Correction added on 19 January 2015, after first online publication: the affiliation and email address of Shu-Jung Sunny Yang have been changed.

Original Article

# The Timing of Capacity Investment with Lead Times: When Do Firms Act in Unison?

Article first published online: 13 MAR 2014

DOI: 10.1111/poms.12204

© 2014 Production and Operations Management Society

Additional Information

#### How to Cite

Anderson, E. J. and Sunny Yang, S.-J. (2015), The Timing of Capacity Investment with Lead Times: When Do Firms Act in Unison?. Production and Operations Management, 24: 21–41. doi: 10.1111/poms.12204

#### Publication History

- Issue published online: 21 JAN 2015
- Article first published online: 13 MAR 2014
- Accepted manuscript online: 1 FEB 2014 12:06PM EST
- Manuscript Accepted: DEC 2013
- Manuscript Received: NOV 2010

### Keywords:

- capacity investment timing;
- lead time;
- volume flexibility;
- existing capacity;
- operations strategy

### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

We study competitive capacity investment for the emergence of a new market. Firms may invest either in capacity leading demand or in capacity lagging demand at different costs. We show how the lead time and other operational factors including volume flexibility, existing capacity, and demand uncertainty impact equilibrium outcomes. Our results indicate that a type of bandwagon behavior is the most likely equilibrium outcome: if both firms are going to invest, then they are most likely to act in unison. Contrary to much received wisdom, we show that leader–follower behavior is very uncommon in equilibrium where firms do not have volume flexibility, and will not occur at all if lead times are sufficiently short. On the other hand, if there is volume flexibility in production, then the likelihood of this sequential investment behavior increases. Our findings underscore the importance of operational characteristics in determining the competitive dynamics of capacity investment timing.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

The timing of investment in new capacity has long been recognized as a critical strategic decision. To protect their competitive position, firms often need to invest when their rivals are investing; but at the same time, they want to avoid a situation in which the combination of their own and their rivals' capacities is too great, leading to poor returns. This dilemma is familiar in capital-intensive sectors such as semiconductors, hospitals, and telecommunications (Lieberman 1987b). Empirical evidence suggests that firms often have difficulty resolving this dilemma and poorly timed capacity expansions can lead to a “bandwagon” or “herd” behavior (Henderson and Cool 2003a). Indeed, Achi et al. (1996) find that return on investment increased by 3%–4% when firms are more successful in timing capacity expansion.

The decision on the timing of capacity investment involves subtle considerations when there is a long lead time to construct new capacity and bring it on line (Gaimon and Burgess 2003). This is because the decision on the amount of capacity expansion may be changed during the lead time and competitors may force such a change by their actions (Dearden et al. 1999). In a case study of the US bulk chemical industry, Ghemawat (1984) points out that the second mover in the industry, Kerr-McGee, announced its own investment plan before the expansion of the first mover, Du Pont, had fully materialized. The presence of a significant lead time enabled Kerr-McGee to force its competitor, Du Pont, to revise its initial capacity plan. This prevented Du Pont from obtaining a first mover advantage and allowed Kerr-McGee to avoid being in the strategically disadvantaged position of investment follower. Also in the North America pulp and paper industry, Christensen and Caves (1997) point out that 38% of major announced plans for capacity expansion were abandoned during the period of their empirical study. Thus, the effect of the lead time is clear: capacity announcement plans are less likely to be fully implemented when there is a longer lead time. However, the relationship between lead time and capacity timing competition has received comparatively little attention in most analytical work (Ghemawat and Cassiman 2007).

Volume flexibility entails the ability to underutilize the installed capacity. We describe an installed capacity as *flexible* in production if a firm has the ability to profitably decrease production below capacity (sometimes called a *holdback* strategy: Van Mieghem and Dada 1999; or, referred to as *downside* flexibility: Goyal and Netessine 2011). For instance, in semiconductor manufacturing, a firm may need to cut production when demand levels are falling (Wu et al. 2005), to avoid surplus production pushing prices to very low levels. In contrast, a firm without volume flexibility always produces to its capacity limit (Goyal and Netessine 2011). In other words, the inflexible firm follows a production *clearance* strategy (see Van Mieghem and Dada 1999). In an extreme situation, firms may even be forced to sell product at less than cost if demand is very low. This can occur in capital-intensive sectors when there are large *fixed costs* associated with production ramp-up; for example, it is not uncommon for automobile and DRAM manufacturers to sell products below cost to maintain high capacity utilization (Goyal and Netessine 2007, Wu et al. 2005). Additionally, such inflexibility may be generated through the economic costs, political efforts, and structure changes incurred in building capacity, making it very costly to withdraw or hold back some capacity in production, see Chen and MacMillan (1992) for a detailed discussion.

In many cases, the industry structure will determine the amount of volume flexibility in the operational processes. For example, in the chemical and automobile industries, firms frequently produce up to their capacity limit due to the high fixed costs of starting and stopping production (Goyal and Netessine 2007). In this case, exercising a clearance policy in production due to volume flexibility may be thought of as the outcome of selling products at a series of different prices until capacity is exhausted (in which case the market price is actually an average price). Note that in this paper, we distinguish between flexible and inflexible capacity in terms of production volume; and, we view volume flexibility or inflexibility in production (either holdback or clearance) as a fixed characteristic of the industry studied. From an operations perspective, volume flexibility plays a crucial role in capacity decisions (Goyal and Netessine 2011); however, most oligopoly models in capacity investment ignore volume flexibility to obtain analytical results (e.g., Goyal and Netessine 2007, Swinney et al. 2011).

We attempt to fill these gaps in the literature by developing a simple game-theoretic model to derive insights into the joint impact of lead time and other operational factors including volume flexibility and existing capacity on competitive capacity investment timing. Although the findings from our stylized model are somewhat restricted by the model assumptions, we are able to develop managerial insights and we believe that our efforts contribute to the operations management literature by showing how operational factors determine firm behavior in a competitive, uncertain environment. The specific research questions we address in this paper are:

- What is the effect of lead time on the timing of capacity investment when firms enter new markets?
- How does the presence of volume flexibility in production impact on capacity investment timing in entering new markets?
- Why is it rare to observe sequential investment behavior in capacity timing competition, so that we do not often see one firm taking a leadership role, and the other firms following?

We begin by illustrating the linkages between investment timing and the lead time required to bring new production capacity on line, and motivate the above research questions using several industrial examples.

#### 1.1 Competitive Capacity Investment Timing: Illustrations

An important element in operations strategy is the ability of firms to make ‘commitments” to expand capacity (Ghemawat and Cassiman 2007). Researchers have argued that by acting quickly and decisively, a firm may be able to preempt its competitors from taking action (Lieberman and Montgomery 1988). In a survey of over a thousand managers (Song et al. 1999), the advantages of being first to market are generally perceived as strong by managers of manufacturing firms. This strategy gains its value from the possibility that when a firm invests earlier and on a larger scale than its rivals, then the rivals will be discouraged from investing (Lieberman 1987c) and potential competitors will be deterred from entering (Porter 1980). The empirical evidence, however, suggests that successful use of this strategy is rare (Lieberman 1987b). In their study of the US chemical industry, Gilbert and Lieberman (1987) discover that firms appeared to bunch their capacity addition, in a type of investment *bandwagon* effect. Rather than deterring their competitors, capacity expansion announcements tended to attract competitors' announcements (Lieberman 1987a). It seems that, in practice, it is hard for firms to make a successful preemptive capacity expansion.

Consider a recent example from the semiconductor industry. In 1997 TSMC (Taiwan Semiconductor Manufacturing Company), the world's largest semiconductor foundry supplier, announced that it would invest more than $14.5 billion over 10 years to build six advanced wafer plants in Taiwan. Meanwhile, UMC (United Microelectronics Corporation), the number two semiconductor foundry supplier, issued a US $18.5 billion investment plan, which was even more ambitious (Liao 1998). At the beginning of 2002, the market thought that there would exist at least ten 300-mm-wafer plants within 5 years, which would imply serious overcapacity in the semiconductor manufacturing industry. However, TSMC sliced its projected capital expenditure for 2002 by 20%, to about $2 billion and delayed the ramp-up to full-scale production of its first 300-mm-wafer plant in Hsinchu until 2003. Almost at the same time, its rival UMC cut back too, and lowered its 2002 capital expenditure to $1.3 billion from the originally planned US $1.6 billion (Robertson 2002). By the end of 2005, TSMC had only completed two 300-mm-wafer plants and UMC had the same number of 300-mm-wafer plants. The serious overcapacity which had been predicted did not eventuate. This instance demonstrates how capacity expansion in a competitive environment encourages firms to announce expansions more or less in *unison* when there are long construction lead times.

#### 1.2 Overview of Our Model and Results

In our model, each of the competing firms can invest in capacity either before or after learning about the size of the market demand. This corresponds to the two well-known timing strategies of capacity expansion: *capacity leading demand* and *capacity lagging demand* (see Van Mieghem 2003). In the absence of competition, a monopolist will adopt a strategy of capacity lagging demand if maintaining a high utilization of capacity is important. However, the choice for a firm in a competitive market is much more complicated. In the settings we consider, a firm needs to weigh the preemption value gained by an early move against the cost of an unnecessary investment if a market expansion does not materialize when expected. Moving early with a new market can also be valuable because of the monopoly power that can be exercised during the period before the other firm's entry; the magnitude of this advantage will depend on the length of the lead time.

When one firm makes an investment decision, rival firms will watch closely and shape their own actions on the basis of what they see their competitors doing. Firms can estimate their rivals' future capacity level and conjecture their future demand forecast. Because of the construction lead time, a competitor can make a decision to react well before the first firm's capacity becomes available (Ghemawat 1984). Thus, we model this competitive capacity investment as a simultaneous-move game, rather than a sequential-move one. In our model, we consider two identical firms both aiming to maximize profits by selling homogeneous products to customers in a multiple period setting. This symmetric framework is for mathematical tractability (and has been used by others interested in multi-period and endogenous timing games, e.g., Pacheco-de-Almeida and Zemsky 2003, Saloner 1987, Swinney et al. 2011, Van Mieghem and Dada 1999).

It is important to pay attention to the exact sequence of decisions made. We suppose that a firm first *announces* whether it will make a capacity investment leading or lagging demand, but cannot yet credibly set the actual investment level. The firm observes its rival's announced strategy before finally deciding on an investment level. The eventual investment levels are the equilibrium outcome of competitive interactions between the firms' announced timing strategies of investment rather than being the firms' decision variables. In real-world situations, there may be several rounds of announcements and manoeuvring in response to the announcements before firms reach the final investment outcome (Christensen and Caves 1997). We do not attempt to model successive rounds of announcements; but, we introduce additional realism by pulling apart the investment decision into a first stage in which the basic strategic posture is settled, which is then followed by more detailed decisions on the size of investment.

We will discuss four factors that drive equilibrium behaviors of investment in a capital-intensive sector: (i) the industry outlook, (ii) the potential market size, (iii) the investment cost, and (iv) the lead time. Industry outlook is represented by the probability that the potential market for the product occurs. We will show that in our model, the combined impact of the size of the potential market demand and the cost of investment can be described by a single structure parameter, which we call the *degree of capital intensity*. This measure is the ratio of the square root of the cost to the potential market size. This then gives three parameters that are varied in our analysis: capital intensity, industry outlook, and lead time.

The interplay of these characteristics in our model leads to the first of our main insights: *Firms usually invest at the same time*. It is well known that a monopolist invests in capacity leading (lagging) demand when industrial outlook is sufficiently optimistic (pessimistic). We find that this is still true under competition: when industry outlook is poor, firms will choose to wait until demand realizations are known. In most cases when the market is worthwhile for investment to take place, the competition and industry outlook effects interact to yield an equilibrium in which both firms invest. When industry outlook is optimistic, both firms invest ahead of demand (i.e., capacity expansion bandwagon behavior), but if industry outlook is more pessimistic, both firms invest after demand has occurred. Despite all the discussion in the literature of credible commitment (e.g., Ghemawat and Cassiman 2007) and first-mover advantage (e.g., Lieberman and Montgomery 1988), we show that it is rare for a firm to move early and successfully force a potential rival to invest later. In fact, we will show that this feature is most pronounced with a short lead time: this behavior never occurs when the lead time is sufficiently short. The symmetric early investment outcome matches the investment bandwagon effect that has been found in empirical investigations in different industry sectors, such as global petrochemicals (Henderson, and Cool 2003a, b), US chemical processing industry (Lieberman 1987c), US paper industry (Christensen and Caves 1997), and UK brick industry (Wood 2005). All these industries are characterized by lumpy capacity increments and high fixed costs.

Throughout this paper, the *leader–follower* behavior refers to the equilibrium outcome where one firm employs a capacity leading demand strategy and the other employs a capacity-lagging demand strategy. Our main insight amounts to saying that this leader–follower behavior is rare. We extend the analysis to a variety of relaxations of our base model, including the possibility of volume flexibility (section 'Volume Flexibility'), existing capacity endowment (section 'Existing Capacity'), and more general stochastic demand behaviors (section 'Continuous Stochastic Demand'), and find that the managerial insights are consistent. In general, we find that variations in the model do not impact our conclusion that leader–follower behaviors are unlikely. For example, any asymmetry in the amount of existing capacity tends to decrease the likelihood of leader–follower behavior and favors the early investment bandwagon behavior. The exception, however, occurs with volume flexibility and this leads to our second main insight: *Sequential (leader–follower) investment equilibria are more likely to occur with volume flexibility*. We show that leader–follower behavior is much more likely to occur when there exists flexibility in production volume. Thus, when the lead firm cannot commit to fully utilize its capacity and the capital intensity is not too high, the other firm is more likely to delay investment rather than reacting by investing early. There are many examples of firms that are successful by letting other firms pioneer and then enter new markets later. A recent example has occurred with netbook computers (see Huang and Sošić 2010). In October 2007, ASUS unveiled its first netbook computer; at that time, other laptop manufacturers did not follow ASUS by offering similar products although most of them have the capability of making netbooks and raising capacity utilization is always a key concern in that industry. Thus, ASUS enjoyed its monopoly position during the first quarter after its release and the sales of Eee PC 701 surged up to 350,000 units. The establishment of this market then triggered other laptop manufacturers, such as Acer and Dell, to offer their own netbooks. By the end of the third quarter of 2008, the sales of netbooks grew 160% and ASUS could not deter Acer and Dell from entry. Our results suggest that these types of outcome are most likely to occur when there is volume flexibility in production.

The remainder of this paper is structured as follows. Section 'Literature Review' gives a review of related literature and shows how our work fits within this. We introduce our model and discuss the key assumptions in section 'Model'. Section 'Analysis' gives the analysis of our model; first for a monopoly, and then for a duopoly. In section 'Robustness of the Model and Insights' we discuss a number of different extensions of the model. We conclude in section 'Conclusion'. The proofs and technical details are provided in Online Appendix S1 if they are not reported in the main body of the paper.

### 2 Literature Review

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

Two streams of literature are relevant to our study: the first from operations management considers optimizing the firm's resource accumulation (e.g., of capacity or inventory, etc.) when increasing returns of investment exist, and the second from industrial organization investigates first/second mover advantage and explores the effect of resource accumulation strategies on the firm's competitive advantage.

There is a substantial body of research in operations management investigating capacity expansion with uncertain demand, see Van Mieghem (2003) for a comprehensive review. In particular, literature on postponement looks at the value of flexibility obtained through delaying operational activities until demand information is available; see, e.g., Anupindi and Jiang (2008) and Van Mieghem and Dada (1999). However, these works differ from ours, in that they do not consider the lead time and fixed costs of investment. Most research in this stream has considered a monopoly setting. For example, Chaouch and Buzacott (1994) and Gaimon and Burgess (2003) consider the impact of the lead time on capacity investment timing and size and other authors give consideration to fixed costs in the investment decision (see Van Mieghem 2003). Variations in the capacity expansion problem that include technology choice and volume flexibility issues appear in Anupindi and Jiang (2008), Huang and Sošić (2010), Rajagopalan et al. (1998), and Goyal and Netessine (2007, 2011). While these papers study various effects of operational factors on capacity decisions, to the best of our knowledge, no operations paper considers the joint impact of the lead time and operational factors such as volume flexibility and existing capacity on competitive capacity investment timing. Indeed, besides Bashyam (1996) and Swinney et al. (2011), little attention has been given to competitive capacity investment timing in the operations management literature. We complement this stream of research by incorporating lead time, volume flexibility (or production policy), fixed costs, and existing capacity into competitive capacity studies.

Our paper is closely related to the industrial organization literature that uses analytical models in which the critical decision on whether firms move simultaneously or sequentially is determined endogenously to investigate first or second mover advantage in various duopoly settings. The seminal paper in this area is Saloner (1987) who analyzes an extended market game allowing for two production periods before the market clears. In Saloner's model, there is flexibility with respect to the timing decision of resource accumulation. He finds a continuum of equilibria that include Cournot-Nash and Stackelberg outcomes in the case of a well-behaved inverse demand function and constant marginal costs of resource accumulation. Many papers (see Dewit and Leahy 2006) suggest that Stackelberg outcomes are more plausible than Cournot outcomes if firms can choose when to move.

Pacheco-de-Almeida and Zemsky (2003) explore capacity accumulation strategies in the presence of time compression diseconomies by giving a treatment of capacity expansion in a duopoly in which there is a substantial lead time. This means that a decision to invest in capacity in advance of the market uncertainty being resolved will enable product to be sold earlier than if the firm decides to wait. This gives rise to a model with two production periods. Their model has some similarities to ours, but they do not consider a fixed investment cost, or separate the timing and quantity decisions in the way that we do. They find that incorporating a lead time gives rise to an incremental equilibrium where firms invest in both periods, which is different from our results. Dewit and Leahy (2006) explore capital accumulation strategies in the presence of scale economies by considering the presence of a time lag between the decision on when to invest and the decision on how much to invest, arguing that this reflects actual practice. Although their model setup is different, their paper is closely related to our work in the treatment they give of the effect of partial commitment, in which a firm commits to the timing of its capacity investment without determining the capacity level to be built. Our paper complements the industrial organization research by making a more realistic set of modeling assumptions regarding the firms' production decisions.

We contribute to the literature on capacity expansion and investment timing by explicitly studying the effect of operational factors on competitive timing decisions. Besides Pacheco-de-Almeida and Zemsky (2003), most analytical works model the sequence of events as exogenous to delineate the “strategy dynamics” and they do not consider the effect of the lead time on the firm's timing decisions (see Ghemawat and Cassiman 2007). By working with an endogenous timing model, we are able to consider timing as an explicit part of the firm's strategic operations decision and gain a better understanding of the effect of competition on the firm's capacity timing strategy.

### 3 Model

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

We consider two identical firms, indexed by *i*, *j* = 1, 2, with *i* ≠ *j*, both of which are risk-neutral and profit maximizers. The firms compete in supplying a potential market opportunity. Their products are perfectly substitutable (homogenous) and quantity-setting (Cournot) competition is assumed in the marketplace. Cournot competition is appropriate for the description of the medium- to long-term equilibrium determination and is supported by the theoretical result of Kreps and Scheinkman (1983). Anupindi and Jiang (2008) further suggest that price competition should be modeled as a Cournot game if firms have some flexibility in exercising control on quantity through production or delivery. So, the Cournot setting is commonly adopted in the model-based capacity literature on operations management and industrial organization (Goyal and Netessine 2007, Pacheco-de-Almeida and Zemsky 2003). In our model, investment decisions are made simultaneously, so that every investment opportunity is a simultaneous-move non-cooperative game. We suppose that both firms are fully aware of one another's cost structure so that there is no private information in the model.

*Capacity timing and construction lead time*. The sequence of events is shown in Figure 1. In the beginning, the firms choose to invest in capacity before or after the emergence of a new market. If only one firm invests ahead of the demand realization, this lead firm has a period of monopoly profits due to the lead time that a lag firm needs to build capacity prior to supplying its products to the market. We normalize the length of the market opportunity to 1 and write *α* for the period of lost market opportunity for a lag firm. This will be the lead time minus the gap between the time at which market uncertainty is resolved and the start of the market opportunity. If both firms invest prior to the demand realization, they will compete head-to-head in the market. If both choose to delay investing, and only commit funds when they are certain, a new market will emerge, they will compete in production but both of them lose the possibility of meeting any demand during the time to build.

*Market uncertainty and price behavior*. Fundamental to the capacity decision is the uncertainty that exists on the size of the market. There are many ways to model this uncertainty; but, for analytical tractability, we adopt the simplest setup of Bernoulli-type probability model to depict the potential market emergence that has been commonly adopted in the operations, marketing and industrial organization literature such as Pacheco-de-Almeida and Zemsky (2003) and Yang et al. (2009).

To model the potential market emergence, we use *λ* to denote the probability that demand occurs where 0 ≤ *λ* ≤ 1 (so 1 − *λ* thus represents the probability that a new market does not eventuate). We refer to parameter *λ* as the *industry outlook* since it reflects the degree of optimism about the new market emergence.

The market price of the product is given by a linear demand curve if the market exists; and without loss of generality, the units of quantity are normalized so that the product price is given by *p*(*Q*) = *A*− *Q* if demand occurs, where *Q* is the total product quantity put on the market by the two firms combined. If demand does not occur, then neither firm produces and the cost of investment will be lost for firms investing early. The production quantity put on the market by firm *i* is *y*_{i} (so that *Q* = *y*_{i} + *y*_{j}), where *y*_{i} is restricted to be no greater than the investment level in capacity, *x*_{i} (and is zero if demand does not occur). The demand intercept *A* is a positive constant.

The price drops to zero when there is a quantity *A* offered and so *A* is the maximum amount that can be sold and we refer to it as the *potential market size*. Note that *A* can also be characterized as the maximum price that can occur if there is a shortage. Due to its analytical simplicity, the choice of linear demand is widely used in capacity competition studies and is often a good approximation to more general demand functions (see, e.g., Anupindi and Jiang 2008, Goyal, and Netessine 2007, 2011). This downward sloping linear inverse demand curve can also be thought of as the result of utility-maximizing behavior by customers with quadratic, additively separable utility functions.

*Cost structure for investment and operations*. Now we turn to the cost structure of the firms: a fixed capacity development cost is incurred at the time point where a firm installs capacity. This cost may include construction and space costs, as well as the development costs for a new technology or for new resources, see Rajagopalan et al. (1998) and Krishnan and Zhu (2006) for details. So, the cost function of investment is concave to reflect potential economies of scale; this is a common assumption in both the capacity expansion literature (e.g., Hendricks et al. 1995, Rajagopalan et al. 1998) and the product/resource development literature (e.g., Grahovac and Miller 2009, Krishnan and Gupta 2001, Lacourbe et al. 2009). It has been validated in a variety of industries (Asano 2002).

We allow a cost difference for investment with different timings: a firm that adds capacity prior to the market demand incurs a fixed cost of *K*, while a firm that waits until after the market demand is confirmed incurs a cost *mK*, where *m* is referred to as the relative cost of building capacity, *m* ∈ (0, ∞). Normally we will take *m* ≤ 1 implying a greater cost for a firm investing early. In this way, we can model three factors that may occur in practice. First, the timing of the market demand may be uncertain. In this case, firms investing prior to the demand realization may have their production capacity sitting idle for some period, and a higher investment cost reflects the cost of doing this. Second, the use of *m* < 1 allows us to model the time value of money, with later investment being cheaper. Finally, there are some circumstances when a delayed investment is cheaper because of improvements in technology. This has been called “time compression diseconomies” in the strategy literature (Pacheco-de-Almeida and Zemsky 2007) and broadly reflects a situation in which the faster a firm seeks to develop a resource, the greater will be the development costs. Nevertheless, for generality, we also allow *m* > 1 which reflects a situation with rising investment costs, as do Swinney et al. (2011).

Without loss of generality, we normalize the variable cost of production to zero (see, e.g., Anupindi and Jiang 2008, Goyal and Netessine 2007). Moreover, we assume following Swinney et al. (2011) and Boyabatli and Toktay (2011) that capacity investment for a firm can occur at most once, so a firm cannot invest in capacity both before and after the emergence of market demand. In fact, in our model, it will never be optimal to invest both before and after the demand realization, since there is an extra cost to doing so and there are no circumstances when we prefer to have greater capacity for the second part of the market opportunity (after time *α* ).

It will be convenient to define a structure parameter:

which we will call the *capital intensity*. When Γ is small, entering the market requires a small investment cost (in comparison with the profit available), and the reverse is true when Γ is large. The form of the expression for Γ arises from the fact that the maximum profit varies as the square of the potential market size *A* (as we will see in the later analysis), so Γ is related to the ratio between fixed costs and profit. By using Γ instead of *A* and *K* separately we are able to represent more information in a single diagram. Although Γ^{2} will occur very frequently, using Γ as a parameter (rather than Γ^{2}) gives a more convenient scale in these diagrams.

*Assumptions and their justification*. For simplicity, we need the following two mild assumptions in the subsequent sections:

Assumption 1. *λm* ≤ 1.

This rules out some extreme cases where it is much more costly to invest after demand occurs than to invest prior to demand. For example, this assumption implies *m* < 3.34 for *λ* = 0.3, and *m* < 1.43 for *λ* = 0.7. A cost disadvantage from capacity lagging demand could occur if production equipment manufacturers offer a discount for early investment or if some resource has increasing scarcity over time. However, the actual values of *m* implied by Assumption 1 are quite large and seem unlikely to be a restriction in practice.

Assumption 2. *α* ≤ 0.5.

The assumption on *α* simply implies that no more than half the market opportunity can be lost while capacity is built, i.e., the product life cycle is long enough for a lag firm to supply its products to the bulk of the market after the time-to-build.

These two assumptions rule out the trivialities that it is never profitable in expectation to invest either before or after the demand realization, and many analytical operations management studies implicitly make similar assumptions by scaling demand functions, investment costs, and operational costs (e.g., Anand and Girotra 2007). These two assumptions will be satisfied in the great majority of real-world situations.

#### 3.1 Rules of the Game

There are two times at which investment in capacity may be initiated: either before or after demand uncertainty is resolved. Since capacity takes a significant time to build, we need to be careful in specifying the timing of decisions that are made. We will assume that the decision on whether or not to build capacity is separated from the decision on the size of the capacity investment. In fact, we assume that both firms have an opportunity to see whether or not their opponent is building capacity, before deciding how large their own investment should be. In practice, the decision of a firm to make an investment of this type will normally be “announced” some time before the construction or investment actually takes place. Sometimes, an early announcement will be liable to later change at least as far as the size of the investment is concerned. We suppose that announcements of investment strategies made before demand occurs are credible and inflexible in terms of the investment *timing*, but do not include information on the investment amount. Hence, each firm has only two possible options in the announcement phase in terms of timing: to announce that it will invest prior to the demand realization, which we call *capacity leading demand* (*C*); or, to announce that it will wait until the demand uncertainty is resolved, which we call *capacity lagging demand* (*D*). Our announcement setting fits with Camerer's (1991) argument that preplay communication prior to tactical decisions is typical in strategic decisions such as pre-announcements of new computers or airline fare changes. The setting is also supported by the empirical capacity expansion literature such as Dearden et al. (1999) and has been adopted in the analytical capacity expansion literature such as Dewit and Leahy (2006).

The announcement must match the firm behavior prior to demand occurring, so that a firm builds capacity prior to the demand realization (*ex ante*) if and only if it announces a lead strategy (i.e., *C*). A firm that announces a lag strategy (i.e., *D*) will only invest in capacity after demand occurs (*ex post*). Hence, the lag strategy can be broken down further into a decision to build capacity if demand occurs, which we call *wait-and-see* (*N*); or, not to invest at all, which we call *exit* (*E*). In our model, it does not matter when the decision between *N* and *E* is made. This is because once a lag strategy has been announced, an investment decision is only needed if demand occurs, and so can be made prospectively.

Our model is a multi-stage simultaneous-move game, depicted in Figure 1. The first-stage game (ahead of the emergence of the market) is referred to as the *announcement game*. There are four possible pure-strategy outcomes to the announcement game. Two are symmetric: both firms announce lead, i.e., (*C*, *C*) pair, and both firms announce lag, i.e., (*D*, *D*) pair. There are two asymmetric outcomes in which one firm leads and the other lags, i.e., (*C*, *D*) or (*D*, *C*) pair. The *capacity game* then unfolds according to the sequences of moves determined by the announcement game. The final stage is the *production game* to determine the output quantity to be put in the market, given the first two decisions: investment timing (decided in the announcement game) and investment level (decided in the capacity game). The production decision is *ex post* since at the time of production, the firm is informed about the market condition and its rival's capacity.

#### 3.2 Problem Formulation

The announcement game is schematically represented in Table 1 as a 2 × 2 matrix typical for strategic-form games where firm *i* is the row player and firm *j* is the column player. In the table, each row–column intersection signifies a subgame in the announcement game, while the matrix entries Π signify expected total profits (given the decisions on leading or lagging). These profits are themselves the results of equilibrium behavior in the capacity and production games. When we use this type of table, the first entry is for firm *i* and the second for firm *j* in each cell of the matrix representation of the game. The superscript *XY* denotes the focal firm employing capacity timing strategy *X* and its competitor employing capacity timing strategy *Y*, where *X*, *Y* ∈ {*C*, *D*}.

Capacity leading (C) | Capacity lagging (D) | |
---|---|---|

Capacity leading (C) | , | , |

Capacity lagging (D) | , | , |

We seek a “subgame perfect Nash equilibrium in pure strategies” (SPNE) for the announcement game. An equilibrium solution is a strategy pair in which neither firm has an incentive to unilaterally deviate. To determine the equilibria of the announcement game, we need to analyze the decisions on capacity amount and production quantity in all subgames of the announcement game in reverse chronological order.

We begin with the *ex post* investment stage by supposing the timing and level of capacity investment have already been made. We write *π*_{i}(*x*_{i}, *x*_{j}) for the expected operating profit available to firm *i*, without including the cost of capacity investment, given that firm *i* has capacity *x*_{i} and firm *j* has capacity *x*_{j}. In the general scheme of Figure 1, this is the result of a production game where each firm optimizes its production quantity *y*_{i} subject to a capacity constraint, *y*_{i} ≤ *x*_{i}. For the moment, we will concentrate on the case where there is no volume flexibility (so firm *i* follows a production clearance strategy) and *y*_{i} = *x*_{i}. (This is also assumed in most analytical competitive capacity studies including Swinney et al. (2011).) In a later section, we will consider what happens when we introduce volume flexibility. The operating profits depend on the timing announcements as well as on the prior capacity decisions:

Here is an index variable, which equals 1 with probability *λ* and 0 with probability 1 − *λ*. The expected value of gives the probability that the market demand occurs.

After the announcement stage, a capacity game is played, in which each firm makes a decision on the size of the capacity investment to make, subject to the timing decisions made at the announcement game stage. All of this is discussed in more detail below.

### 4 Analysis

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

In this section, we first analyze the monopoly model of capacity investment timing in a new market with uncertain demand and then move to the duopoly model of capacity investment announcement. The monopoly model will serve as a baseline for comparison with the subsequent duopoly model, so we retain our basic modeling assumptions.

#### 4.1 Monopoly Model

A monopolist firm's optimal expected profit from capacity leading demand is

The firm's optimal expected profit from capacity lagging demand is

In both cases, the optimal choice of capacity *x* is *A*/2. The following result gives the optimal investment timing under different conditions.

Theorem 1. A monopolistic firm

- invests prior to demand occurring (
*C*) if ; - invests only after demand occurs (called wait-and-see,
*N*) if ; - makes no investment at all (
*E*) if .

Note that if Γ lies on the boundary between two of these regions, then the monopolist is indifferent between these two choices. Throughout this paper, this indifference at the boundaries between Γ regions holds in all our analytical results for both optimal and equilibrium solutions.

Figure 2 shows the optimal investment timing for a monopolist as a function of the capital intensity and the industry outlook when *α* = 1/3 and *α* = 1/6 given that *m* = 1 due to the fact:

Remark 1. Without loss of generality, *m* = 1 is fixed in the graphical illustrations due to the fact that the direction of impact of lead time *α* on the optimal and equilibrium solutions is similar to that of the relative cost of building capacity, *m*.

There are three regions in the figure: early capacity leading demand resulting in early investment (*C*), capacity lagging demand resulting in late investment (*D* *N*), and capacity lagging demand resulting in exit (*D* *E*). Throughout this paper, the notation “” indicates whether strategy *D* in the first-round announcements becomes strategy *N* or *E* in the optimal solution or equilibrium of the second round announcements after demand occurs. Figure 2a and b will help us better understand the impact of competition on capacity timing behavior by comparing the monopoly case to the duopoly cases.

As we might expect, this proposition shows that a monopoly firm prefers early investment if the capital intensity is low and the industry outlook is optimistic, corresponding to the bottom right hand corner of these figures. It is easy to see that increasing the lead time will make delaying investment less attractive. The same is true if we increase the probability of the market occurring. Summarizing these observations, we have the following corollary, which can be established simply by looking at the direction of change of the region boundaries.

Corollary 1. For a monopolistic firm, the likelihood of investing in capacity leading demand (C) is increasing in *λ*,* m* and *α*.

#### 4.2 Duopoly Model

Now we return to the duopoly model and analyze the equilibria that occur for each of the four announcement subgames given in Table 1, assuming that both firms invest in production facilities with a technology that does not allow volume flexibility, or they need to maintain high capacity utilization due to the industry structure, so they always produce to their maximum capacity. In this setting, the lead firm can credibly commit to maintain the same production quantity after the lag firm's entry as before. Once having derived the equilibria in the capacity game, we will analyze the equilibrium to the announcement game.

Suppose that announcements have already been made and consider the resulting capacity game. With inflexible firms, the output of each firm if demand occurs is equal to its capacity and so we set *y*_{i} = *x*_{i} for *i* = 1, 2 in the production game. We consider the three possible cases in the capacity game corresponding to the announcement pairs: (*C*, *C*) in which both firms invest *ex ante*; (*D*, *D*) in which neither firm invests *ex ante*; and (*C*, *D*) (or (*D*, *C*)) in which one firm invests *ex ante*, and one does not. Upon observing the prior timing choices of capacity made by both firms and conjecturing the rival firm *j*'s capacity of *x*_{j}, firm *i* solves the following decision problem in the capacity game:

Here we write *q*^{+} = max{0, *q*} for . In the capacity game, the firms play a Cournot game in the (*C*, *C*) and (*D*, *D*) pairs, but play a Stackelberg game in the (*C*, *D*) and (*D*, *C*) pairs. The following lemma describes the resulting equilibrium in the corresponding announcement subgames. We analyze the game assuming that each player has chosen *C* or *D*. Our analysis includes a second stage of the announcement game in which a player who has announced *D*, chooses to announce (and commit to) either *N* or *E*. This extra stage of announcement serves to reduce the range of possible equilibrium solutions that can occur for certain values of Γ. For each possible asymmetric equilibrium given in the following table, there is another in which the roles of the two players are reversed.

Lemma 1. Let

Given an announcement pair with two inflexible firms, the equilibrium behavior of *ex post* investment depends on the value of Γ and the initial announcement, and Table 2 sets out the different equilibrium solutions that occur for different ranges of Γ, together with the equilibrium capacities and the equilibrium profits.

Announcements | Γ range | Capacities (x_{i}, x_{j}) | Expected profits (Π_{i}, Π_{j}) |
---|---|---|---|

(C, C) | |||

(D, D) (N, N) | |||

(D, D) (N, E) | |||

(D, D) (E, E) | (0, 0) | (0, 0) | |

(C, D) (C, N) | Γ^{2} ≤ R | ||

(C, D) (C, E) | |||

(C, D) (C, E) |

Having derived an equilibrium for each of the capacity subgames, we now start to analyze the announcement game. To do this, we will need to consider different ranges for Γ, according to the results in Table 2. Our aim is to identify the regions in the (*λ*, Γ) plane that correspond to different possible equilibrium pairs of announced strategies. It is not hard to see that if an announcement pair is an equilibrium in some set *S* in the (*λ*, Γ) plane, then it is also an equilibrium at any point on the boundary of *S*. This follows immediately from the fact that all expected profits in Table 2 are continuous functions of *λ* and Γ if announcements are held constant, and the definition of a Nash equilibrium. Thus in establishing the equilibrium behavior over different ranges of Γ, it is enough to look at intervals for Γ without considering the endpoints, and allow the behavior at these boundaries to be determined afterwards.

Our next result describes the equilibria in the announcement game for different values of Γ, using a case by case analysis. All the detailed formulations and analysis are given in Online Appendix S1.

Theorem 2. Let *ω* ≡ 2*α* + *α*^{2} + 6, and let *H*_{1} and *H*_{2} be given by the two roots:

with *H*_{1} being the smaller value. Then the following pure strategy equilibria to the announcement game exist for different ranges of Γ:

- For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if ; and, the (*C*,*N*) pair is a SPNE if neither inequality holds. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if , or, when , if either Γ ≤*H*_{1}or Γ ≥*H*_{2}; and, the (*C*,*E*) pair is a SPNE if neither the (*C*,*C*) or (*N*,*N*) pairs are equilibria. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE if neither inequality holds. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*E*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE if neither inequality holds. - For : the (
*E*,*E*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE otherwise.

Using this result, we can plot the possible equilibria that occur for different values of *α* and *m* and this is done for two examples in Figure 3. According to Remark 1, we assume that there is no cost difference between early and late investments (i.e., *m* = 1), and we show the equilibrium outcomes in the announcement game when *α* = 1/3 and *α* = 1/6 as regions in the parameter space defined by industry outlook (*λ*) and the degree of capital intensity (Γ).

The general behavior that we observe in Figure 3, and that can be established from Theorem 2, is that an equilibrium in which at least one firm uses a lead strategy *C* is more likely to occur as the industry outlook becomes better or the capital intensity decreases. To be more specific, we can show that provided the capital intensity is low enough, then the equilibrium behavior in which both firms invest early, (*C*, *C*), will occur whenever the industry outlook is sufficiently optimistic. This matches the prevalence of bandwagon investment patterns for capital-intensive sectors as discussed by the empirical capacity expansion literature.

We can also use Theorem 2 to obtain the comparative statics by looking at the signs of the first-order derivatives of the equilibrium conditions for the (*C*, *C*) pair with respect to different parameters. The result matches Corollary 1 for the monopoly case:

Corollary 2. The likelihood of the (*C*, *C*) pair being a SPNE is increasing in *λ*,* m* and *α*.

The most striking observation that can be made from Figure 3 is the very limited range of circumstances in which we find one firm investing early and the other investing late (i.e., the (*C*, *N*) pair). Moreover, this equilibrium behavior is increasingly unlikely as the lead time *α* decreases, with the equilibrium region for the (*C*, *N*) pair, i.e., the leader–follower behavior, disappearing entirely when the lead time is sufficiently short. We have the following corollary to Theorem 2 demonstrating that when lead times are short in comparison with the time for which the market exists, then the firms will either end up acting in unison or one of the firms will not invest at all.

Corollary 3. If *α* < 0.2 then the (*C*, *N*) pair is not a SPNE and the region Ω of multiple equilibria (*C*, *C*) and (*N*, *N*) occurs.

This corollary shows that the leader–follower behavior does not occur as an equilibrium when more than four fifths of the total market demand occurs after the lagging firm enters the market. Hereafter, we use the notation “Ω” to refer to the region of multiple equilibria of the (*C*, *C*) and (*D*, *D*) pairs co-existing. Simply speaking, when the lead time is short, the bandwagon behaviors of the (*C*, *C*) and (*N*, *N*) pairs are far more likely. This is an unexpected result and demonstrates that the leader–follower behavior arises partly as a result of the loss of market opportunity from delaying investment.

At first sight, the fact that short lead times make leader–follower behavior less likely may seem counter-intuitive. With a short lead time, the loss from moving late is also small and so we could expect that investing late is correspondingly more attractive. Our explanation is that the leader–follower equilibrium requires *both* parties to be satisfied with the outcome. One firm moves early and gets an advantage both from selling when the other firm is not in the market and through dictating the quantity sold. The lag firm has the advantage of not needing to invest when the market does not eventuate, but loses sales both in the first part of the selling season and also through having to accommodate a selling quantity set by the lead firm.

It is interesting that multiple equilibria may exist when the lead time is sufficiently short, see Figure 3b. In the region of multiple equilibria, both firms simultaneously invest either early or late; thus, it is hard to predict which announcement pair will take place. In other words, there is “strategic uncertainty” emerging endogenously from the strategic decisions of firms (Porter 1980). We thus conclude that strategic uncertainty increases as the lead time decreases. There are also other multiple equilibria in all the regions where an asymmetric pattern occurs (e.g., (*C*, *N*) or (*C*, *E*)), since as is common with “battle of the sexes” types of game there is no way to determine who will take which role given symmetric firms. With some prior communication or previous competing history, we might expect that a particular equilibrium will be chosen through some “focal point” effect (Schelling 1960). See Cachon and Netessine (2004) for a discussion of the implications of multiple equilibria.

Figure 4 depicts a direct comparison between Figures 2 and 3 to examine the effect of competition on investment timing. In region *C*, both the monopoly and duopoly solutions suggest capacity leading demand; in region *N*, both solutions suggest capacity lagging demand; in region *E*, both suggest an exit to the market; and in region *B*, the monopoly solution suggests late investment, but the duopoly solutions (including equilibria (*C*, *C*) and (*C*, *E*)) suggest at least one firm makes an early investment. Note that the region Ω of multiple equilibria (*N*, *N*) and (*C*, *C*) in Figure 3b has been labeled as *N* in Figure 4b (since the monopoly solution matches at least one of the duopoly solutions). Thus, when there is competition, capacity leading demand is more likely to occur due to the existence of region *B*. Furthermore, Figure 4a and b suggest that the size of region *B* is decreasing with *α* and for small values of Γ we can prove this formally by considering the various derivatives:

Corollary 4. If , then the region *B* is decreasing in *α*.

Note that even outside these specific Γ values, numerical calculations for different *α* values tend to confirm our conclusion on the region *B* decreasing in size as *α* increases.

To summarize our analysis in this section: The main conclusion is that leader–follower behavior of the (*C*, *N*) or (*N*, *C*) pair does not usually occur, and become increasingly unlikely when lead time is short. Indeed, the lead time variable is critical in a number of areas. Not only is a leader–follower solution impossible with short lead times, but we also find that short lead times, although tending to make delay more likely (Corollary 2), do not do so to the same extent in the duopoly as in the monopoly. Thus, we have the result (Corollary 4) that short lead times make it more likely that a duopoly equilibrium brings forward the investment in capacity in comparison with the monopoly solution.

### 5 Robustness of the Model and Insights

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

Up to this point, we have looked at a simple model. In this section, we will examine some variants of this model with the aim of investigating whether our main insights summarized at the end of the last section are robust to these variations (and identifying some critical characteristics that are important in our results). We will see that loosening the restrictions of our original model will, in some cases, make the analysis quite complex, and so, to understand the different implications, we will consider variations on our original model one at a time.

#### 5.1 Volume Flexibility

In this section, we consider the equilibrium outcomes in timing competition between firms with volume flexibility. This will enable us to make a comparison with an inflexible duopoly and allow us to see how volume flexibility has an impact on competitive outcomes. In this case, a lead firm is not able to credibly commit to maintaining the same production quantity after the other firm enters, and a lag firm will form an expectation about the leader's incentive to hold capacity back (Anand and Girotra 2007, Dixit 1980). We now analyze this case in detail.

A firm will only choose to produce less than its capacity (a production holdback strategy) in one of the asymmetric announcement pairs, (*C*, *D*) and (*D*, *C*). In the (*C*, *C*) pair, both firms invest capacity in the Cournot outcome *ex ante* and produce the Cournot outcome if the demand occurs. In the (*D*, *D*) pair, the investment occurs after the resolution of demand uncertainty; hence, the firms would never invest in more capacity than necessary. It is obvious that a firm with a lag strategy always produces to full capacity. In sum, production holdback strategies (utilizing volume flexibility) can only occur in the (*C*, *D*) or (*D*, *C*) pair when a lag firm can force a lead firm to hold back its capacity. Formally,

Lemma 2. Suppose that both firms are flexible in production volume and Assumptions 1 and 2 hold. If the *ex ante* announcement is (*C*, *D*), the equilibrium outcomes after the demand realization are:

- if , the
*ex post*equilibrium announcement is (*C*,*N*) with equilibrium capacities and equilibrium profits are ; - if , the
*ex post*equilibrium announcement is (*C*,*E*) with equilibrium capacities and equilibrium profits are .

Because the equilibrium outcomes of (*C*, *C*) and (*D*, *D*) are identical to those in the setting with inflexible firms, using this lemma, we can obtain the various equilibrium solutions that occur for different ranges of Γ. These are given in Table 3.

Announcements | Γ range | Capacities (x_{i}, x_{j}) | Expected profits (Π_{i}, Π_{j}) |
---|---|---|---|

(C, C) | |||

(D, D) (N, N) | |||

(D, D) (N, E) | |||

(D, D) (E, E) | (0, 0) | (0, 0) | |

(C, D) (C, N) | |||

(C, D) (C, E) |

Following the logic used to prove Theorem 2, we obtain the following theorem that describes all of the possible SPNE for the announcement game between flexible firms.

Theorem 3. Suppose that both firms are flexible in production volume and Assumptions 1 and 2 hold. Then, the following pure strategy equilibria to the announcement game exist for different ranges of :

- For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if ; and, the (*C*,*N*) pair is a SPNE if neither inequality holds. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*E*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE if neither inequality holds. - For : the (
*E*,*E*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE otherwise.

Figure 5 shows the equilibrium solutions graphically as a function of the industry outlook and capital intensity when *m* = 1. There are a number of observations we can make. First, we note that the leader–follower equilibrium, (*C*, *N*), has a much larger role here than in the inflexible situation. Moreover, rather than disappearing when the lead time *α* is small, this equilibrium is more likely to emerge as *α* decreases. The reason for this is that the volume flexibility reduces the advantage of being the first mover and gives the second mover a better outcome. Hence, it becomes easier to choose parameters under which *N* is an optimal response to *C* in the announcement game for flexible firms. With the same parameters and inflexible firms, the optimal response is much more likely to be either to exit *E* or to join the leader in moving early, *C*. Thus, the ikelihood of leader–follower behaviors increases if the lead firm is unable to commit to full production. This observation is opposite to the commitment-timing-based argument (see Ghemawat and Cassiman 2007). In particular, we note from Theorem 3 that whenever , the (*C*, *N*) pair is an equilibrium for some choices of *λ*.

We can also see that the bandwagon equilibrium solution (*C*, *C*) is more likely in Figure 5 than in Figure 3. This is another example of the reduced first mover advantage of flexible production lines, making firms less likely to invest ahead of the resolution of uncertain demand. Applying Theorem 3, the comparative statics for the bandwagon equilibrium of the (*C*, *C*) pair can be obtained in exactly the same way as for the inflexible case in Corollary 2. Finally, we note that there are no multiple equilibria occurring in the flexible duopoly (other than pairs like (*C*, *N*) and (*N*, *C*)). Roughly speaking, there is less strategic uncertainty under volume flexibility.

#### 5.2 Existing Capacity

The second variation from our base model that we consider is the possibility that at the start of the game, there is some pre-existing capacity: we write *r*_{i} for firm *i*'s starting capacity, *i* = 1, 2. For example, this may be from production of a similar type of product prior to exploring the new market. Without loss of generality, we assume that *r*_{j} ≤ *r*_{i}, thus we allow for an *ex ante* asymmetry in capacity with firm *i*'s existing capacity level being greater than firm *j*'s.

Our aim here is to explore the consequences of existing capacity on our main conclusion concerning the rarity of a leader–follower equilibrium, so we focus on the case where this equilibrium occurs. We will show that existing capacities make a leader–follower equilibrium even more unlikely.

With consideration of existing capacity, the capacity game is modified and we obtain:

where

The unwieldy expressions here in the (*D*, *D*) and (*D*, *C*) cases correspond to the calculation of whether anything is gained from capacity investment over against continuing with just an amount of capacity *r*_{i}.

There are two conditions necessary for the (*C*, *N*) or (*N*, *C*) pair to occur. First the existing capacity must not be so large in comparison with the potential market size that the best choice of investment for the lagging firm turns out to be zero. It can be shown that this condition translates into the following

The second condition required for a (*C*, *N*) or (*N*, *C*) equilibrium involves Γ and relates to the possibility that the lead firm by overinvesting can force the lag firm to exit. Explicitly, we require

This is the condition that corresponds to Γ^{2} ≤ *R* in Theorem 2a when there is no initial capacity. It is a stronger condition than necessary to ensure so that the second terms in is non-zero.

Following the logic used to prove Theorem 2, our main result for existing capacities (*r*_{i},*r*_{j}) is obtained.

Theorem 4. Suppose that existing capacities are (*r*_{i},*r*_{j}), with *r*_{i}, *r*_{j}, and Γ required for a leader–follower equilibrium, (*C*, *N*) or (*N*, *C*), to occur, i.e., and Γ ≤ *Z*. Let

Then, the following pure strategy equilibria to the announcement game exist for different ranges of Γ: the (*C*, *C*) pair is a SPNE if ; the (*N*, *N*) pair is a SPNE if ; the (*C*, *N*) pair is a SPNE if ; and the (*N*, *C*) pair is a SPNE if .

Since both *ρ* and *ψ* reduce to zero in the absence of existing capacities, Theorem 4 shows that existing capacities will tend to decrease the likelihood of the (*C*, *C*) pair being a SPNE, but increase the likelihood of the (*N*, *N*) pair being a SPNE due to the fact that *ρ* > 0 and *ψ* < 0, ∀*i*, *j*, as intuition would suggest. However, the direction of impact of the model parameters on the likelihood of the (*C*, *N*) or (*N*, *C*) pair being a SPNE is not straightforward since both *r*_{i} and *r*_{j} are involved in the magnitude of *ψ*. To gain greater insight, we plot the equilibrium solution as a function of industry outlook and capital intensity in Figure 6. Note that in this figure, we do not consider cases with Γ > *Z*, where the (*C*, *N*) and (*N*, *C*) equilibria are not possible.

The figure shows that when *r*_{i} and *r*_{j} are similar and both at a low level, then the equilibrium solution is similar to that in Theorems 2 and 3 for low capital intensities. In Figure 6a–c we look at what happens as *r*_{i} and *r*_{j} are both increased but held at the same value. We see that the region of leader–follower equilibrium is reduced, and disappears entirely by the time *r*_{i} = *r*_{j} = 10. As the existing capacity increases, there is a growing region Ω where both (*C*, *C*) and (*N*, *N*) pairs may occur. In Figures 6d to 6f, we look at increasing *r*_{i} while leaving *r*_{j} = 0. This has an opposing effect on the likelihood of the (*C*, *N*) or (*N*, *C*) pair being a SPNE, with the (*C*, *N*)pair no longer appearing in the equilibrium solution when firm *i* (the strong firm) has too much existing capacity. This change of equilibrium behavior results from the fact that as *r*_{i} increases, the strong firm's incentive to make an early investment decreases. In fact, *there is a certain range of existing capacity in which the weak (or start-up) firm can take a leadership role and invest first, with the stronger (or established) firm following*.

#### 5.3 Continuous Stochastic Demand

Our basic model has demand occurring for a fixed period, or not occurring at all. Now we want to explore some other possible stochastic models of demand. It is convenient to do this through looking at the “intercept” *A* that we have described as the potential market size. One possibility is to have *A* varying over time, so that rather than the demand remaining fixed throughout the period where sales take place it first increases and then decreases. Another possibility is that *A* (whether or not it is a function of time) is stochastic. In this situation, we do not know what value of *A* will occur if the market does eventuate. In this case, we need to carefully distinguish between two different possibilities relating to the information available when the capacity is determined (for a firm that delays). One possibility is that a firm can delay until it is sure that the market demand will occur and then makes a capacity decision based on knowledge of the distribution of demand, but not its actual value. A second possibility, when firms have more information available, is that a delaying firm is able to choose the capacity size according to the size of the market *A*. In our basic model, the delaying firm is able to avoid investing if the market does not occur, but in this case, there is an additional advantage to delay because the firm can then “right size” their capacity if the market does occur, and this will increase the option value from delay.

##### 5.3.1 Demand as a Stochastic Process

We begin with an investigation of the first possibility and consider relaxing the assumption we made that the demand intercept *A* is constant over time. Instead, we will look at a much more general stochastic process model for the evolution of *A*. We suppose that both firms know in advance that information about whether a new market will emerge will become available at a given time *T*. Because of the uncertainty, the firms may decide either to wait until after time *T* to invest (*ex post*) or not (*ex ante*). The process of investment is inflexible and also is not instantaneous: there is a (fixed) *lead time t*_{L} to build the capacity. A firm deciding to gamble by investing early will start at time *T* − *t*_{L}. On the other hand, if a firm decides to delay building capacity until after the market emerges, it will lose the possibility of meeting any demand that occurs prior to *T* + *t*_{L}. To match our previous discussion, we normalize the units of time so that the market opportunity, which is of finite length, starts at time 0 and finishes at time 1 (hence we assume *T* < 0) and we also set *α* = *T* + *t*_{L}

The product price at some time *t* in the period [0, 1] is given by *A*(*t*) − *q*(*t*) where *A*(*t*) > 0 is essentially the size of the market at time *t*, and *q*(*t*) is the total amount of product offered to the market by the firms combined at time *t*. In the event that the market does not emerge (so demand is zero), then no firm produces and the cost of investment will be lost for firms investing early. We allow *A*(*t*) to evolve as a general stochastic process—with the simple restriction that the expectation of its integral exists. As before, 1 − *λ* is taken as the probability that the market does not emerge in which case *A*(*t*) = 0 for *t* ∈ [0, 1]. We will continue to assume that there is no volume flexibility, so once a capacity level *x*_{i} is set, then the production quantity put on the market at time *t* by firm *i* is either *x*_{i} or zero.

We define a quantity

where the expectation is taken conditional on the fact that the market emerges (*A*_{0} is the average demand level and plays a similar role to *A*). This leads to a revised definition of the capital intensity

Let

(again the expectation is conditional on market demand occurring) and *A*_{1} gives the average demand occurring after a lagging firm enters the market. Finally, we define

Observe that *γ* is close to zero when the lead time is small (since this also makes *A*_{1} ≃ *A*_{0} ); and, *γ* is increasing in *α* (since *A*_{1} is decreasing in *α*). Moreover, *γ* approaches 1 when *α* 1 if *A*(*t*) is bounded. So *γ* can be thought of as modified lead time; and when there is uniform average demand throughout the period [0, 1], then *A*_{1} = *A*_{0}(1 − *α*) and we have an exact equivalence *γ* = *α*.

As in section 'Duopoly Model', upon observing the prior timing choices of capacity made by both firms and conjecturing the rival firm *j*'s capacity of *x*_{j}, firm *i* solves the following decision problem in the capacity game:

For consistency, we make assumptions on the parameter values in this model that correspond closely to our previous assumptions on *m* and *α* in our base model (i.e., Assumptions 1 and 2).

Assumption 3. (a) . (b) *A*_{1} ≥ (1 − *α*)*A*_{0}.

In this assumption, part (a) is similar to *λm* ≤ 1 (Assumption 1) and makes sure that the effective cost delaying a decision until after *T* is not too large. On the other hand, part (b) implies that the demand is not front loaded, so that the proportion of demand missed is no more than *α*. The qualitative implications of this assumption are similar to those implied by Assumptions 1 and 2, and the argument for it being appropriate are also similar.

Using the same approach as in the proof of Theorem 2, the investment equilibrium behavior for the inflexible duopoly under stochastic demand rate *A*(*t*) can be obtained analytically:

Theorem 5. Let

and let and be given by the two roots:

with being the smaller value and where . Suppose that demand intercept *A* evolves as a general stochastic process and Assumption 3 holds. Then the following pure strategy equilibria to the announcement game exist for different ranges of :

- For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if ; and, the (*C*,*N*) pair is a SPNE if neither inequality holds. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if*ψ*≤ 0, or, when*ψ*> 0 , if either or ; and, the (*C*,*E*) pair is a SPNE if neither the (*C*,*C*) or (*N*,*N*) pairs are equilibria. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*N*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE if neither inequality holds. - For : the (
*C*,*C*) pair is a SPNE if ; the (*N*,*E*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE if neither inequality holds. - For : the (
*E*,*E*) pair is a SPNE if ; and, the (*C*,*E*) pair is a SPNE otherwise.

It is not hard to show that when *A*_{1} = *A*_{0}(1 − *α*), we have an exact equivalence *γ* = *α*, , , and , so that this result exactly matches the result of Theorem 2. The conclusions we can draw are also closely matched with our earlier results. We hence confirm that when demand intercept *A* is a general stochastic process, the (*C*, *N*) equilibrium is still rare and becomes increasingly unlikely as the lead time gets shorter. Similar to Corollary 3 for our base model, we obtain:

Corollary 5. Suppose that demand intercept *A* evolves as a general stochastic process and Assumption 3 holds. If *α* < 0.2 then the (*C*, *N*) pair is not a SPNE and the region Ω of multiple equilibria (*C*, *C*) and (*N*, *N*) occurs.

##### 5.3.2 Demand as a Continuous Random Variable

Now we turn to the second arrangement for stochastic demand: If the market emerges (as happens with probability *λ*), then market demand has a demand intercept *A* that is a continuous random variable with positive support, mean *μ*, and variance *σ*^{2}. Within this model framework, by setting *λ* = 1, we can also deal with the case where the market is certain to occur but is of unknown size, as do the past studies including Anand and Girotra (2007), Goyal and Netessine (2007), Anupindi and Jiang (2008), and Swinney et al. (2011).

Besides, we will make the assumption that if the market does emerge, then its size is large enough to make it profitable to invest for both players. Without this assumption, then a delaying firm may decide not to invest when the demand that emerges is low enough. To deal with this more general case, we would need to consider integrals over the distribution of the random variable *A* (with results that then depend on that distribution) and this adds much more complexity that is necessary for our aim of a confirmation of the essential results from the base model. The actual requirement for the smallest *A* value is hard to define precisely (it depends on the complete distribution of *A*). Essentially, we are interested in the equivalent to the condition Γ^{2} ≤ *R* in Theorem 2, since we are interested in the possibility of the (*C*, *N*) outcome.

Assumption 4. Suppose that demand intercept *A* is a continuous variable with mean *μ* and variance *σ*^{2}. The smallest demand intercept *A* is large enough (and the capital intensity small enough) that there is no incentive for a lead firm to offer an amount such that the lag firm will not invest at this smallest demand level.

The smallest value of *A* must be at least , but the actual value needed is greater (or the capital intensity smaller) in the same way that to avoid the possibility of the lead firm deliberately over investing to persuade the lag firm not to invest.

The capacity game is modified as follows:

We have written this with *x*_{i}(*A*) and *x*_{j}(*A*) wherever these quantities depend on the random variable *A*. Moreover, we have dropped the “positive sign” in the expression for the profits, since under Assumption 4 this is not required.

Two structure parameters are defined for the ease of expression:

We write for a revised definition of Γ with *A* replaced by its mean value and *v* for the squared coefficient of variation of the demand intercept, reflecting the degree of *demand uncertainty*.

Following the same procedure in section 'Existing Capacity', we can obtain the equilibrium solution under a continuous random demand model:

Theorem 6. Suppose that the demand intercept *A* is a continuous variable with mean *μ* and variance *σ*^{2}, and Assumption 4 holds. Then, the following pure strategy equilibria to the announcement game exist for different ranges of : the (*C*, *C*) pair is a SPNE if , the (*N*, *N*) pair is a SPNE if , and the (*C*, *N*) pair is a SPNE if neither inequality holds.

This result is a close match to Theorem 2 because we can see the role of the demand uncertainty in the terms involving demand uncertainty *v*. As might be expected, uncertainty here acts to reduce the region in which the (*C*, *C*) equilibrium occurs and increase the region in which (*N*, *N*) occurs. This is because of the increased option value of delay through being able to respond to the actual demand level in the market.

In the continuous random demand model, a short lead time will also lead to the leader–follower behavior disappearing and the multiple equilibrium region appearing. From Theorem 6, we can obtain:

Corollary 6. Suppose that demand intercept *A* is a continuous variable with mean *μ* and variance *σ*^{2}, and Assumption 4 holds. If , then the (*C*, *N*) pair is not a SPNE and the region Ω of multiple equilibria (*C*, *C*) and (*D*, *D*) occurs.

This preserves our main result about the impact of lead time and demand uncertainty and demonstrates that it is not sensitive to the Bernoulli random demand setting. Note that with larger values of demand uncertainty the threshold lead time at which the (*C*, *N*) pair no longer appears is reduced. In other words, *uncertain market demand increases the chance of a leader–follower equilibrium*.

### 6 Conclusion

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

In capacity investment competition, firms face uncertainty about the evolution of market demand, which is typically exogenous to the industry. But it is also critical to see any action within the competitive framework. Thirty years ago, Michael Porter argued that “Because capacity additions can involve lead times measured in years and capacity is often long lasting, capacity decisions require the firm to commit resources based on expectations about conditions far into the future. Two types of expectations are crucial: those about future demand and those about competitors' behavior. The importance of the former in capacity decisions is obvious. Accurate expectations about competitors' behavior are essential as well” (Porter 1980, p. 324). With the spotlight on demand uncertainty, strategic uncertainty (emerging endogenously from the strategic interaction between competing firms) has been given short shrift in the operations literature. We establish that even without being able to reduce demand uncertainty, factors like volume flexibility and the lead time to build capacity can be critical in mitigating strategic uncertainty. So, operational characteristics significantly impact the competitive dynamics of capacity investment timing.

In this paper we provide a systematic examination of how the lead time required to build capacity determines the capacity investment and timing decisions of firms entering new markets. Our model allows firms the freedom to invest in capacity at either of two timing choices: before or after demand uncertainty is resolved. We derive the equilibrium solutions of capacity investment timing for various related settings discussed in the operations literature. One feature of our model is the fact that the timing of capacity expansion is determined before the size of the capacity is fixed. We have argued that this is a more realistic assumption in many industries than a more conventional approach, which allows firms to fix the size of the capacity investment prior to giving any indication to the other firm that a capacity expansion is underway. From a game theory modeling perspective, splitting apart the capacity decisions on timing and size turns out to make the problem simpler, since it eliminates a large number of situations in which there are multiple equilibria of different structures. With the model we have chosen, in almost all cases, knowing the parameters of the problem will determine a unique equilibrium solution (unique up to the allocation of the firms to the two roles if these differ). When the quantity and timing of investments are chosen together, the equilibrium regions often overlap, making deductions less clear cut.

Capacity investment is an important aspect of operations strategy and construction lead time is its essential attribute. We examine how the lead time influences capacity investment timing behavior. We formally demonstrate that firms often act in unison, i.e., they simultaneously employ the same timing strategy, either capacity leading demand or capacity lagging demand when the lead time is long. More surprisingly, perhaps, we prove that the sequential investment outcome that one firm leads and the other lags is unlikely to take place when the lead time is short and there is no volume flexibility. Following the findings in our paper, we argue that the operational elements such as lead time and volume flexibility (holdback) are vital in predicting competitive capacity investment timing decision. This is in strict contrast to the findings of Goyal and Netessine (2007), who show that volume (in-)flexibility does not have a large impact on competitive outcomes, and Swinney et al. (2011), who show that the difference between the start-up's and established firm's objective functions primarily lead to the sequential investment outcome. Moreover, opposite to the received wisdom in business strategy, we find that the ability to make a credible commitment in production may lead to an investment bandwagon outcome, with both firms investing early, rather than achieving first-mover advantage (Ghemawat and Cassiman 2007). Our work provides a theoretical justification for the role of operations in making strategic decisions. Overall, our results speak of the importance of the link between a firm's operations (production policy and volume flexibility) and its competitive environment (industry structure, investment costs, and demand uncertainty).

Our findings are robust to various variants of the model. For instance, we extend our setting to incorporate existing capacity and show that the higher the asymmetry in existing capacities between competing firms, the more likely they are to act in unison, contrary to intuition. Moreover, no matter whether demand uncertainty is continuous over time or size, a decrease in the lead time favors the investment bandwagon behavior. Notice that our extension to a continuous random demand model is more general than many other capacity investment studies (e.g., Goyal and Netessine 2007, Swinney et al. 2011) since we consider not only unknown markets of unknown size but also whether the new market emerges or not. While the algebra and notation become significantly involved, the chief managerial insights of our simple model do not change.

Managerially, blindly following a mantra of seizing the first-mover advantage misses most of what is important. Being the first mover may work if the economics of the market will ensure that the other firm exits completely, but, in other circumstances, managers may expect that similar rational competitors will end up following the same strategy as each other. Even in a competitive environment, the decisions to be made are not driven by first mover advantage. However, if the lead time to build capacity building is not too long compared with the length of the product life cycle, then in weighing the costs of delay (missing a selling opportunity) against the benefits (being able to assess the size of the market), the ability of firms to commit to production makes an early move more likely.

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- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

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### Supporting Information

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Model
- 4 Analysis
- 5 Robustness of the Model and Insights
- 6 Conclusion
- References
- Supporting Information

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poms12204-sup-0001-Appendix.pdf | PDF document | 114K | Appendix S1: Proofs. |

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