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This paper studies the effects of sudden events on the optimal timing and capacity choice in a duopoly market. According to the characteristics of economic environment, we assume that the product demand follows geometric Brownian motion with a Poisson jump process. Under the settings, the firms face the risk of a sudden drop in demand which is caused by sudden events. We develop the real option game model to derive the investment equilibrium strategies. Moreover, the effects of sudden events on investment decisions are obtained by numerical analysis.

We develop the real option game model to discuss the effects of sudden events on the optimal timing and capacity choice in a duopoly market. When sudden events occur, such as the financial crisis, economic policy from government, and the emergence of new products, discontinuous change in product demand appears. We use jump- diffusion process to capture the discontinuous changes of product demand.

Most real option game models suppose that the uncertainty variables such as asset price or product demand follow the geometric Brown motion (GBM) to describe the characteristics of continuous changes (e.g. Smets [

However, the GBM cannot explain some important empirical features of asset price or product demand dynamics. Jorion [

The large majority of real option game models focus on the investment timing without considering production capacity choice. However, in reality, production capacity decision is a key factor when one firm invests products. Few studies have considered the interaction between the investment timing and the production capacity in a real option framework. Besanko et al. [

The remainder of the paper is organized as follows. Section 2 introduces the basic assumption of the real option game model. In Section 3, we derive the equilibrium strategies in a duopoly market. Section 4 exercises numerical analysis. Section 5 concludes the paper.

In the section, we assume two firms have the chance to produce the homogeneous products in a duopoly market. Time is continuous and horizon is infinite. So every firm can defer the investment timing until the optimal moment to enter the market. The firm that enters first is known as the leader and the other as the follower. The product price at time t in market is given as follows:

where

Among the above factors,

In the section, we develop the model to determine the values and the investment decisions of two firms in a duopoly market, facing the risk of random sudden drop in product demand.

We need to consider the game backwards. When the leader has invested the project, the follower can make his decisions optimally in response to capacity of the leader. Suppose that the leader has invested the project with capacity q_{l}, the investment threshold and capacity of the follower are chosen as X_{f} and q_{f}, so the investment costs are

According to Itô’s Lemma, the value of the follower

The general solution of (1) is of the form:

Among them,

Moreover, the value of the follower

Condition (4) says that the value will be 0 if

Under these conditions, the value

where

We assume that the initial demand level is sufficiently low, the follower will not start production immediately. According to (7), the value of the follower

See from (10),

Combining (8) and (11), we obtain the optimal threshold and the capacity of the follower:

Substitute (12) into (9) and (10), the value of the follower

Let

When the follower is out of the market, the leader earns profits

ters the market, the leader’s profits decreases to

the leader is

The general solution of (15) is of the form:

The value of the leader

If we apply (17), (18) to (16),

where

Before the leader invests the project, the value of the leader

The general solution of (21) is of the form:

In addition, the value of the leader

Condition (24) says that:

If we apply (23), (24), (25) to (22), we obtain:

where

We assume that the initial demand level is sufficiently low, the leader will not start production immediately. To maximize the value, we apply (28), (29) to (27), and substitute (12) into (27), the value of the leader is calculated as:

Here, in order to examine impacts of pre-emptive competition, we assume that the roles of the leader and the follower are designated exogenously. The follower enters the market only after the leader has entered. This means that one firm is designated as the leader beforehand. So, the risk of pre-emption is eliminated, two firms can delay their investment to maximize their values. When the initial demand is sufficiently low, we suppose the leader (follower) select the optimal capacity

of the leader (follower) is denoted as

Combining (28) and (31), we have:

So, by substituting (32) into (12), the optimal threshold and the capacity of the follower are given by:

Comparing (14) and (32), Proposition 1 is obtained.

Proposition 1. The investment threshold of the designated leader is the same as that of the monopolist.

The designated leader has valuable option to defer investment at the optimal threshold of the monopolist as he need not face the risk of being preempted.

However, two firms are allowed to invest first in reality. This means that firm roles are endogenous. So, the risk of pre-emption exists. We assume that the initial demand level is sufficiently low, two firms are induced to delay their investment. When firm roles are endogenous, according to Fudenberg and Tirole [

Proposition 2 describes the sequential equilibrium. For the proof of Proposition 2, see the Appendix.

Proposition 2. (sequential equilibrium). If the initial demand level is lower than

the optimal capacity

the optimal capacity

Proposition 3, 4 describe the impacts of pre-emptive competition. The proofs are in Appendix.

Proposition 3. When the risk of pre-emption exists, the leader reduce its capacity to invest early, the follower increases its capacity to invest early. That is,

Proposition 4. If the initial demand level is lower than

The subsection describes the impacts of pre-emptive competition on the firms. The parameters are as follows:

Based on the given parameters, we calculate to obtain the optimal capacities and thresholds respectively as the roles of the firms are endogenous or exogenous:

So,

In the subsection, we perform a comparative static analysis, focusing on the impacts of different parameter values such as the Poisson jump process of intensity λ, the deterministic amplitude of the jumps θ, the volatility

λ. As we mentioned above, higher λ makes the investors reduce output to invest later. On the other hand, higher price which due to less output encourages the follower to invest earlier. As λ increases, the values of both the leader and the follower will decline.

In this paper, we examine the impact of sudden events on the investment timing and production capacity decisions of a firm that faces competition. We obtain the investment equilibrium strategies.

0.1 | 0.2745 | 0.3067 | 1.5456 | 2.6776 | 2.1887 | 2.1887 |

0.15 | 0.2657 | 0.3017 | 1.5747 | 2.6727 | 1.6971 | 1.6971 |

0.2 | 0.2572 | 0.2965 | 1.6054 | 2.6723 | 1.3385 | 1.3385 |

0.25 | 0.2491 | 0.2914 | 1.6377 | 2.6762 | 1.0688 | 1.0688 |

0.3 | 0.2414 | 0.2862 | 1.6715 | 2.6839 | 0.8612 | 0.8612 |

−0.1 | 0.2745 | 0.3067 | 1.5456 | 2.6776 | 2.1887 | 2.1887 |

−0.15 | 0.2660 | 0.3019 | 1.5815 | 2.6861 | 1.6994 | 1.6994 |

−0.2 | 0.2581 | 0.2971 | 1.6239 | 2.7086 | 1.3447 | 1.3447 |

−0.25 | 0.2510 | 0.2925 | 1.6727 | 2.7447 | 1.0800 | 1.0800 |

−0.3 | 0.2445 | 0.2883 | 1.7279 | 2.7939 | 0.8783 | 0.8783 |

0.2 | 0.2495 | 0.2916 | 1.0963 | 1.7928 | 2.0273 | 2.0273 |

0.25 | 0.2633 | 0.3003 | 1.3053 | 2.2034 | 2.1005 | 2.1005 |

0.3 | 0.2745 | 0.3067 | 1.5456 | 2.6776 | 2.1887 | 2.1887 |

0.35 | 0.2835 | 0.3116 | 1.8177 | 3.2168 | 2.2819 | 2.2819 |

0.4 | 0.2908 | 0.3154 | 2.1227 | 3.8225 | 2.3746 | 2.3746 |

We find that pre-emptive competition and sudden events have great influence on investment decisions; pre-emptive competition makes firms accelerate investment. Higher uncertainty for market demand increases the values of both the leader and the follower. When sudden events occur more frequently or product demand declines in greater magnitude, the values of both firms will decline.

This paper considers the case of two firms. Consequently, a natural idea is to consider the case of a number of firms. Future research can also be concerned with the application of a different random process, e.g., arithmetic Brownian motion.

This research is supported by NSFC (71271127, 10971127).

The proof for Proposition 2:

When pre-emptive competition exists, the follower makes his decisions reacting to the capacity of the leader. According to (28), we can obtain (36). According to (12), we can obtain (37).

The value is the same for both firms at the threshold

Simplifying the above equation, we conclude that (35) stands and

The proof for Proposition 3:

We proof that when

Let

Substitute

The proof for Proposition 4:

If

ing function of