ଶ Y(t)0Panel (i)small-scale

‫ܫ‬ଵ

0

‫ܫ‬ଶ Y(t)Intermediate

Panel (iv)

‫כ‬
‫ݔ‬ଶଶ

small-scale

‫ܫ‬ଵ

0

small-scale

‫כ‬
‫ݔ‬ଶଵ
‫ݔ‬ଵ‫כ‬‫ܫ‬ଶ Y(t)‫ܫ‬ଵ0‫ܫ‬ଶ Y(t)Panel (vi)large-scalelarge-scale‫ݔ‬ଶ‫כ‬‫ܫ‬ଶ Y(t)large-scale

Panel (v)‫ݔ‬ଶ‫כ‬‫ܫ‬ଵ0Panel (iii)

large-scaleLow‫ܫ‬ଶ Y(t)large-scale‫ݔ‬ଶ‫כ‬0small-scale

‫כ‬
‫ݔ‬ଶଵ
‫ݔ‬ଵ‫כ‬Panel (ii)‫ݔ‬ଶ‫כ‬large-scale
‫כ‬
‫ݔ‬ଶଶ

large-scale

C

High

‫כ‬
‫ݔ‬ଶଶ

small-scale

small-scale

‫כ‬
‫ݔ‬ଶଵ
‫ݔ‬ଵ‫כ‬0Panel (vii)‫ܫ‬ଶ Y(t)0‫ܫ‬ଵPanel (viii)‫ܫ‬ଶ Y(t)0‫ܫ‬ଵ‫ܫ‬ଶ Y(t)Panel (ix)Figure 2: The exercise regions SC,1,E , SC,1,I , SC,2,E , and SC,2,I . This figure summarizes theproperties proved in Propositions 3–6. The first, second, and third rows correspond to the caseswith high, intermediate, and low cost of external financing, respectively. The first, second, andthird columns correspond to the cases with high, intermediate, and low values of I2 /I1 (or(A1 − 1)/(A2 − 1)). The hatched regions correspond to the entirely internal financing regions,
SC,1,I and SC,2,I , while the other regions correspond to the partially external financing regions,
SC,1,E and SC,2,E .

25

large-scale‫ݔ‬ଶ‫כ‬0small-scale

‫ܫ‬ଵ

‫ܫ‬ଶ

Y(t)

Figure 3: The minimum output price for which the firm invests. This figure corresponds toPanel (viii) in Figure 2.

26

50

50
45

σ=0.1

40

45
40

SC,2,E

SC,2,I

35
X(t)

X(t)

20

15

15
10
SC,1,E40

SC,1,I
60

80
Y(t)

5
100

45
SC,2,E

80
Y(t)

100

120

σ=0.213
SC,2,E40

SC,2,I35

SC,2,I

35
30
X(t)

30
X(t)

60

50
σ=0.2

40

25

25

20

20

15

15

10

10

SC,1,I

5
40

60

80
Y(t)

100

120

40

60

80
Y(t)

100

120

50

σ=0.2545
SC,2,E40

SC,1,I550
45SC,1,I

SC,1,E
40

120

50

σ=0.3
SC,2,E40

SC,2,I35

SC,2,I

35

30

30
X(t)

X(t)

25

20

45

SC,2,I30

25

5SC,2,E3530

10

σ=0.15

25

25

20

20

15

15

10

10

5

5
40

60

80
Y(t)

100

120

40

60

80
Y(t)

100

120

Figure 4: The comparative statics. The figure plots the exercise regions in the case with aproportional cost of external financing with varying levels of σ. The parameter values otherthan σ are set at the base case (13).

27

50

50
45

σ=0.1

45
40

40

SC,2,I

SC,2,E

35

30
X(t)

X(t)

25

20

15

15

10

10

SC,1,I

5
40

60

80
Y(t)

100

120

40

60

80
Y(t)

100

120

50
σ=0.2

45

40

SC,2,E

σ=0.213
SC,2,E40

SC,2,I35

SC,2,I

35
30
X(t)

30
X(t)

SC,1,I

5

50

25

25

20

20

15

15

10

10

SC,1,I

5
40

60

80
Y(t)

100

120

40

45
SC,2,E

SC,2,I

80
Y(t)

100

120

30

30
X(t)

35

25

20

15

15

10

10

5

5
80
Y(t)

100

SC,2,E

120

SC,2,I

25

20

60

σ=0.3

40

35

40

60

50

σ=0.2540

SC,1,I

5

50

X(t)

25

20

45

SC,2,I

SC,2,E353045

σ=0.15

40

60

80
Y(t)

100

120

Figure 5: The comparative statics. The figure plots the exercise regions in the case withboth fixed and proportional costs of external financing with varying levels of σ. The fixed
cost is set at 1, and the other parameter values are set at the base case (13). Note that
SC,2,E := {(x, y) ∈ R2+ | V (x, y) = (A2 − 1)x/(r − µ) − I2 − C(I2 − y) − 1, y < I2 } because of
the fixed cost.

28

X(t)

‫כ‬

‫ݔ‬஼ ሺܻ ሺ‫ ݐ‬ሻሻ

x
‫כ‬

‫ݔ‬ଶ

y ‫ܫ‬ଶ0Y(t)Figure 6: The assumption of x∗2 < limy↑I2 x∗C (y). The dot represents the initial point (x, y)satisfying x∗2 < x < limy↑I2 x∗C (y) and y ≈ I2 .

29