investment timing and capacity choice in duopolistic

Mathematics and Financial Economics (2022) 16:125–152https://doi.org/10.1007/s11579-021-00303-3

Investment timing and capacity choice in duopolisticcompetition under a jump-diffusion model

Xiaoqin Wu1 · Zhijun Hu2

Received: 25 August 2020 / Accepted: 3 August 2021 / Published online: 21 August 2021© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021

AbstractThis paper aims to apply the real options game theoretic to study the impact of sudden eventson the optimal investment timing and capacity choice in a duopoly market. We model themarket demand and investment cost as the geometric Brownian motions with jumps drivenby the Poisson processes. A new computing method independent on specific distributionfunctions is proposed for the real option models with jump processes of random frequencyand amplitude. Based on this method, we find that both firms delay investment with a largercapacity as uncertainties of demand and investment cost increase. We also demonstrate thattwo firms both invest later and the optimal capacity relationship between them is ambiguousin the presence of two sources of uncertainty. Numerical simulation reveals that upward(downward) jump in demand and downward (upward) jump in investment cost cause thefirms to invest earlier (later) with a larger (smaller) capacity. Finally, in a duopoly withsymmetric firms, the first investor invests earlier than in an asymmetric duopoly due to thethreat of preemption.

Keywords Duopoly market · Real option game · Jump-diffusion process · Investmentdecision

1 Introduction

Nowadays, uncertainty is a main characteristic of the business environment. When firms planto introduce a new product or a new technology to the market, two common and difficultdecisions are howmuch capacity to invest in andwhen to do it. The size of a firm’s investmentusually reflects the level of future production capacity of the firm, that is, the scale of projectoutput formed through investment, which then affects the quality of the firm’s investment.By investing with a large capacity, the firm takes a risk when demand and investment cost are

B Zhijun [email protected]

Xiaoqin [email protected]

1 School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

2 School of Management, Guizhou University, Guiyang 550025, China

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http://orcid.org/0000-0001-9690-5936

126 Mathematics and Financial Economics (2022) 16:125–152

uncertain. On the one hand, if the ex post demand is too low and investment cost is too large,the production capacity of the firm will be idle and the income may be too low to cover theinvestment cost. On the other hand, large-scale investment would enable the firm to capturea larger share of the market and make more money out of it if the realized demand is highand investment cost is low.

In the context of investment decisions under uncertainty, making “optimal” decisions maybe a challenging task due to the unknown impacts of uncertainties and the strategic interac-tions between firms. The real world of investment is characterized by strategic competitionamong competitors, where each firm evaluates its own strategic competitiveness among thecompeting parties. Therefore, it is interesting to analyze how strategic interactions in a com-petitive environment affect both the optimal investment timing and the investment capacitylevels under uncertain market conditions.

In this paper, we assume that there are two sources of uncertainty: market demand andinvestment cost. Thus, the firm needs to consider random fluctuations in income caused byuncertain demand and investment cost, and maximizes the value of firm by optimizing theinvestment timing and production capacity. Investment decision-making issues regardingthese two sources of uncertainty can be found in many literatures, such as [1–4]. However,these studies only discuss how a single firm determines the optimal timing of an investmentprojects of a given size.

In recent two decades, the stream of research for real options has been extended to invest-ments under competition by combining the game theoretic analysis with real options theory.The founding work on real options game theory was developed by Smets [5]. Dixit andPindyck [1] proposed a continuous-time duopolistic real options game model that simplifiesthe original approach of Smets [5]. Nielsen [6] analyzed a duopoly stochastic entry gamewith both positive and negative externalities. Weeds [7] studied a real options game modelwith R&D competition under a winner-takes-all patent system. Huisman and Kort [8] studiedinvestment strategy in a duopoly market with future availability of new technology. Siddiquiand Takashima [9] developed a sequential two-stage game-theoretic real options model forcapacity expansion investments under output price uncertainty. A detailed review of realoptions game-theoretic models can be found in Azevedo and Paxson [10].

Within a strategic real options framework, Huisman and Kort [11] firstly considered theinvestment decisions for both timing and capacity of two symmetric entrants in a newmarketand evaluated the impact of competition on social welfare. Lavrutich et al. [12] extendedthe duopoly model by considering the hidden competition of a third firm, and found that thefollower of the two is more eager to invest in order to avoid being squeezed out of the marketby the hidden competitor. Lavrutich [13] and Huberts et al. [14] also examined a duopolyinvestment game in which the firms choose the capacity and timing of their investment. Inparticular, Lavrutich [13] added an exit option to investigate entry and exit decisions undercapacity sizing, Huberts et al. [14] assumed that one of the firms already has some capacityin place.

The existing real-options literature on capacity-timing games usually assumes the pro-jected cash flow stream or product demand follows a geometric Brownian motion (GBM)process as in the Black and Scholes model, that is, the internal and external environment ofa firm changes within a predictable range. The advantage of using the GBM is that it leadsto a more simplified and closed-form solution which can be effectively analyzed for firm’sinvestment decision. Besides, under the GBM framework, investment decision rules are clearand intuitive. However, a large body of empirical and theoretical studies already showed thatthe GBM is not rich enough to capture the patterns commonly found in most financial andcommodity markets. For example, Mason [15] found that large sudden changes in the spot

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Mathematics and Financial Economics (2022) 16:125–152 127

price of Henry Hub natural gas during the period from January 2004 to January 2010. Nunesand Pimentel [4] found that the number of worldwide sales of iPhones from 3rd quarter 2007to 3 rd quarter 2016 do not fit GBM.

With the increasing complexity of current economic, social and environmental systems,unexpected sudden events such as financial crisis, plague (e.g., COVID-19 pandemic), andnatural disasters (e.g., hurricane or earthquake), equipment and technology updates, insti-tutional reform may lead to a sudden increase or decrease in demand and investment costfor certain products and cause major shifts in markets. Obviously, these shocks will greatlyaffect the investment decisions of firms. Considering that the involved processes may exhibitsample-path discontinuities, some real options literatures suggest using jump-diffusion pro-cess (e.g., [4,16,17]), in which discrete value changes are superimposed on the Brownianmotion. As pointed out by [1], a GBM process with discrete jumps is more realistic to modelthe cash flow stream of a firm’s project.

In the present study, we analyses the optimal investment decision of duopolistic firmsunder the two-factor jump-diffusion model with the first factor being the market demandand the second factor being the investment cost. The jumps may represent uncertaintiescaused by unexpected future events, including financial crisis, natural catastrophes, plague,technological innovation (e.g., [18]) and other sources. We develop a real option gamemodelto examine the effects of sudden events on firms’ optimal investment thresholds and optimalinvestment capacities.

In addition, most of the existing literatures assume that the amplitude of “jump” followsa specific probability distribution (e.g., [19–23]), and its limitation is that the analysis resultswill greatly depend on the assumption of this specific distribution. To avoid this limitation,we propose a new numerical method, which relaxes the assumption that the jump amplitudeis constant or follows a specific distribution by introducing several simple statistical charac-teristic parameters to replace the complex and unmeasurable probability density function.

By numerical analysis, we obtain the following conclusions: First, the optimal investmentthresholds of duopolistic firms under the two-factor diffusionmodel are larger than that of theHuisman & Kort model [11], whereas the optimal capacities cannot be compared. Second,under the two-factor jump-diffusion model, the upward jumps in demand and downwardjumps in investment cost have a positive impact on the optimal investment timing and capacitysize of the firms, whereas the downward jumps in demand and upward jumps in investmentcost have a negative impact. Finally, due to the threat of preemption, the two firms withsymmetric costs invest earlier than those with asymmetric costs, and the first investor (leader)enters the market earlier with a larger capacity than the second investor (follower). Therefore,the jump-diffusion model can incorporate different scopes of investment uncertainty andincrease the evaluation of investment opportunity.

The contribution of our paper can be seen from two angles. First, to the best of ourknowledge, this is the first paper to consider the optimal investment problem of duopolisticfirms with two sources of uncertainty. We extend the model considered by [11] to the jump-diffusion uncertainty, in which the stochastic demand and investment cost of project aredriven by jump-diffusion processes. Meanwhile, we extend the monopolistic firm’s timingdecision model of [4] to a duopolistic framework, and we further consider the firms’ optimalcapacity decision problem. Second, this paper proposes a new numerical solution methodfor the real option game model. An important improvement of this method is that underthe framework of preserving the assumption of random distribution of “jump” amplitude, itprovides a unified calculation rule for the valuation of real options with “jump”, which doesnot depend on its specific probability distribution. Based on this method, the influence ofsudden events on project investment is discussed.

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The remainder of this study is organized as follows. Section 2 describes the dynamics ofstochastic processes involved. Section 3 introducesmodel valuation framework and discussesthe monopoly investment decision problem. Section 4 studies the optimal investment timingand capacity decision in a duopolymarket with two sources of uncertainty. Section 5 providesnumerical analysis for the optimal investment decisions. We conclude in Sect. 6 and discusssomepotential extensions. The proofs of allmathematical statements in the paper are providedin Appendix. The appendix contains all proofs.

2 Basic setup and assumptions

In this section, we mainly introduce the mathematical model and associated assumptions toderive the firm’s investment decisions.We extend the approach used inHuisman andKort [11]to deal with two sources of uncertainty and we assume both market demand and investmentcost evolving according to jump-diffusion processes.

Generally speaking, stochastic process S = {St , t ≥ 0} is described by the jump-diffusionprocess, which includes the GBM (continuous process) and compound Poisson process (dis-crete process) as follows

dStSt−

= μdt + σdBt + d

( Nt∑i=1

Ui

), S0 = s0 > 0 (1)

where s0 is the initial value of the process,μ is the drift coefficient, and σ is the instantaneousvolatility (excluding the impact of jumps). {Bt , t ≥ 0} is a standard Brownian motion ona filtered probability space (�, F, {Ft }t≥0, P) satisfying the usual properties. {Nt , t ≥ 0}is a time-homogeneous Poisson process with intensity λ, and the value of Nt representsthe cumulative number of sudden events up to time t , that is, Nt is the number of suddenchanges in the stochastic process St till time t . And thus, dN = 1 with probability λdt anddN = 0 with probability 1 − λdt . The percentage of jumps are random variables, denoted

by{Uj}j≥1

i .i .d.∼ U , which means that though each jump is different, they are independentof each other and follow the same probability law. The notation St− indicates that wheneverthere is a jump, the value of the process before the jump is considered. Furthermore,

{Uj}j≥1

is independent of the processes dNt and dBt .The Eq. (1) implies that St follows a geometric Brownian motion, but over each time

interval dt there is a small probability that it will fall or go up to (1+Uj ) times as its originalvalue, then it will continue fluctuating until another event occurs [1]. Note that, if the suddenevent does not occur (i.e., either λ = 0, or U = 0, with probability one), then we obtain thestandard GBM [8–11].

Following Merton [20], the solution of Equation (1) is given by

St = s0 exp

[(μ − σ 2

2

)t + σ Bt

] Nt∏i=1

(1 +Ui ) (2)

Considering the investment problem, we assume that there are two sources of uncertainty.More specifically, we assume that both the exogenous demand shock X = {Xt , t ≥ 0}and unit investment cost (including the cost of operating per unit of production capacity)

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I = {It , t ≥ 0} follow jump-diffusion processes

dXt

Xt−= μXdt + σXdB

Xt + d

⎛⎝ N X

t∑i=1

UXi

⎞⎠ , X0 = x0 > 0, (3)

and

d ItIt−

= μI dt + σI d BIt + d

⎛⎝ N I

t∑i=1

U Ii

⎞⎠ , I0 = i0 > 0. (4)

We distinguish the two processes by using index X or I for each parameter and assumethat these processes are independent of each other, and the jump sizes of stochastic demandand unit investment cost are independent of each other. In particular, if there are no jumps inmarket demand and investment cost, i.e., λX = λI = 0, then processes (3) and (4) are calledtwo-factor diffusion processes.Ul (l = X , I ) take positive (negative) values indicates upward(downward) jumps. Nunes and Pimentel [4] assume that the jumps in demand process arenegative and the jumps in investment cost are positive. However, Nunes et al. [18] assume thatthe cost of investment decreases with technology innovation bymeans of decrease jumps. Letμ1 and μ2 represent the expected jump amplitudes in stochastic demand and unit investmentcost, respectively, i.e., E

(UX) = μ1, E

(U I) = μ2. Here, μ1 > 0(< 0) means an upward

(downward) jump in demand, and μ2 > 0(< 0) means an upward (downward) jump ininvestment cost. Following [4,18], we consider the two scenarios: (1) μ1 > 0, μ2 < 0, and(2) μ1 < 0, μ2 > 0.

The risk-neutral discount rate is given by a constant r , and in order to guarantee conver-gence of firm value so that the option is exercised within a finite period of time (e.g., [17]),we make the following assumption.

Assumption 1 r − μX − λXμ1 > 0 and r − μI − λIμ2 > 0.

3 Benchmark case: monopoly market

To better understand how the model works, we first discuss the benchmark case where amonopoly firm holds a single investment opportunity. Suppose a risk-neutral firm that isconsidering to undertake an irreversible investment to enter a market with uncertain demandand investment cost. The decision-making problem concerns both the investment timing andthe capacity size of the production plant. When the current level of market demand is lowand investment cost is high, the firm is idle, waiting for a better investment opportunity.Once the current levels of market demand and investment cost reach the optimal investmentthresholds, it makes the investment, the production starts immediately and the firm becomesactive on the market. If the firm invests in a plant with capacity Q, it will therefore have topay total investment cost It Q. We assume that the output price at time t is

Pt = Xt (1 − ηQt ), (5)

where Qt is total market output and η > 0 is a fixed price sensitivity parameter. Note that inthe multiplicative inverse demand function, if Qt ≤ 1/η holds, then the market is boundedabove and it corresponds to amarket with a limited number of potential customers. Following[11] and [13], we assume that after investing the firm always operates at full capacity.

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3.1 Investment timing and capacity decisions

The expected value of the firm can be stated as follows

J (x0, i0) =E(x0,i0)

[∫ ∞

τ

e−r t(Xt Q(1 − ηQ) − It Q

)dt

](6)

which indicates the value of the option when the firm executes it at time τ < +∞. Following[24], we refer to J as the performance criterion. Moreover, E(x0,i0) denotes the expectationoperator conditional on the available information at time 0 with (X0, I0) = (x0, i0), τ is themoment of investment, and Q is the acquired capacity level or quantity at time τ .

Since Brownian motion and Poisson processes are Markovian, it follows that both pro-cesses Xt and It are also Markovian. Using the strong Markov property, we can state andprove a result of the performance criterion.

Theorem 1 The performance criterion can be rewritten as

J (x, i, Q) = E(x,i)[e−rτG (Xτ , Iτ , Q)

], (7)

with

G(x, i, Q) = xQ(1 − ηQ)

mX− i Q

mI, (8)

where

mX = r − μX − λXμ1, mI = r − μI − λIμ2.

The goal of the monopoly firm is to maximize performance criterion with respect to theinvestment timing and capacity level, i.e., the firm solves the following optimal stoppingproblem

V (x, i) = supτ≥0,Q≥0

J (x, i, Q) = J τ∗(x, i), x, i ≥ 0,

where τ ∗ is called the optimal stopping time. According to Theorem 1, given the currentlevels of demand x , unit investment cost i and capacity Q, the profit of the monopoly firm is

G(x, i, Q) = xQ(1 − ηQ)

mX− i Q

mI.

Therefore, when the value of the monopoly firm, V (x, i), is equal to the profit that thefirm obtains by investing, G(x, i, Q), the firm should choose to invest immediately. Onthe contrary, if V (x, i) > G(x, i, Q), it is more profitable for the monopolist to delay itsinvestment. Intuitively, low investment cost and high market demand will prompt monopolistto invest immediately, while high investment cost and low market demand make monopolistpostpone investment.

Furthermore, by applying optimal stopping theory (see, [24]), we derive that V (x, i)satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:

1

2σ 2X x

2 ∂2V (x, i)

∂x2+ 1

2σ 2I i

2 ∂2V (x, i)

∂i2+ μX x

∂V (x, i)

∂x+ μI i

∂V (x, i)

∂i+ λX E[

V((

1 +UX)x, i)]

+ λI E[V(x,(1 +U I

)i)]

− (r + λX + λI ) V (x, i) = 0,(9)

where E[V((1 +UX )x, i

)]and E[V(x, (1 +U I )i

)]represent the expected value func-

tions when the jumps occur in market demand (UX ) and unit investment cost (U I ).

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Additionally, V (x, i) must satisfy the following boundary conditions:

V (x∗, i∗) = x∗Q(1 − ηQ)

mX− i∗Q

mI,

∂V (x, i)

∂x

∣∣∣∣x=x∗,i=i∗ = Q(1 − ηQ)

mX,

∂V (x, i)

∂i

∣∣∣∣x=x∗,i=i∗ = − Q

mI.

where (x∗, i∗) denote the trigger values in terms of demand and unit investment cost at theoptimal stopping time τ ∗.

Following [1], we can apply a variable transformation method to reduce the problem fromtwo-dimensional to one-dimensional. To this end, we let

G(x, i, Q) = i

[xQ(1 − ηQ)

imX− Q

mI

]= ig(y, Q), (10)

and assume that

V (x, i) = iv (x/i) = iv(y), (11)

where y = x/i represents the demand-to-cost ratio, v is a function to be determined. Substi-tuting (11) into (9) and it can be expressed as follows

1

2

(σ 2X + σ 2

I

)y2v′′(y) + (μX − μI ) yv

′(y) + λX E[v((

1 +UX)y)]

+

λI E

[(1 +U I

)v

((1 +U I

)−1y

)]− (r − μI + λX + λI ) v(y) = 0.

(12)

In order to solve Eq. (12), we need to deal with the two terms E

[v

((1 + UX )y

)]and

E[(1 + U I )v

((1 +U I )−1y

) ]. According to [25], we assume that λX = λI = 0, then the

general solution of Eq. (12) is of the form v(y) = Ayβ . Therefore, we assume

E[v((1 +UX )y

)]= E

[(1 +UX

)β]v(y), (13)

E

[(1 +U I

)v

((1 +U I

)−1y

)]= E

[(1 +U I

)1−β]

v(y). (14)

Substituting (13) and (14) into (12) yields

1

2

(σ 2X + σ 2

I

)y2v′′(y) + (μX − μI ) yv

′(y) + λX E

[(1 +UX

)β]

v(y) + λI E

[(1 +U I

)1−β]

v(y) − (r − μI + λX + λI ) v(y) = 0.

(15)

To solve Eq. (15), we derive a new numerical integrationmethod for arbitrary jumpmagni-tudesUX andU I in the Eq. (15) and with sufficiently high accuracy. FollowingWestman andHanson [26], we use the principle of Gauss-Statistics quadrature to replace the complex andunmeasurable probability density function of UX and U I with several statistical moments.For the continuous functions (1+UX )β and (1+U I )1−β , if the probability density functions

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ofUX andU I are ϕ(y1) andψ(y2), respectively, then E[(1 +UX )β

]and E[(1 +U I )1−β

]can be expanded using a two-node approximation as follows

E

[(1 +UX

)β] =∫Y1

(1 + y1)βϕ(y1)dy1 � a1 (1 + y11)

β + a2 (1 + y12)β ,

E

[(1 +U I

)1−β]

=∫Y2

(1 + y2)1−βψ(y2)dy2 � b1 (1 + y21)

1−β + b2 (1 + y22)1−β ,

where a1, a2, b1 and b2 are the weights, y11, y12 ∈ Y1 and y21, y22 ∈ Y2 are the nodes. Aftersome algebra, these parameters can be expressed as

a1 = 1

2

(1 + Sk1/2

√(Sk1/2)2 + 1

), a2 = 1

2

(1 − Sk1/2

√(Sk1/2)2 + 1

)

y11 = μ1 +(Sk12

−√

(Sk12

)2 + 1

)σ1, y12 = μ1 +

(Sk12

+√

(Sk12

)2 + 1

)σ1,

b1 = 1

2

(1 + Sk2/2

√(Sk2/2)2 + 1

), b2 = 1

2

(1 − Sk2/2

√(Sk2/2)2 + 1

)

y21 = μ2 +(Sk22

−√

(Sk22

)2 + 1

)σ2, y22 = μ2 +

(Sk22

+√

(Sk22

)2 + 1

)σ2,

where μ1, μ2 are the expected values of UX and U I respectively, μ1 means average actionstrength of jump in demand, andμ2 means average action strength of jump in unit investmentcost. Meanwhile, σ1 and σ2 are the standard deviations of UX and U I respectively, σ1represents the volatility of action strength of jump in demand, and σ2 represents the volatilityof action strength of jump in unit investment cost. Furthermore, Sk1 and Sk2 are, respectively,the skewnesses of UX and U I , Sk1 implies the action direction of jump in demand, and Sk2implies the action direction of jump in unit investment cost. If Sk1 > 0 (Sk2 < 0), it indicatesthat there is a higher probability that the demand (unit investment cost) jump will cause anabnormal increase in the project value; On the contrary, if Sk1 < 0 (Sk2 > 0), then thereis a higher probability that the demand (unit investment cost) jump will cause an abnormaldecrease in the project value.

Now, the weights and nodes are no longer depend on the the specific form of the dis-tributions of UX and U I , but only depend on three statistical moments: means, standarddeviations, and skewnesses. Following [26], after some simple manipulations, Eq. (15) canbe rewritten as

1

2

(σ 2X + σ 2

I

)y2v′′(y) + (μX − μI ) yv

′(y) + λX[a1(1 + y11)

β + a2(1 + y12)β]v(y)

+ λI[b1 (1 + y21)

1−β + b2 (1 + y22)1−β]v(y) − (r − μI + λX + λI ) v(y) = 0.

(16)

Eq. (16) is the Cauchy-Euler equation, whose solution is v(y) = A1yβ1 + A2yβ2 , whereβ1 and β2 are, respectively, the positive and negative roots of the following equation

1

2

(σ 2X + σ 2

I

)β(β − 1) + (μX − μI ) β + λX

[a1(1 + y11)

β + a2(1 + y12)β]

+λI[b1 (1 + y21)

1−β + b2 (1 + y22)1−β]− (r − μI + λX + λI ) = 0.

(17)

Since β2 < 0, if y → 0, then yβ2 → ∞, which implies that A2 = 0. Thus, the solutionof Eq. (16) must take the form v(y) = A1yβ1 .

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In this dynamic setting, the optimal investment decision is composed of two parts: timingand capacity. Let the trigger value y∗ = x∗/i∗ denotes a point at which the monopolistis indifferent between choosing to invest immediately or delay investment, and Q∗ is thecorresponding capacity level. Hence, we can describe the optimal investment timing as τ ∗ : =inf {t > 0 |yt ≥ y∗ }. Namely, the optimal investment timing is equal to the moment whenthe exogenous demand-to-cost ratio yt = xt/it first reaches y∗.

Combined with the boundary condition of Eq. (16), we can derive the optimal investmentstrategy and the value of firm as follows

Theorem 2 The value function of the monopoly firm is equal to

v(y) =

⎧⎪⎨⎪⎩A1yβ1 , y < y∗(mI y − mX )2

4yηmXm2I

, y ≥ y∗ , (18)

where

A1 = 1

(β12 − 1)mIη

[(β1 + 1)mX

(β1 − 1)mI

]−β1

, (19)

and β1 is the positive root of Eq. (17).The optimal investment threshold y∗ and the corresponding capacity level Q∗ are given

by

y∗ = (β1 + 1)mX

(β1 − 1)mI, (20)

Q∗ = 1

(β1 + 1)η. (21)

3.2 Consumer surplus and social welfare

To investigate the impacts of investment timing and size on social welfare, we first evaluatethe consumer surplus. Following [11], given the demand shock X and the capacity Q of thefirm, the instantaneous consumer surplus is equal to∫ X

P(Q)

D(P)dP = 1

2ηXQ2, (22)

where P(Q) = X(1− ηQ) and D(P) = 1

η

(1 − P

X

). The total expected consumer surplus

(CS) is

CS(x, Q) = E

[∫ ∞

t

1

2XsQ

2ηe−r(s−t)ds

∣∣∣∣Xt = x

]= xηQ2

2mX. (23)

The expected producer surplus (PS) corresponds to the monopolist’s value is

PS(x, i, Q) = xQ(1 − ηQ)

mX− i Q

mI. (24)

Now, we can calculate the total expected social welfare (TS) as their sum:

T S(x, i, Q) = CS(x, Q) + PS(x, i, Q) = xQ(2 − ηQ)

2mX− i Q

mI. (25)

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Let

T S(x, i, Q) = i SW (y, Q) = i

(yQ(2 − ηQ)

2mX− Q

mI

), (26)

Substituting (20) and (21) into (26), we can obtain that at the moment of investment, theexpected social welfare is given by

SW (y∗, Q∗) = 3

2(β1 + 1)(β1 − 1)ηmI. (27)

On the other hand, the investment threshold and size that maximize social welfare in amonopoly market are

y∗W = (β1 + 1)mX

(β1 − 1)mI= y∗, (28)

Q∗W = 2

(β1 + 1)η= 2Q∗. (29)

This result is the same as in Huisman and Kort [11], that is, the welfare-maximizinginvestment threshold is consistent with the optimal investment threshold of monopoly firm,while the welfare-maximizing capacity is twice that of the monopolist’s optimal capacity.

In addition, the expected social welfare with a welfare-maximizing policy at the momentof investment is given by

SW (y∗W , Q∗

W ) = 2

(β1 + 1)(β1 − 1)mIη. (30)

Therefore, in a monopoly market, the loss of welfare at the moment of investment equals

SWL ≡ SW (y∗W , Q∗

W ) − SW (y∗, Q∗) = 1

2(β1 + 1)(β1 − 1)mIη. (31)

4 Duopoly market

Now, we investigate a duopoly market in which two firms compete with homogeneous goodsor serve a particular demand. Both firms are assumed to be risk neutral and profit maximizerswith constant time discount rate r . Further, the two firms have the option to wait for theiroptimal timing to enter the market. Similar to literature [11,27,28], we call the firm thatenters the market first as the leader, and the other as the follower. Denote by QL and QF

the investment sizes of the leader and the follower, respectively. When both firms are activein the market, the total market output is equal to Q = QL + QF . Following [11–13], weassume that the firms invest that always produce at full capacity and sell all their products tothe market.

To provide asmuch intuition as possible, we analyze two situations inwhich two firms facethe same or different investment costs. Several causes can lead to a situation that investmentcosts are heterogeneous among firms. For instance, the firms may have different accesses tocapital markets, or show different degrees of organizational flexibility in implementing a newproduction technology [29]. The two firms face a multiplicative demand curve as specifiedin Eq. (5). Following the standard procedure in real option games, we analyze the problembackwards in time.

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4.1 Asymmetric costs

We first consider a situation in which the investment cost is strongly asymmetric. Withoutloss of generality, we make the following assumption about the unit investment costs I1t , I2t(t ≥ 0) of two firms.

Assumption 2 I1t = It < κ It = I2t , where κ > 1.

where the fixed unit investment cost parameters of firm 1 and 2 are 1 and κ , respectively, andthe process It satisfy Eq. (4).

Assumption 2 indicates that the unit investment costs of both firms follow the samedynamic, and firm 1 has a significant cost advantage over firm 2. Therefore, we assume firm1 is the leader and firm 2 is the follower in the following analysis.

Denote by yF = xF/i and yL = xL/i the investment timing of the leader and follower,respectively. It is assumed that the leader investment has already taken placewhen the followermakes investment decision, therefore the follower cannot influence the investment decisionof the leader. For a given level of the leader’s investment capacity QL , the optimal investmenttiming y∗

F and the investment capacity Q∗F of the follower are functions of QL . The optimal

investment threshold and capacity level of the follower are described in the following theorem.

Theorem 3 Given the current level y = x/i and the capacity QL of the leader, the optimalcapacity level Q∗

F (y, QL) of the follower is equal to

Q∗F (y, QL) = 1

2η

(1 − ηQL − κmX

ymI

). (32)

The follower’s value function v∗F (y, QL) is

v∗F (y, QL ) =

⎧⎪⎨⎪⎩AF (QL)yβ1 , y < y∗

F (QL)[ymI (1 − ηQL) − κmX

]24yηmX

, y ≥ y∗F (QL),

(33)

where

AF (QL) = 1 − ηQL

(β12 − 1)mIη

[(β1 − 1)(1 − ηQL)mI

(β1 + 1)κmX

]β1, (34)

y∗F (QL) = (β1 + 1)κmX

(β1 − 1)(1 − ηQL)mI, (35)

accordingly

Q∗F (QL) ≡ Q∗

F (y∗F (QL), QL) = 1 − ηQL

(β1 + 1)η. (36)

Compared with Theorem 2, the results of Theorem 3 show that the factor 1−ηQL appearsin y∗

F (QL) and in Q∗F (QL). This is because the leader has already invested in capacity, which

means that, according to equation (5), when QF = 0, the maximal output price is reducedby the factor ηQL .

In the next step we analyze the leader’s investment decision. Similar to [11], the leadingfirm can adopt two possible strategies: entry deterrence or entry accommodation. Entry deter-rence corresponds to sequential investment and enables the leading firm to gain a monopolyprofit for a period of time starting from its investment until the follower enters. On the con-trary, entry accommodation leads to an immediate investment of the follower, that is, the

123


follower invests at the same time as the leader. The leader can use its optimal capacity QL

as a tool to enforce either one of these two strategies. According to Eq. (35), it can be seenthat the deterrence strategy occurs when the leading firm invests in a capacity size QL thatis larger than QL(y), such that

QL(y) = 1

η

(1 − (β1 + 1)κmX

(β1 − 1)ymI

). (37)

Therefore, in the opposite case, that is, QL ≤ QL(y), the leader and the follower investsimultaneously.

Notice that y∗F (QL) is increasing in QL (see Eq. (35)), this means that the leading firm can

expand its monopoly period by investing in a lager capacity. According to Eq. (36), anothermotivation for the leader to invest in a large capacity is that the capacity Q∗

F (QL) decreaseswith QL .

We first discuss the leader’s deterrence strategy. The leader’s value function with an entrydeterrence strategy is given by

gdetL (y, QL) = y(1 − ηQL)QL

mX− QL

mI−(

y

y∗F (QL)

)β1 y∗F (QL)ηQ∗

F (QL)QL

mX. (38)

Since the leader uses the entry deterrence strategy, it generates monopoly profits fora certain period of time, given by the first item of the value function. The second itemrepresents the investment cost required to install capacity with a quantity of QL . Moreover,as the leader cannot be in a monopoly position all the time, the follower will enter the marketat some point in time, which decreases the profit of the leader. Therefore, we make a negativecorrection to the value function of leader by subtracting the third term, where

(y/y∗

F (QL))β1

is the stochastic discount factor that discounted from themoment of the follower’s investmenty∗F (QL) back to y.The following Theorem 4 summarizes the leader’s optimal investment decision when it

uses the entry deterrence policy. Substituting (35) and (36) into (38) gives (41) in Theorem4.

Theorem 4 The leader will consider the deterrence strategy whenever the current level ofdemand-to-cost ratio y lies within the interval (ydet1 , ydet2 ), where ydet1 is the positive root ofthe following nonlinear equation

ydet1

mX− 1

mI−[ydet1 (β1 − 1)mI

(β1 + 1)κmX

]β1κ

(β1 − 1)mI= 0, (39)

and

ydet2 = (β1 + 1)mX

(β1 − 1)mI

[β1(κ − 1) + κ + 1

]. (40)

Given that the leader invests at y, the value function of the leader is as follows

gdetL (y) = y(1 − ηQdet

L (y))Qdet

L (y)

mX− Qdet

L (y)

mI

−[y(β1 − 1)

(1 − ηQdet

L (y))mI

(β1 + 1)κmX

]β1κQdet

L (y)

(β1 − 1)mI,

(41)

123


and the optimal investment capacity QdetL (y) is implicitly determined by

y(1 − 2ηQdet

L

)mX

− 1

mI−[y(β1 − 1)

(1 − ηQdet

L

)mI

(β1 + 1)κmX

]β1

κ(1 − (β1 + 1)ηQdet

L

)(β1 − 1)

(1 − ηQdet

L

)mI

= 0.

(42)

Furthermore, given that y < ydetL , the optimal investment threshold ydetL and the corre-sponding capacity Qdet

L under the entry deterrence strategy are

ydetL = (β1 + 1)mX

(β1 − 1)mI, (43)

QdetL = 1

(β1 + 1)η. (44)

The alternative for the leader is to apply an entry accommodation strategy, where it allowsthe follower to immediately invest once it has invested itself. Specifically, if the leaderchooses its capacity QL ≤ QL(y), which will trigger the follower to make invest imme-diately. Theorem 5 gives the optimal investment decision of the leader when it uses the entryaccommodation policy.

Theorem 5 The entry accommodation strategy will be considered if the current level ofdemand-to-cost ratio y is larger than or equal to yacc1 , where

yacc1 =[2(1 − β1) + (1 + 3β1)κ

]mX

(β1 − 1)mI. (45)

The value of the entry accommodation strategy, when investment takes place at y, is equalto

gaccL (y) =[ymI − (2 − κ)mX

]28yηmXm2

I

. (46)

For the entry accommodation strategy, the optimal investment threshold and correspondingcapacity level are given by

yaccL = (β1 + 1) (2 − κ)mX

(β1 − 1)mI, (47)

QaccL ≡ Qacc

L (yaccL ) = 1

(β1 + 1)η. (48)

We find that the leader will make corresponding strategic adjustments when the currentlevel y is in different regions. For y < yacc1 , the leaderwill definitely apply an entry deterrencestrategy. When y ∈ (yacc1 , ydet2 ), the leader chooses an entry deterrence or accommodationstrategy that maximizes its value. Moreover, if y > ydet2 , the leader can only adopt theentry accommodation strategy. The leader will wait to invest until y reaches ydetL for the firsttime. Accordingly, the leader’s optimal investment strategy is summarized in the followingTheorem.

Theorem 6 The optimal investment timing of the leader is

y∗L ={ydetL , y ∈ [0, ydetL

),

y, y ∈ [ydetL ,∞) , (49)

123


and the corresponding optimal capacity level is

Q∗L(y) =

⎧⎪⎨⎪⎩Qdet

L (ydetL ), y ∈ [0, ydetL

),

QdetL (y), y ∈ [ydetL , y

),

QaccL (y), y ∈ [y,∞) ,

(50)

where y = min{y ∈ (yacc1 , ydet2

) |gaccL (y) = gdetL (y)}. Furthermore, the value of the leader

is

v∗L(y) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(y

ydetL

)β

gdetL (ydetL ), y ∈ [0, ydetL

),

gdetL (y), y ∈ [ydetL , y),

gaccL (y), y ∈ [y,∞) ,

(51)

4.2 Symmetric costs

In this subsection, we suppose that both firms have the same unit investment costs, thatis, I1t = I2t = It . Considering the competition in the duopoly market, the firms mighthave incentives to preempt the market. Under the endogenous firm role, both firms have theopportunity to become market leader, and the advantage of becoming leader is that the firmcan enjoy monopoly profit for a period of time. Thereafter, once it is known which of the twofirms invests first, the other will become a follower. After the first investor invests, the secondinvestor behaves as if the market position is exogenous, because its investment decision nolonger involve strategic aspects. Therefore, we can refer to Theorem 3 (with κ = 1) for thesecond investor’s optimal investment decision in the case of endogenous firm roles.

The preemption threshold, denoted by y∗Le, is the moment for which the firms are indif-

ference between being a leader or a follower. Fudenberg and Tirole [27] first studied thepreemptive trigger decision of two firms within a deterministic framework. Following [30],we can obtain the preemptive threshold in stochastic timing game by solving the followingequation for y∗

Le

v∗F

(y∗Le, Q

∗L(y∗

Le)) = v∗

L

(y∗Le

). (52)

The intuition behind (52) is that when y < y∗Le, no firm wants to invest since in this case

it would be more profitable to be a follower (because the leader’s value is lower than thefollower’s value). On the other hand, when y > y∗

Le, the return of first investor is bigger, soit is more profitable for a firm to invest immediately and become a leader than to wait forinvestment. Suppose that firm 1 wants to invest at level y, then firm 2 would preempt firm 1and invest at y − ε, which will cause firm 1 to invest even earlier, at y − 2ε. This preemptivemechanism continues until y − nε = y∗

Le, where one of the firms will invest. Because thefirms are symmetric, they have equal chances of becoming market leader at the preemptivetrigger. The result of the preemption game: the first investor, as the leader, invests at thepreemptive equilibrium point y∗

Le in capacity Q∗Le = Q∗

L

(y∗Le

), and the second investor, as

the follower, invests at y∗Fe = y∗

F

(Q∗

L

(y∗Le

))in capacity Q∗

Fe = Q∗F (y∗

Le).

4.3 Welfare analysis

Now we proceed to the welfare analysis in a duopoly market. In order to examine the invest-ment outcome of the duopoly from the perspective of social welfare, we suppose a social

123


planner who is allowed to make an investment at two moments in time. For these two invest-ments, the social planner is free to choose the investment timing and capacity size. We usebackward induction to solve the social optimal investment strategy.

In order to compare the investment behavior of the monopolistic firm and the socialplanner, we first briefly examine the impact on investment policy when the monopolistic firmhas two investment opportunities. For the monopolist problem, the first investment increasesthe firm’s capacity from 0 to Q1, and the second one from Q1 to Q1 + Q2. We solve theproblem backwards. Firstly, for a given capacity level Q1, the second investment is solved.Thereafter, given the optimal investment behavior of the second investment, we solve thefirst investment. When a monopolistic firm can invest twice in time, the following theoremdescribes its optimal investment strategy.

Theorem 7 In a market with multiplicative demand, if a monopolist has two investmentopportunities, then the optimal investment thresholds y∗

1 , y∗2 and the corresponding optimal

capacities Q∗1, Q

∗2 are implicitly determined by the following equations:

1 − β1ηQ∗1

1 − ηQ∗1

− 2

[β1(1 − 2ηQ∗

1

)(β1 + 1)

(1 − ηQ∗

1

)]β1

= 0, (53)

y∗1 (Q1) = β1mX

(β1 − 1)(1 − ηQ1)mI. (54)

y∗2 (Q1) = (β1 + 1)mX

(β1 − 1)(1 − 2ηQ1)mI, (55)

Q∗2(Q1) = 1 − 2ηQ1

(β1 + 1)η. (56)

Now we consider the optimization problem of the social planner. Regarding the secondinvestment, we know that the follower’s investment decision in the duopoly model is essen-tially the same as the monopolist’s investment decision. Therefore, similar to the monopolymodel in Sect. 3, if we assume that we have the same quantity of the first investment in thecase of social welfare and duopoly, then the capacity level of the second investment in thewelfaremaximization policywill be twice asmuch as the capacity chosen by the follower. Lety∗F,W denote the investment threshold of the second investment in the welfare maximizing

policy, and the corresponding optimal capacity level is denoted by Q∗F,W . Following [11],

we can derive that

y∗F,W (QL) = (β1 + 1)mX

(β1 − 1)(1 − ηQL)mI, (57)

Q∗F,W (QL) = 2(1 − ηQL)

(β1 + 1)η. (58)

As for the first investment, according to Theorem 7, the capacity level for maximizingwelfare, which is denoted by Q∗

L,W , is twice the capacity chosen by the monopolist who canmake two investments. This capacity level is the solution of the following equation:

1 − 1/2β1ηQ∗L,W

1 − 1/2ηQ∗L,W

− 2

⎡⎣ β1

(1 − ηQ∗

L,W

)(β1 + 1)

(1 − 1/2ηQ∗

L,W

)⎤⎦

β1

= 0. (59)

The optimal trigger for the first investment of the social planner is given by:

y∗L,W (QL) = β1mX

(β1 − 1) (1 − 1/2ηQL)mI. (60)

123


Table 1 Related variables and numerical assumptions

Basic parameters: r = 0.06, η = 0.05, μX = 0.038, μI = 0.028, σX = 0.3, σI = 0.2, λ = 0.1, σ1 = 0.1,σ2 = 0.1, Sk1 = 0, Sk2 = 0.

Huisman & Kort (2015): demand uncertainty model without jump (μI = σI = λ = 0). two-factor diffusionmodel: demand and investment cost uncertainties model without jumps (λ = 0). μ1 = 0.1, μ2 = −0.1:upward jump in demand and downward jump in investment cost. μ1 = −0.1, μ2 = 0.1: downward jump indemand and upward jump in investment cost.

Consequently, if the capacity level in the case of welfare maximizing is twice that of themonopolist who has two opportunities to invest, then the investment thresholds of welfaremaximizing policy are the same as that of the monopolist. According to Eqs. (57) and (58), itcan be found that the secondwelfare investmentwill occur later than the follower’s investmentin the duopoly model, and the corresponding capacity size will be less than twice as high.Furthermore, from a welfare point of view, the firms invest too early in too small capacitiesin the duopoly of symmetric costs.

The total social welfare is given by

SW (yL , QL , yF , QF , y) =(

y

yL

)β1(yL QL(2 − ηQL)

2mX− QL

mI

)+(

y

yF

)β1

(yF(QL + QF

)(2 − η(QL + QF )

)2mX

− QF

mI− yF QL(2 − ηQL)

2mX

).

which obviously depends on the investment moments, yL and yF , the capacities QL and QF

chosen by the leader and the follower, and the level of demand-to-cost ratio y.

5 Numerical illustration

In this section, we perform a numerical analysis to investigate the impact of combininguncertainty about market demand and investment cost on investment decision of firms. Forthe numerical illustration, we assume that the discontinuous jumps in demand shock and unitinvestment cost in our model are caused by the same sudden events, so they have the samejump intensity, i.e., λ = λX = λI .

The basic parameter values of numerical simulation are given in Table 1. When analyzingthe influence of the change of a specific jump parameter on the optimal investment decision,we assum that other parameters remain unchanged.

5.1 Monopoly case

Wefirst examine the impact of σX and σI on the optimal investment threshold y∗ and capacityQ∗ for four different cases in the monopoly market. As shown in Fig. 1, we observe thaty∗ and Q∗ both increases with σX and σI . This implies that a higher uncertainty of marketdemand or unit investment cost delays the investment but promotes a larger scale productioncapacity. In other words, a monopolistic firmwill adopt a waiting strategy and choose a largercapacity level to make greater profits in a more volatile market. This is consistent with theconclusion in [11]. It is worth noting that in the Huisman & Kort (2015) model ( [11]), theunit investment cost is a constant (σI = 0), so the optimal investment timing and optimal

123


0 0.1 0.2 0.3 0.41

4

7

10

13

Huisman & Kort (2015)

0 0.1 0.2 0.3 0.46

7

8

9

10


(a) Optimal investment threshold and capacity as function of σX .

0 0.1 0.2 0.3 0.43

6

9

12

15


0 0.1 0.2 0.3 0.47.7

8.2

8.7

9.2

9.7


(b) Optimal investment threshold and capacity as function of σI .

Fig. 1 Optimal investment decisions ofmonopolist for different uncertainties ofmarket demand and investmentcost

capacity depend only on market demand and are not affected by the change of investmentcost.

The top-left and bottom-left panels in Fig. 1 show that when the market demand andinvestment cost of the project are both diffusion processes (jump is not considered), theoptimal investment threshold of the monopolistic firm is larger than that of Huisman & Kort(2015) model. Moreover, if we assume that both X and I evolve according to the jump-diffusion processes, the optimal investment threshold of the monopolistic firm is also largerthan that of Huisman & Kort (2015) model. However, the top-right and bottom-right panelsin Fig. 1 illustrate the relationship of optimal investment capacities between the Huisman &Kort and two-factor diffusion models is ambiguous. According to Eq. (21), the monopolist’soptimal investment capacity Q∗ depends on the value of β1, and the larger β1 is, the smallerQ∗ is. However, the value of β1 in Huisman & Kort (2015) model is given by

1

2− μX

σ 2X

+√√√√(1

2− μX

σ 2X

)2+ 2r

σ 2X

,

123


whileas, from Eq. (17), the value of β1 in the two-factor diffusion model is equal to

1

2− μX − μI

σ 2X + σ 2

I

+√√√√(1

2− μX − μI

σ 2X + σ 2

I

)2+ 2(r − μI )

σ 2X + σ 2

I

.

Obviously, the β1 of these two models cannot be compared. Therefore, the relationshipbetween the optimal investment capacities of the two models is ambiguous.

In addition, fromFig. 1,we can derive that, compared to the two-factor diffusion processes,in the two-factor jump-diffusion processes, the upward jump in demand and downward jumpin investment cost enable the monopoly firm to accelerate investment with greater capacity.On the other hand, the downward jump in demand and upward jump in investment cost enablethe monopoly firm to delay investment with smaller capacity. This is due to the fact that whendemand jumps up and investment cost jumps down, the NPV of the investment project willincrease, whereas when demand jumps up and investment cost jumps down, the NPV of theinvestment project will decrease.

5.2 Duopoly case

We now proceed to investigate the optimal investment decisions in a duopoly market. Table2 lists the optimal investment thresholds and optimal capacities of tow firms with differentlevels of σX and σI . Specifically, Table 2(a) reports the optimal investment strategies in anasymmetric duopoly market (κ = 1.5), while Table 2(b) reports preemption equilibrium in asymmetric duopoly market (κ = 1) by solving Eq. (52). We see from Table 2 that the optimalinvestment thresholds and optimal capacities of both the leader and the follower increasewith each parameter in σX and σI . This confirms Huisman &Kort ([11]), who concludes thatwhen the economic environment becomes more uncertain, both firms will delay investmentwith greater capacity.

Furthermore, as seen in Table 2, in contrast to the Huisman & Kort’s model, in our two-factor diffusion model, both the leader and the follower will delay investment as the σX or σI

increases, but the relationship between the optimal investment capacities of these twomodelsis ambiguous.

As we expected, Table 2 states that, compared to the two-factor diffusion processes, inthe two-factor jump-diffusion processes, if the demand jumps up and investment cost jumpsdown, both the leader and the followerwill invest earlier inmore capacity,whileas if the jumpsin demand are downwards and the jumps in investment cost are upwards, the duopolistic firmswill invest later in less capacity. This is because the upward jump in demand and downwardjump in investment cost indicate that the economic environment is upbeat, which encouragesthe leader and the follower invest earlier with higher capacities, and vice versa. Finally, fromTable 2, we conclude that the first investor should invest earlier in a higher capacity than thesecond investor due to the threat of pre-emption.

6 Concluding remarks

In this paper,we investigate the impact of unexpected sudden events on the optimal investmentthresholds and optimal capacities in a duopoly market by using the real option game method.Based on the duopoly game model, we introduce the Poisson jump process to describe theeffect of sudden events on the product market. Since our jump-diffusion model allows us to

123


Table2

Equilibrium

investmentstrategiesin

aduopolymarketw

ithtwosourcesof

uncertainty

Parameters

Huism

an&

Kort(20

15)

two-factor

diffusionmod

elμ1

>0,

μ2

<0

μ1

<0,

μ2

>0

y∗ LQ

∗ Ly∗ F

Q∗ F

y∗ LQ

∗ Ly∗ F

Q∗ F

y∗ LQ

∗ Ly∗ F

Q∗ F

y∗ LQ

∗ Ly∗ F

Q∗ F

(a)Equilibrium


anasym

metricduopoly.

σX

01.63

337.75

514.00

174.74

803.23

677.87

598.00

904.77

442.99

069.04

468.00

004.95

444.45

526.73

5210

.076

04.46

71

0.1

1.88

078.05

044.72

164.81

003.61

018.09

569.09

784.81

873.25

469.12

218.97

574.96

154.99

367.08

7211

.601

44.57

58

0.2

2.52

648.54

866.61

854.89

474.67

478.52

9312

.225

94.89

194.02

639.29

0411

.278

74.97

486.53

407.77

3916

.032

84.75

22

0.3

3.48

608.94

829.46

264.94

476.35

988.91

9017

.218

04.94

165.27

619.45

8515

.015

24.98

538.97

958.38

0223

.183

24.86

88

0.4

4.74

659.22

7513

.218

44.97

028.64

289.20

4524

.017

84.96

846.99

109.59

1320

.149

54.99

1712

.297

28.81

7232

.989

64.93

01

σI

03.48

608.94

829.46

264.94

475.01

808.62

9913

.240

14.90

624.27

909.33

2312

.033

64.97

777.03

187.93

1517

.479

74.78

61

0.1

3.48

608.94

829.46

264.94

475.35

768.71

6814

.244

84.91

774.53

019.36

9312

.784

14.98

017.52

458.06

6918

.916

64.81

32

0.2

3.48

608.94

829.46

264.94

476.35

988.91

9017

.218

04.94

165.27

619.45

8515

.015

24.98

538.97

958.38

0223

.183

24.86

88

0.3

3.48

608.94

829.46

264.94

477.99

589.14

0222

.088

24.96

306.50

369.56

0718

.689

64.99

0411

.356

88.71

9230

.202

14.91

80

0.4

3.48

608.94

829.46

264.94

4710

.248

59.32

9228

.812

84.97

758.20

389.65

1723

.783

14.99

3914

.631

99.00

5939

.926

74.95

06

Parameters

Huism

an&

Kort(20

15)

two-factor

diffusionmod

elμ1

>0,

μ2

<0

μ1

<0,

μ2

>0

y∗ Le

Q∗ Le

y∗ Fe

Q∗ Fe

y∗ Le

Q∗ Le

y∗ Fe

Q∗ Fe

y∗ Le

Q∗ Le

y∗ Fe

Q∗ Fe

y∗ Le

Q∗ Le

y∗ Fe

Q∗ Fe

(b)Equ

ilibrium


asymmetricdu

opoly

σX

00.59

907.75

512.66

784.74

801.13

837.87

595.33

934.77

440.56

189.04

465.45

964.95

442.15

776.73

526.71

744.46

71

0.1

0.61

958.05

043.14

784.81

001.16

808.09

566.06

524.81

870.57

099.12

215.98

384.96

152.22

477.08

727.73

434.57

58

0.2

0.66

218.54

864.41

234.89

471.23

798.52

938.15

064.89

190.59

319.29

047.51

914.97

482.38

117.77

3910

.688

54.75

22

0.3

0.70

778.94

826.30

844.94

471.31

988.91

9011

.478

74.94

160.62

019.45

8510

.010

14.98

532.56

408.38

0215

.455

54.86

88

0.4

0.74

999.22

758.81

224.97

021.39

879.20

4516

.011

84.96

840.64

639.59

1313

.433

04.99

162.74

208.81

7221

.993

14.93

00

σI

00.70

778.94

826.30

844.94

471.25

698.62

998.82

674.90

610.59

939.33

238.02

244.97

772.42

357.93

1511

.653

24.78

61

0.1

0.70

778.94

826.30

844.94

471.27

448.71

689.49

654.91

770.60

519.36

938.52

274.98

012.46

268.06

6912

.611

14.81

32

0.2

0.70

778.94

826.30

844.94

471.31

988.91

9011

.478

74.94

160.62

019.45

8510

.010

14.98

532.56

408.38

0215

.455

54.86

88

0.3

0.70

778.94

826.30

844.94

471.37

909.14

0214

.725

54.96

300.63

979.56

0712

.459

84.99

042.69

748.71

9220

.134

84.91

80

0.4

0.70

778.94

826.30

844.94

471.44

109.32

9219

.208

54.97

750.66

029.65

1715

.855

44.99

392.83

859.00

5926

.617

84.95

06

123


investigate the duopoly competition in a more realistic economic setting, therefore, we arelikely to obtain more accurate and more reliable results than those reported by [11].

We find that whether in the monopoly or duopoly market, uncertainty of market has apositive impact on the optimal investment timing and optimal capacity. Based on the two-factor jump-diffusion model, this paper demonstrates that the sudden events will affect theinvestment thresholds and production capacities of the firms. More specifically, comparedto the two-factor diffusion processes, if the jumps in demand are upwards and the jumps ininvestment cost are downwards, the duopolistic firmswill invest earlier in higher capacity, andvice versa. Additionally, in order to capture the market, the optimal investment threshold forthe first investor in a symmetric duopoly is relatively smaller than that of the first investor inan asymmetric duopoly. Therefore, when the market changes abruptly due to sudden events,the firm can apply the jump-diffusion model to analyze the project’s optimal investmentdecisions and adjust its strategies in time to avoid unnecessary losses or miss the investmentopportunities.

Our model studies the investment decisions in a duopoly market with two-factor uncer-tainties, both following jump-diffusion processes. However, investment in practice is muchmore complex. Several extensions are possible for this paper. For instance, the competitionusually involves more than two firms and a third player can adopt either a follower, leader ormixed strategy (see [31]). A direction for future research is to study the optimal investmentstrategy of the middle player. Moreover, in order to understand whether the results are sensi-tive to the choice of specific demand structure, or if they can be extended, it is also interestingto investigate different demand models. Finally, the firms only invest once in our model. Ifthe firms can invest multiple times, that is, have several investment options, future researchshould address their strategic interactions.

Funding This research was funded by the National Natural Science Foundation of China (No. 71361003) andNatural Science Foundation of Guizhou Province (No. [2018]3002).

Declarations

Conflict of interest The authors declare that there are no conflicts of interest regarding the publication of thispaper.

Availability of data andmaterial All data generated or analyzed during this study are included in this paper.

Appendix

Proof of Theorem 1

Using a change of variable in the performance function, we obtain

J (x, i, Q) = E(x,i)

[e−rτ(∫ ∞

0e−r t(Xt+τ Q(1 − ηQ) − It+τ Q

)dt

)]. (A1)

According to the tower property of conditional expectation and strong Markov property, wehave

J (x, i, Q) = E(x,i)

{e−rτ

E

[∫ ∞


)dt∣∣∣ (Xτ , Iτ )

]},(A2)

123


where

E

[∫ ∞


)dt∣∣∣(Xτ , Iτ )

]

=∫ ∞

0e−r t(Q(1 − ηQ)E(Xt+τ |Xτ ) − QE(It+τ |Iτ )

)dt

= Xτ Q(1 − ηQ)

r − μX − λXμ1− Iτ Q

r − μI − λIμ2.

(A3)

Substituting (A3) into (A2) yields Eq. (7).

Proof of Theorem 2

Given the demand shock and the unit investment cost are x and i , respectively, if the firm’sinvestment capacity is Q, then the profit of the firm is equal to,

G(x, i, Q) = xQ(1 − ηQ)

mX− i Q

mI. (A4)

Let g(y, Q) = G(x, i, Q)/i , the firm’s profit can be rewritten as

g(x, i, Q) = yQ(1 − ηQ)

mX− Q

mI. (A5)

Maximize (A5) with respect to Q yields the optimal capacity level Q∗

Q∗(y) = 1

2η

(1 − mX

ymI

). (A6)

Furthermore, the value of the firm, v(y), is given by the partial differential equation (15),with the boundary conditions

A1y∗β1 = y∗Q(1 − ηQ)

mX− Q

mI, (A7)

β1A1y∗β1−1 = Q(1 − ηQ)

mX, (A8)

where β1 is the positive solution of the nonlinear equation (17).The value of β1 can be obtained by numerically solving Eq. (17). Combining (A7) and

(A8) yields Eq. (20). From (A6) and Eq. (20), we can obtain the results in Theorem 2.

Proof of Theorem 3

Given the current level of the stochastic demand and unit investment cost denoted by x andi , and the capacity level QL and QF of the leader and the follower, respectively, the profitof the follower is given by

GF (x, i, QL , QF ) = xQF (1 − η(QL + QF ))

mX− κi QF

mI. (A9)

123


Further, letGF (x, i, QL , QF ) = i gF (y, QL , QF ), then the follower’s profit can be rewrittenas

gF (y, QL , QF ) = yQF (1 − η(QL + QF ))

mX− κQF

mI. (A10)

Maximizing (A10) with respect to QF yields the optimal capacity of the follower

Q∗F (y, QL) = 1

2η

(1 − ηQL − κmX

ymI

). (A11)

Before the follower has invested, i.e., y < y∗F (QL), the value function of the follower is

vF (X) = AF yβ1 , (A12)

and it satisfies the following value matching and smooth pasting conditions

vF (y∗F ) = y∗

F QF (1 − η (QL + QF ))

mX− κQF

mI, (A13)

v′F (y∗

F ) = QF (1 − η (QL + QF ))

mX. (A14)

Solving the above two equations, we obtain

y∗F (QL , QF ) = β1

β1 − 1

κmX(1 − η(QL + QF )

)mI

, (A15)

AF (QL) = 1 − ηQL

(β12 − 1)mIη

[(β1 − 1)(1 − ηQL)mI

(β1 + 1)κmX

]β1. (A16)

According to (A11) and (A15), the optimal threshold and capacity level of the followerare Eqs. (35) and (36) respectively.

Proof of Theorem 4

For the entry deterrence strategy, the leader’s value function at the moment of investment canbe expressed as

gdetL (y, QL) = y(1 − ηQL)QL

mX− QL

mI−(

y

y∗F (QL)

)β1 y∗F (QL)ηQ∗

F (QL)QL

mX.

(A17)

Substituting (35) and (36) into (A17) yields the following equation

gdetL (y, QL ) = yQL(1 − ηQL)

mX− QL

mI−(y(β1 − 1)(1 − ηQL)mI

(β1 + 1)κmX

)β1 κQL

(β1 − 1)mI.

(A18)

Maximizing gdetL (y, QL ) with respect to QL gives the following first-order necessarycondition

φ(y, QL) ≡ y (1 − 2ηQL)

mX− 1

mI−[y(β1 − 1) (1 − ηQL)mI

(β1 + 1)κmX

]β1

× κ (1 − (β1 + 1)ηQL)

(β1 − 1) (1 − ηQL)mI= 0. (A19)

123


Solving (A19) yields QdetL (y). Let QL = 0 in (A19) gives Eq. (39). Further, let

ψ(y) ≡ φ(y, QL = 0) = y

mX− 1

mI−[y(β1 − 1)mI

(β1 + 1)κmX

]β1 κ

(β1 − 1)mI, (A20)

then we get

ψ(0) = − 1

mI< 0, (A21)

ψ(y∗F (0)) = 1

mI

(β1κ

β1 − 1− 1

)= β1(κ − 1) + 1

(β1 − 1)mI> 0, (A22)

∂ψ(y)

∂ y= 1

mX

[1 − β1

β1 + 1

(y

y∗F (0)

)β1−1]

. (A23)

For y ∈ (0, y∗F (0)), then ∂ψ(y)/∂ y > 0 holds. Therefore ydet1 exists. In addition, the

leader can no longer uses the entry deterrence strategy if y ≥ y∗F

(Qdet

L (y)).

We define ydet2 as

ydet2 = y∗F

(Qdet

L (ydet2 ))

. (A24)

Substitution of (35) into (A19) results in

β1 + 1

β1 − 1

1 − 2ηQL

1 − ηQLκ − 1 − (1 − (β1 + 1)ηQL)

(β1 − 1)(1 − ηQL)κ = 0. (A25)

Solving Eq. (A25) for QL , we obtain

QL = β1κ − (β1 − 1)[β1(κ − 1) + κ + 1

]η

. (A26)

By substituting (A26) into (35), we can obtain Eq. (40).The option value of the leader is given by

vdetL (y) = AdetL yβ1 . (A27)

To determine ydetL , the corresponding value matching and smooth pasting conditions aregiven by

AdetL yβ1 = y(1 − ηQL (y))QL (y)

mX− QL (y)

mI−[y(β1 − 1) (1 − ηQL (y))mI

(β1 + 1)κmX

]β1 κQL (y)

(β1 − 1)mI,

(A28)

β1AdetL yβ1−1 =

QL (y) (1 − ηQL (y)) + y∂QL

∂ y(1 − 2ηQL (y))

mX− 1

mI

∂QL

∂ y−

(y(β1 − 1) (1 − ηQL (y))mI

(β1 + 1)κmX

)β1

[QL (y)

(β1 (1 − ηQL (y)) − (β1 + 1)yη

∂QL

∂ y

)+ y

∂QL

∂ y

]κ

y(β1 − 1) (1 − ηQL (y))mI.

(A29)

123


Substitution of (A29) into (A28) results in

yQL (y) (1 − ηQL (y))

mX−

yQL (y) (1 − ηQL (y)) + y2∂QL

∂ y(1 − 2ηQL (y))

β1mX− QL (y)

mI+

y

β1mI

∂QL

∂ y+(y(β1 − 1) (1 − ηQL (y))mI

(β1 + 1)κmX

)β1

⎛⎜⎜⎝y∂QL

∂ y(1 − (β1 + 1)ηQL (y)) κ

β1(β1 − 1) (1 − ηQL (y))mI

⎞⎟⎟⎠ = 0.

(A30)

Combining (A19) and (A30) leads to

(β1 − 1)y(1 − ηQL)

mX− β1

mI= 0. (A31)

Solving Eq. (A31), the leader’s threshold ydetL is equal to

ydetL (QL) = β1mX

(β1 − 1)(1 − ηQL)mI. (A32)

Substituting (A32) into (A19), the optimal investment threshold and optimal capacity of theleader are, respectively, given by Eq. (43) and Eq. (44) in Theorem 4.

Proof of Theorem 5

For the entry accommodation strategy, the value function of the leader at the moment ofinvestment is given by

gaccL (y, QL) = yQL(1 − η(QL + Q∗

F (QL)) )

mX− QL

mI. (A33)

Substituting (A11) into (A33) and maximize with respect to QL gives

QaccL (y) = 1

2η

(1 − (2 − κ)mX

ymI

). (A34)

The leader will adopt the accommodation strategy only if the optimal quantity QaccL (y))

causes the follower to invest immediately. Thus it holds that

y ≥ y∗F

(Qacc

L (y)), (A35)

We define yacc1 as

yacc1 = y∗F

(Qacc

L (yacc1 )). (A36)

Then, substitution of (36) and (A34) into (A36) yields Eq. (45).Furthermore, by substituting (36) and (A34) into (A33), it gives Eq.(46). The valuematch-

ing and smooth pasting conditions for the entry accommodation strategy are given by

AaccL yβ1 =

[ymI − (2 − κ)mX

]28yηmXm2

I

, (A37)

β1AaccL yβ1−1 = y2m2

I − (2 − κ)2m2X

8y2ηmXm2I

. (A38)

123


Substituting (A38) into (A37) leads to[ymI − (2 − κ)mX

]28yηmXm2

I

− y2m2I − (2 − κ)2m2

X

8β1yηmXm2I

= 0, (A39)

and after some simple calculations, Equation (A39) can be reformulated as[ymI − (2 − κ)mX

][(β1 − 1)ymI − (β1 + 1)(2 − κ)mX

]8β1yηmXm2

I

= 0. (A40)

Acoording to Eq. (A37), y = [(2 − κ)mX]/mI is not a valid solution to Eq. (A40). Hence,

we obtain

yaccL = (β1 + 1)(2 − κ)mX

(β1 − 1)mI, (A41)

and substitution of (A41) into (A34) gives Eq. (48) in Theorem 5.

Proof of Theorem 6

Given in the text.

Proof of Theorem 7

When the firm’s capacity increases from Q1 to Q1 + Q2, the value of the firm at the secondinvestment is equal to

G2(x, i, Q1, Q2) = x(Q1 + Q2)(1 − η(Q1 + Q2))

mX− i Q2

mI. (A42)

Let G2(x, i, QL , QF ) = i g2(y, QL , QF ), the value of the firm at the moment of the secondinvestment can be rewritten as

g2(y, Q1, Q2) = y(Q1 + Q2)(1 − η(Q1 + Q2))

mX− Q2

mI. (A43)

Before the second investment, the value of the firm is

v2(y, Q1) = yQ1(1 − ηQ1)

mX+ B2y

β1 , (A44)

In addition, the value-matching and smooth-pasting conditions are as follows

y∗2Q1(1 − ηQ1)

mX+ B2y

∗2

β1 = y∗2 (Q1 + Q2)

(1 − η(Q1 + Q2)

)mX

− Q2

mI. (A45)

Q1(1 − ηQ1)

mX+ β1B2y

∗2

β1−1 = (Q1 + Q2)(1 − η(Q1 + Q2))

mX. (A46)

Solving the above two equations gives

y∗2 (Q1, Q2) = β1mX

(β1 − 1)(1 − η(2Q1 + Q2)

)mI

, (A47)

and B2 = (y∗2

)−β1 Q2

(β1 − 1)mI.

123


The optimal capacity Q2 of the second investment is determined by solving the followingoptimization problem

maxQ2≥0

y(Q1 + Q2)(1 − η(Q1 + Q2))

mX− Q2

mI. (A48)

The first order condition of (A48) is

y(1 − 2η(Q1 + Q2))

mX− 1

mI= 0, (A49)

which gives

Q∗2(y, Q1) = 1

2η

(1 − 2ηQ1 − mX

ymI

). (A50)

Combining (A47) and (A50) leads to the Eqs. (55) and (56).When the firm’s capacity increases from 0 to Q1, the firm’s value at the moment of first

investment is equal to

g1(y, Q1) = yQ1(1 − ηQ1)

mX− Q1

mI+ B2y

β1 . (A51)

Before the first investment, the value of the firm is given by

v1(X) = B1yβ1 , (A52)

The value matching and smooth pasting conditions leads to the following equations

B1y∗1

β1 = y∗1Q1(1 − ηQ1)

mX− Q1

mI+ B2y

∗1

β1 , (A53)

β1B1y∗1

β1−1 = Q1(1 − ηQ1)

mX+ β1B2y

∗1

β1−1, (A54)

Solving these equations gives

y∗1 = β1

β1 − 1

mX

(1 − ηQ1)mI, (A55)

B1 = B2 +(y∗1

)1−β1

β1

Q1(1 − ηQ1)

mX. (A56)

The optimal capacity Q1 of the first investment can be obtained by maximizing the firm’svalue g1(y, Q1),

maxQ1≥0

yQ1(1 − ηQ1)

mX− Q1

mI+ B2(Q1)y

β1 . (A57)

The first order condition is

y(1 − 2ηQ1)

mX− 1

mI+ ∂B2(Q1)

∂Q1yβ1 = 0, (A58)

123


where

B2(Q1) = (y∗2 (Q1))−β1 Q2

(β1 − 1)mI= 1 − 2ηQ1

(y∗2 (Q1))−β1

(β1 + 1)(β1 − 1)mIη

= 1 − 2ηQ1

(β1 + 1)(β1 − 1)mIη

((β1 + 1)mX

(β1 − 1)(1 − 2ηQ1)mI

)−β1

= 1

(β1 + 1)(β1 − 1)mIη

((β1 + 1)mX

(β1 − 1)mI

)−β1

(1 − 2ηQ1)β1+1 , (A59)

∂A2(Q1)

∂Q1= 1

(β1 + 1)(β1 − 1)mIη

((β1 + 1)mX

(β1 − 1)mI

)−β1

(β1 + 1)(1 − 2ηQ1)β1(−2η)

= − 2

(β1 − 1)mI

((β1 + 1)mX

(β1 − 1)(1 − 2ηQ1)mI

)−β1

= −2(y∗2 (Q1))−β1

(β1 − 1)mI. (A60)

Substitution of (A60) into (A58) gives

y(1 − 2ηQ1)

mX− 1

mI− 2

(β1 − 1)mI

(y

y∗2 (Q1)

)β1

= 0. (A61)

Finally, substituting (A58) and (A55) into (A61) gives Eq. (53).

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