The Pricing of a Supply Contract under Uncertainty ...

International Journal of Information and Management Sciences 25 (2014), 35-49

The Pricing of a Supply Contract under Uncertainty with Long-Range Dependence Po-yuan Chen and Horng-Jinh Chang Jinwen University of Science and Technology and Tamkang University Abstract This paper aims to address a contracting problem between upstream and downstream agents in a supply chain using a stochastic demand process with autocorrelation properties. For example, the quarterly global sales volumes of Apple’s iPhone are highly autocorrelated over time although the time lag is as long as 10 quarters. Based on such empirical evidence, an autocorrelated demand process referred to as fractional Brownian motion is adopted in this paper. It is assumed that there are two echelons in the supply chain: business and consumer markets. The information flows fall into four categories: demand flow, marketing info flow, uncertainty flow, and premium charge flow. The downstream agent can transfer demand uncertainty to the upstream firm (uncertainty flow) by signing a supply contract (contracting agent). The demand in the consumer market is assumed to follow a fractional Brownian motion. Based on the fractional Ito formula for the real option model, the result demonstrates that the real option value can be an increasing or decreasing function of the degree of autocorrelation in which the real option value reaches its maximum at the critical point. As a consequence, the trading price determined in the supply contract without considering the autocorrelation of demand could be significantly undervalued or overvalued. In other words, to ensure a fair game in a contracting activity, the upstream agent should charge more for the trading price depending on the degree of autocorrelation in demand.

Keywords: Supply contract, uncertainty, fractional Brownian motion, autocorrelation. 1. Introduction Channel coordination has been a central topic in studies of supply chain management, and researchers have focused on different aspects, such as pricing, inventory control, and risk sharing. Giannoccaro and Pontrandolfo [16] argued that pricing decisions could be made jointly or separately among agents, depending on their bargaining power along the supply chain. However, Zhou and Yang [38] showed that joint profits could be 2.3 times higher than the profits made independently among agents. Furthermore, Biehl et al. [2] indicated that cooperative decision making contributes to the efficiency and competitiveness of a supply chain. Many mechanisms can be developed to coordinate the operations of a supply chain, such as quantity discounts, price discounts, and

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PO-YUAN CHEN AND HORNG-JINH CHANG

profit-sharing contracts. All of these mechanisms are devised to distribute profits among agents favorably and fairly. Giannooccaro and Pontrandolfo [16] employed a revenuesharing contract to coordinate the supply chain, and they assumed that market demand is independent of the retailing price. In contrast, Zhou and Yang [38] assumed deterministic price-sensitive demand. Battiston et al. [1] investigated the effects of interactions among firms that are connected by production and credit ties on bankruptcy prorogation resulting from supply failures. Xiao et al. [37] proposed a model in the context of one manufacturer and two retailers with contractual arrangements among agents. The authors found that coordination among agents could enhance performance along the supply chain. Sethi et al. [33] formulated a supply chain in a different context with two suppliers and a single retailer. This paper extends the “Capacity Choice” model by Pindyck [30], who employed the contingent claim approach to propose that the market value of a firm can be maximized through the optimal choice of the production capacity, which depends heavily on how the stochastic demand evolves. In contrast, we intend to propose an optimization model by employing the stochastic dynamic programming approach to valuate a dynamic supply contract. Moreover, Pindyck [31] indicated in the framework of “Investment Timing” that the value threshold is chosen to maximize the net payoff of an investment incurred with irreversible sunk costs. The common decision rule for those papers focusing on entry strategies is that the investment will not be undertaken until the project value stochastically evolves to be higher than the value threshold. The first passage of time when the value of the underlying real asset is above the threshold is referred to as the optimal entry time. Under the assumption that the uncertainty of market demands follows a geometric Brownian motion (gBm), the delivery rates for the supplier and the retailer can be simultaneously determined in maximizing the supply chain value using a real option approach. Many discussions on the real option model in continuous time can be found in studies by Brennan and Schwartz [4], McDonald and Siegel [28], Pindyck [30], Dixit and Pindyck [10, 11], Brandao and Dyer [3], Gutierrez [18] and Korn and Korn [24]. Most of these studies are focused on revenue-related uncertainty, whereas other researchers focus on technological uncertainty, such as Grenadier and Weiss [17], Farzin et al. [15], and Doraszelski [12]. Moreover, Murto [29] investigated the interactions between technological and revenue-related uncertainties. The main reason that scholars valuate a real asset (a project or firm value) using the real option approach is that the traditional NPV method underestimates the value of the real asset because of the neglect of uncertainty and manufacturing flexibility (see Pindyck [31], Triantis and Hodder [35]). Most scholars have focused on the effects of price volatility on the output rate and found that an increase in price volatility induces an increase in the output rate. Other effects of price volatility can be found in the works of Van Wijnberger [36] and Ingersoll and Ross [22]. In our model, demand uncertainty is introduced in a more realistic context: auto-correlated demand of a fractional Brownian motion (fBm). Recently, more and more researchers have paid attention to the stochastic process with autocorrelation and replaced a geometric Brownian motion (gBm) with an fBm in their works. For example,

THE PRICING OF A SUPPLY CONTRACT UNDER UNCERTAINTY

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Figure 1: Autocorrelation of global quarterly sales for Apple’s iPhone from Q1 2008 to Q1 2013. (Data Source: http://www.statista.com)

fBm-related models were proposed by Hu and Oksendal [20], Kong et al. [25] and Mao and Liang [27]. This framework aims to address the problem of contracting between upstream and downstream agents in a supply chain using a correlated stochastic model, which plays the role of negotiation agent to ensure fair coordination of the contracting activity, as Jiao et al. [23] proposed. In contrast to Jiao et al. [23], however, we assume that market demands are correlated in time horizons based on the empirical evidence (shown in Figure 1) that the quarterly sales volumes of Apple’s iPhone are highly correlated over time although the lag time is as long as ten quarters. Two echelons exist in the supply chain (shown in Figure 2): the business market and the consumer market. The information flows fall into four categories: demand flow, marketing info flow, uncertainty flow, and premium charge flow. The downstream agent can transfer demand uncertainty to the upstream firm (uncertainty flow) by signing a supply contract (contracting agent). On the other hand, the upstream firm should charge more in excess of the product price as an insurance premium (premium charge) in exchange for assuming the demand uncertainty transferred from the downstream agent. The demands in the market are assumed to follow an fBm. By employing the fractional Ito formula, we can derive the solution for the real option value (ROV), which is embedded in the supply contract. The result demonstrates that the ROV could be an increasing or decreasing function of autocorrelation in demand, depending on the level of the Hurst parameter of an fBm. Consequently, the trading price determined in the supply contract without considering the long-range dependence of demand over time could be significantly undervalued or overvalued. In conclusion, to ensure a fair game in the contracting process, the upstream agent confronting demand uncertainty with autocorrelation over time should charge more or charge less for the trading price, depending on the degree of autocorrelation in demand.

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Figure 2: The framework consists of four different information flows among agents, who can transfer (receive) demand uncertainty at the cost of (in exchange for) price premium by contracting activities.

This paper is organized as follows: Section 1 addresses the necessity and benefits of implementing a coordinated supply chain and reviews the recent research on supply chain coordination. A real option approach is also introduced to explore the effects of demand uncertainty on contracting activity. Section 2 presents a two-echelon supply chain model under the assumption of auto-correlated and stochastically evolved demand in a singlesupplier and single-retailer system. The dynamic maximization in profits is proposed, and analytical solutions are obtained. Section 3 performs sensitivity analyses for the entry threshold and ROV. The analytical results are also depicted in various figures. In addition, managerial implications are discussed. Section 4 concludes the paper and recommends possible further studies. 2. The Model In this model, we intend to optimize the value of a supply contract using the stochastic dynamic programming approach of Dreyfus [13]. We assume that there are two major agents in the supply chain: a single upstream firm (UF, e.g., a supplier) that manufactures and delivers goods to a single downstream firm (DF, e.g., a retailer) that will


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instantly forward the products to the final customers in the consumer market. The UF and the DF coordinately sign a supply contract to predetermine the trading price and to guarantee the supply quantity. The supply contract transfers the demand uncertainty confronting the DF to the UF. To ensure fair contracting activity, the UF should charge more for the risk premium by the amount of the ROV to compensate for its risk taking. We denote the trading price as p. The marginal cost is denoted by c. We assume that the UF owns sufficient capacity to instantly meet the demands of the DF. Variable q denotes the quantity delivered by the UF. Variable π denotes the profit flows of the supply contract. Suppose that the contracting and delivery activities in the supply chain are the core operations for both the UF and the DF. Fama [14] advocated that for the valuation of a project or a firm under uncertainty, a single risk-adjusted discount rate can be applied to the uncertain future cash flows generated by the company whose operational activities are not anticipated to change much over time. Dixit [9] investigated the effect of uncertainty on entry and exit decisions. Sodal [34] also took into account the impact of the discount factor on entry and exit decisions. Hodder et al. [19] showed that risk-adjusted discounting can be made to work for valuing real options, although it requires more information and computations than risk-neutral discounting does. Moreover, Dixit and Pindyck [10] discussed two approaches to valuate real options: dynamic programming and contingent claims. In contrast, the risk-free rate is used as a discount rate under the assumption of no arbitrage in our model. Based on the work of Carruth et al. [5], the unit prices can be obtained by the inversed demand function as follows: 1

p = φt · q − ε

(2.1)

In Equation (2.1), ε denotes the price elasticity of demand and is assumed to be a constant. φt is a common factor of uncertainty (demand shock) at time t. The demand shock was ordinarily used to represent the demand uncertainty as the work of Li and Huang [26] proposed. It can be observed through a procedure proposed by Chen [6]. For example, we can obtain a linear equation by taking natural logarithm on both sides of equation (2.1) as follows: ln p = (− 1ε ) · ln q + ln φt . We can then obtain ε and φ by plugging the observed data p and q into this equation through a regression approach. In contrast to the work of Huang [21] who adopted a gBm as the demand process, our framework adopts an fBm as demand shock φt : dφt = rdt + σdztH . φt

(2.2)

In Equation (2.2), r is the instantaneous growth rate of φt . In neutrality, the growth rate is assumed to be a risk-free rate. σ is the volatility, and dztH is an fBm with zero mean and variance t2H . H is the Hurst parameter denoting a standard Brownian with independent increments when H = 21 or denoting an fBm with positively correlated increments when 12 < H < 1 and with negatively correlated increments when 0 < H < 12 . Compared with the adoption of an fBm in this paper, a gBm, which is a special case with H = 12 , is much more commonly used in previous studies. For example, the study of

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investment strategies for a supply chain under demand uncertainty by Chen [6] adopted a gBm with independent increment as a stochastic demand process. Even though the concept of correlation has been discussed in previous papers, most of them focus on crosscorrelation, instead of autocorrealtion. For example, Chen and Chang [7] investigated the impacts of uncertainties of cross-correlated price and foreign exchange rate on foreign investment strategies. Therefore, this paper makes a major contribution by generalizing previous stochastic models. In equation (2.3), the unit cost of the product is assumed to be proportional to the unit price as per the assumption of Schwartz and Moon [32], who divided the cost structure into two components: a fixed term and a variable term that is proportional to the revenue. We also assume that each agent can control its cost rate of goods sold at a fixed level relative to revenues through its managerial activities, such as cost controls or cost reduction measures. The different cost structures and technology levels lead to the different cost rate, denoted by γ. Therefore, we have the following: c = γ · p,

(2.3)

where 0 < γ < 1. We substitute Equation (2.1) into Equation (2.3) and then rewrite the unit cost as follows: 1

c = γp = φt γq − ε .

(2.4)

Therefore, we can obtain the profit flows for the supply contract: π = (p − c)q.

(2.5)

We now use Equations (2.1), (2.3), and (2.4) to rewrite the profit flows: πt = pq − cq 1

1

1

= φt q 1− ε − γφt q 1− ε = φt (1 − γ)q 1− ε .

(2.6)

According to the valuation model of two-process production within a single firm by Cortazar and Schwartz [8], the profit flow for the supply contract at time t is denoted by πt (φt ), which can be suppressed as πt . Suppose that the UF and the DF will coordinately establish a supply contract at the optimal entry time. The value of the supply contract is denoted by Vt . This value is equal to the net present value (NPV) of the future profits generated from the supply contract; in other words, Vt equals the expected NPV of the future profits as follows (proved in Appendix A): Vt (φt ) = φt · λt , where

Z 1− 1ε λt = (1 − γ)q

(2.7) ∞ t

e[−cu−0.5σ

2 [(t+u)2H −t2H −u2H ]]

du.


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The UF and the DF coordinately determine the optimal entry time for the establishment of the supply contract. The UF should pay an irreversible sunk cost (I) for the installation of production facility to meet the retailer’s demand and assume the demand uncertainty that is specified in the supply contract. The UF obtains the value of the supply contract (Vt ) in return. The investment profit of the UF is Vt −I. Therefore, the optimization of the supply contract could be described as follows: maxs Et [(Vt+s (φt+s ) − I) · e−rs ]. The optimized value is the ROV, denoted by F (φt ) as follows: F (φt ) = G(Vt (φt )) = max Et [(Vt+s (φt+s ) − I)e−rs ], s≥0

(2.8)

where F is a composite function and F (φt ) = G ◦ V (φt ), and Et [·] is a conditional expectation based on the information set at time t. For the optimization, because the expected cost of capital (rF dt) for the supply contract is equal to its expected rate of capital appreciation in a short period of time dt, we have rF dt = Et (dF ). (2.9) Equation (2.9) is the Bellman equation for the option value F . Under the optimal entry time, we can obtain the differential equation (see Hu and Oksendal [20]) as follows: Ht2H−1 σ 2 φ2 Fφφ + rφFφ − rF = 0.

(2.10)

Solving for Equation (2.10), we can finally obtain the option value F (φ). Assume that the option value is in the following form: F (φt ) = c1 · (φt )d1 + c2 · (φt )d2 ,

(2.11)

where c1 , c2

are two constant coefficients.

Equation (2.11) must satisfy the first boundary condition: F (0) = 0. Because d1 is less than 0, we should set c1 = 0. Otherwise, Equation (2.11) will explode when φt approaches zero. As a result, Equation (2.11) can be reduced to the following: F (φt ) = c2 (φt )d2 .

(2.12)

We denote the demand shock threshold as φ∗ . The other two boundary conditions are as follows: F (φ∗t ) = V (φ∗t ) − I,

(2.13)

and Fφ (φ∗t ) = Vφ (φ∗t ).

(2.14)

According to Dixit and Pindyck [10], we can derive coefficients d2 , c2 and threshold φ∗ from Equations (2.10), (2.12), (2.13), and (2.14) as follows (proved in Appendix B): I 1−d2 2 c2 = · d−d · λdt 2 (2.15) 2 d2 − 1

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and

d I 2 · , d2 − 1 λt p (a − b) + (a − b)2 + 4ra > 1, d2 = 2a a = r − c,

φ∗t = where

b = Ht2H−1 σ 2 Z 1− 1ε λt = (1 − γ)q

∞

e[−cu−0.5σ

2 [(t+u)2H −t2H −u2H ]]

(2.16)

du.

t

After the optimization, we can finally obtain the ROV as follows: F (φt ) = c2 (φt )d2 d − 1 d2 −1 V (φ ) d2 t t 2 . · = I d2

(2.17)

The threshold of demand shock φ∗ can be equivalently transformed into the value threshold V ∗ as follows: d 2 · I. (2.18) Vt∗ = φ∗t · λt = d2 − 1 From equation (2.18), it can be shown that the degree of autocorrelation of demand affects the entry threshold in terms of demand shock, which should be jointly monitored by the UF and the DF for the optimal time of supply contract establishment 3. Numerical Results Based on the analytical solutions derived above, sensitivity analyses are performed using a parameter set: ε = 1.2, γ = 0.6, r = 0.02, σ = 0.06, c = 0.01, I = 1000, q = 1000. We illustrate the sensitivity of demand shock and value thresholds in Figures 3 and 4 with varying demand autocorrelation of the Hurst parameter, ranging from 0.05 to 0.95. The results demonstrate that the entry threshold is an increasing function of the Hurst parameter, which implies that greater autocorrelation of demand leads to a higher entry threshold. However, the increasing effect of positive autocorrelation with 12 < H < 1 is much more significant than that of negative autocorrelation with 0 < H < 12 . The managerial implication is that the optimal establishment of the supply contract will be deferred for a longer period of time in a market with positively autocorrelated demand, such as the global iPhone market. Because the entry threshold is much higher in the market with positively auto-correlated demand, it will take much more time for the demand shock to evolve to the level of the entry threshold. In contrast, the entry threshold is lower in the market with negatively auto-correlated demand. As a result, the demand shock could easily and rapidly reach the level of entry threshold. Figure 5 shows that the ROV is an increasing function of demand autocorrelation with a Hurst parameter less than 0.7, and ROV is a decreasing function of demand autocorrelation with a Hurst parameter greater than 0.7. We set the benchmark ROV


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Figure 3: The entry threshold of demand shock as an increasing function of demand autocorrelation in terms of the Hurst parameter of a fractional Brownian motion.

Figure 4: The entry threshold of contract value as an increasing function of demand autocorrelation in terms of the Hurst parameter of a fractional Brownian motion.

as 454 for a standard Brownian motion with a Hurst parameter equal to 0.5. The lower portion of Figure 5, below the level of independent demand, is the price discount region, where the real option value is lower than the benchmark ROV with a standard

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Figure 5: The real option value is a function of demand autocorrelation in terms of the Hurst parameter. The price premium or discount arises as a result of autocorrelation in a fractional Brownian motion, compared with the real option value of a standard Brownian motion.

Brownian motion. On the contrary, the upper portion of Figure 5 is the region of the price premium, where the real option value is greater than the benchmark ROV with a standard Brownian motion. For example, when the Hurst parameter lies between 0.5 and 0.75, the ROV is greater than 454. The value difference arises as a result of the price premium of demand autocorrelation. In contrast, if the Hurst parameter lies outside of the range [0.5, 0.75], then the ROV is less than 454. Similarly, the value discrepancy is caused by the price discount of demand autocorrelation. We also found that the ROV decreases sharply in the region of highly positively auto-correlated demand with a Hurst parameter greater than 0.75. The managerial implication reveals that marketers should target a market with a moderate degree of positive demand autocorrelation to maximize the value of a supply contract. For example, a market with a Hurst parameter equal to 0.7 in demand has maximum ROV in the setting of our study. 4. Conclusion This paper employs a real option approach to investigate the effects of demand autocorrelation on the entry threshold and ROV for a single-supplier and single-retailer system in a two-layer supply chain setting. In our formulation, the uncertainty of contractual profit originates from the stochastic demand shock, as proposed by Carruth et al. [5]. However, Carruth et al. [5] assumed that demand shock evolves as a gBm, whereas we adopt an fBm with auto-correlated demand based on empirical evidence of global iPhone demand. In our model, a DF has the right to place orders, depending on


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the realized demand shock and the corresponding threshold, specified in a signed supply contract. The demand shock threshold can be equivalently transformed into the value threshold (Dixit and Pindyck [10]), which is more economically meaningful. It can be shown that a higher threshold allows a firm to defer investment because it will take more time to reach the threshold level. As a result, the optimal entry time in a real option approach is later than in the NPV approach, which requires investors to make decisions immediately. To obtain maximum contract value, the UF and DF should jointly monitor the demand shock (or the contract value) in a coordinated manner to determine whether it reaches the threshold level, at which time the supply contract is established and goods delivery begins. In other words, the DF owns a real option to place any order. However, in addition to the intrinsic price of a product, the DF should pay a greater premium to the UF in exchange for the real option. Although the demand shock (or the contract value) is below the threshold level, the contract is not activated until the optimal entry time appears. The real option approach that is employed in this paper has several managerial implications. The downstream firm can transfer the demand uncertainty to the upstream firm at the cost of the premium charge, which can be considered as an insurance premium. Therefore, the real option value is equal to the value of revenue flows generated from the product price plus the value of the insurance premium. As a result, the supply contract embeds a right to place an order for the insurance premium payer (DF) and an obligation to fulfill the order for the insurance premium receiver (UF). That is why the real option value is greater than the conventional net present value, which is only equal to the value of revenue flows. The investment opportunity provided by the supply contract embedded with a real option permits the supplier and the retailer to wait for the time when the maximum supply contract value appears. However, when the supplier and the retailer make a collaborative decision for investments, the degree of autocorrelation in demand should be took into account and its significant impact on the investment timing should be observed carefully. Moreover, if the demand trend is reversed downward, then the supplier and the retailer could save the losses from the delay and suspension of goods deliveries. For those readers who are interested in strategies for supply contracting activity, it is recommended that the assumptions made in this model concerning a singlesupplier, single-retailer system in the two-echelon supply chain context be read as the setting in the work by Sethi et al. [33], who use game theory to formulate a supply chain with a single retailer and two suppliers that compete for the retailer’s purchases and who conclude using the Nash solutions. Some extensions of this paper could be further studied. Acknowledgement We appreciate the financial support from National Science Council (NSC) of Taiwan with reference number NSC 101-2410-H-228-001 to support the development of this framework and the participation in international academic conferences.

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Appendix Appendix A Let Et [∗] denote the conditional expectation based on the information set at time t. hZ ∞ i πt+u e−ru du Vt (φt ) = Et t i hZ ∞ 1 = Et (1 − γ)φt+u q 1− ε e−ru du Z ∞ t 1 (1 − γ)q 1− ε e−ru Et [φt+u ]du. = (A1) t

Based on the derivations of Hu and Oksendal [20], we have the following: φt+u = φ0 e{(r−c)(t+u)−0.5σ

2 (t+u)2H +σz H }

(A2)

.

We can obtain the conditional expectation of (A2) by several derivations as follows: Et [φt+u ] = φt e{(r−c)u−0.5σ

2 [(t+u)2H −(t)2H −(u)2H ]}

(A3)

.

Substituting (A1) with (A3), we have the following: Z ∞ 1 (1 − γ)q 1− ε e−ru Et [φt+u ]du Vt (φt ) = Zt ∞ 1 2 2H 2H 2H = (1 − γ)q 1− ε e−ru φt e{(r−c)u−0.5σ [(t+u) −t −u ]} du Zt ∞ 1 2 2H 2H 2H (1 − γ)q 1− ε φt e{−cu−0.5σ [(t+u) −t −u ]} du = t Z ∞ 2 2H 2H 2H 1− 1ε = φt (1 − γ)q e{−cu−0.5σ [(t+u) −t −u ]} du t n Z ∞ o 2 2H 2H 2H 1− 1ε = φt (1 − γ)q e{−cu−0.5σ [(t+u) −t −u ]} du .

(A4)

t

By denoting in (A4) Z n 1 λt = (1 − γ)q 1− ε

∞

e{−cu−0.5σ

2 [(t+u)2H −t2H −u2H ]}

t

o du ,

we have the following: Vt (φt ) = φt · λt .

(A5)

Appendix B By using two boundary condition equations, we have the following: F (φ∗t ) = V (φ∗t ) − I = φ∗t · λt − I = c2 (φ∗t )d2 and

(B1)


Fφ (φ∗t ) = Vφ (φ∗t ) = λt = c2 d2 (φ∗t )d2 −1 .

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(B2)

Multiplying (B1) by d2 , we have the following: d2 (φ∗t · λt − I) = c2 d2 (φ∗t )d2 .

(B3)

Multiplying (B2) by φ∗t , we have the following: λt φ∗t = c2 d2 (φ∗t )d2 .

(B4)

Subtracting (B4) from (B3), we have the following: d2 (φ∗t · λt − I) − λt φ∗t = 0.

(B5)

Rewriting (B5), we have the following: φ∗t =

I d2 . λt d2 − 1

Substituting (B2) with (B6), we have the following: d − 1 d2 −1 λ d2 t 2 · . c2 = I d2

(B6)

(B7)

Substituting Equation (12) with (B7), we obtain the following: d − 1 d2 −1 V (φ ) d2 t t 2 · . F (φt ) = I d2 Similarly, substituting Equation (7) with (B6), we have the following: d 2 · I. Vt∗ (φ∗t ) = φ∗t · λt = d2 − 1

(B8)

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[33] Sethi, S. P., Yan, H. and Zhang, H. (2005). Analysis of a duopoly supply chain and its application in electricity spot markets, Annals of Operations Research, Vol.135, 239-259. [34] Sodal, S. (2006). Entry and exit decisions based on a discount factor approach, Journal of Economic Dynamics and Control, Vol.30, 1963-1986. [35] Triantis, A. J. and Hodder, J. E. (1990). Valuing flexibility as a complex option, Journal of Finance, Vol.45, 549-565. [36] Van Wijnbergen, S. (1985). Trade reform, aggregate investment and capital flight, Economic Letters, Vol.19, 369-372. [37] Xiao, T., Yu, G., Sheng, Z. and Xia, Y. (2005). Coordination of a supply chain with one-manufacturer and two-retailers under demand promotion disruption management decisions, Annals of Operations Research, Vol.135, 87-109. [38] Zhou, Y. W. and Yang, S. (2008). Pricing coordination in supply chains through revenue sharing contracts, International Journal of Information and Management Sciences, Vol.19, 31-51. Department of Financial and Tax Planning, Jinwen University of Science and Technology, No. 99, Anchung Rd., Sindian Dist., New Taipei City 23154, Taiwan, R.O.C. E-mail: [email protected] Major area(s): Stochastic models in finance and marketing, real options. Department of Management Sciences, Tamkang University, No.151, Yingzhuan Rd., Tamsui Dist., New Taipei City 25137, Taiwan, R.O.C. E-mail: [email protected] Major area(s): Statistical theory, sampling and survey, operation research.

(Received December 2013; accepted March 2014)