Adoption of cost-reducing technology by asymmetric duopolists in stochastically evolving markets.

1. Introduction

Large monopolists and duopolists routinely evaluate opportunities to acquire cost-reducing technologies. We follow the definition of technology offered by Balcer and Lippman (1984) in that we are considering, "all currently implementable technological information useful in reducing production costs." Clearly, the fact that a new technology is expected to reduce the unit production cost is not a sufficient condition to guarantee immediate adoption in the face of uncertainty. Potential adopters may defer investment awaiting a stochastic change in some market characteristic, or the actions of a rival firm. These firms will continue to make routine, tactical decisions during this time such as output rates, and product prices. Since changes in market size, the actions of rivals and ongoing tactical decisions all alter the payoff from technology adoption, it is important to account for each of these factors when modeling investment decisions.

This work focuses on the optimal timing of the adoption of a cost-reducing technology by one or two competing firms. We assume a "Cournot"-style setting in which market clearing prices are determined by market size (which evolves stochastically) and output rates, which are consistent with a unique Nash equilibrium at each moment in time. We allow the initial unit production costs, the required investment for technology adoption and the change in unit production costs to all be firm-specific. We discuss profit-maximizing behavior in three distinct settings. Settings in which the rival's flexibility is ignored can be treated as the game of a monopolist. Regulatory issues, the presence of proprietary knowledge or asymmetric involvement with technology development may lead to instances in which the order of adoption is known a priori. We will model such settings as two players in a Stackelberg, or "leader-follower" game. Finally, the more general problem involving two asymmetric firms with no preset order of adoption is considered.

This work explicitly deals with production costs to an extent that is absent from the industrial organization literature. We explicitly model the competitive interplay in ways that are absent from the financial economics literature to date. The combined effect is that we provide insights for the Operations Manager that are under-developed in previously published works. Speaking more technically, this work is the first which focuses on production costs, reflects adjustments made continuously, accounts for a real option to defer investment and accommodates asymmetric production and technology acquisition costs simultaneously. The inclusion of these elements leads to several counter-intuitive results not fully developed in the extant literature.

Consider a simple example in which firms A and B sell identical products to a market that is expected to grow 4% per year. Firm A's production cost is $1.05 per unit, and firm B's cost is $1.10 per unit. Both firms consider an investment which will reduce per unit cost to $1.00. This produces savings in per unit cost of [S.sub.A] = $0.05 and [S.sub.B] = $0.10. Let us assume that the required investments to bring about this cost reduction are [K.sub.A] = $5 and [K.sub.B] = $8. The ratios of the cost savings to the required investment suggests that the technology should be more attractive to firm B since

([S.sub.B]/[K.sub.B]>[S.sub.A]/[K.sub.A]).

Clearly, the value of this investment will depend upon the market size M and that the value of M which makes the Net Present Value (NPV) of investment for firm B positive is lower than the corresponding value for firm A. However, we shall show that the optimal behavior by both players results in firm A adopting this cost-reducing technology first even though A's expected payoff from this investment is in fact negative. This behavior is rational because firm A is defending its position in a growing market and "early" adoption effectively delays the adoption of firm B so that firm A retains a temporary cost advantage. We note that even though the net result is negative, this span of time with a cost advantage results in a payoff that is better than the more negative alternative of remaining idle while the rival firm adopts the new technology. This work provides a rigorous explanation for such behavior.

The remainder of this paper is organized as follows. Section 2 presents background information and provides a brief discussion of related literature. Section 3 formally defines the problem. Section 4 presents the analysis of the setting for a monopolist. Sections 5 and 6 consider two players in a Stackelberg game, and two players with no preset order of adoption respectively. Section 7 concludes the work by highlighting several managerial insights, and discussing future works and extensions.

2. Background and literature

A large class of technology adoption problems involves riskneutral, profit-maximizing firms considering an investment in technology when the payoff from that investment is uncertain. Let us assume that a technology becomes available which will reduce the production costs for the adopting firm (firm i,) from some initial (firm-specific) level [c.sub.i] to a lower level [[c.bar].sub.i]. We formulate the problem as a non-cooperative game in which each moment in time is associated with equilibrium tactical decisions about output rates along with an option to invest a firm-specific amount [K.sub.i] for the innovation which lowers production cost. We note that [C.sub.i], [[c.bar].sub.i] and [K.sub.i] may all be firm-specific even when the competing products are direct substitutes and a single third party is offering the same technology to both players for several reasons. For example, Astebro (2002) discusses the adoption of computer numerically controlled machines in metal working industries in the US. These machines are typically developed by third-party vendors and offered to competing firms simultaneously. Once acquired, the machines may perform equally well for any buyer. However, the expected change in unit production cost and the required investment may differ dramatically between competing firms for several reasons. The firms may differ in scale and experience. They may be considering partial replacement of their stock of machines because they have differing investment histories and the benefit of replacing older machines may be greater than the benefit of replacing newer models.

However, even if both firms are of equal size and replacing an equal number of machines the required investment may differ because this value includes more than the sale price of the machines. Astebro (2002) discusses several "non-capital costs" such as the costs of learning how to use and program the machines, and how to troubleshoot and perform problem solving using these machines. There are also costs of organizational adaptations, including changes in reporting routines, authority and spans of control. (See Siegel (1999).) The net result is that even when these machines are offered at the same "price" to multiple potential customers and the customers are considering the replacement of the same number of older machines, the actual cost of adoption may vary dramatically from one firm to the next.

An increasingly common implementation of technological knowledge to reduce production costs involves outsourcing or offshoring. Aron and Singh (2005) discusses the facts that a significant investment is often required to establish the needed relationships or offshore capacity, and that the outcomes can vary dramatically among firms in the same industry depending on how they manage the relationships involved. On the other hand, Kouvelis and Niederhoff (2007) documents efforts by multiple suppliers in the garment industry who invest to acquire offshore capacity with the objective of achieving identical production costs. Therefore, we discuss cases in which post-investment production costs differ as well as those in which these costs are equal.

Games between firms involving competition in quantities are often referred to as Cournot competition. (See Fudenberg and Tirole (1991) for a thorough introduction.) Such games typically assume that market clearing prices are described by a downward sloping price curve where price is predicted given output rates selected by non-cooperative firms. When the market size is fixed, it is trivial to normalize the problem such that the market size is equal to one. We extend the logic of this setting by including a stochastic evolution of market size.

Shackleton et al. (2004) considers strategic investments by Airbus and Boeing. This competition is frequently modeled as a Cournot-style game. If we consider investments in cost-reducing technology by such firms we have several additional characteristics which must be considered. First, the demand for the firms' products is heavily influenced by a number of economic variables, and the potential market size changes continuously as a result. Second, production scheduling also reflects long-term contracts which are virtually certain. Thus, while total demand evolves over time, much of the firm's capacity is already "booked." Third, the actions of each firm affect market prices and demand. Consequently, each firm must be concerned with the investment and output related behavior of the other.

Evaluating investment opportunities when the firm has the ability to select the time of the initial outlay has provided much incentive for early works on real options. (See Dixit (1993), Dixit and Pindyck (1994), Trigeorgis (1996), Amram and Kulatilaka (1999) and Copeland and Antikarov (2001).). The single-firm perspective of this early work has been expanded to include competitive pressures by combining real options and games. (See Brennan and Trigeorgis (2000), Grenadier (2000), Huisman (2001) and Smit and Trigeorgis (2004)).

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