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Differential Games in Industrial Organization

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Handbook of Dynamic Game Theory

Abstract

In this chapter, we provide an overview of continuous-time games in industrial organization, covering classical papers on adjustment costs, sticky prices, and R&D races, as well as more recent contributions on oligopolistic exploitation of renewable productive assets and strategic investments under uncertainty.

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Notes

  1. 1.

    Continuous time can be interpreted as “discrete time, but with a grid that is infinitely fine” (see Simon and Stinchcombe 1989, p.1171).

  2. 2.

    An interesting area where dynamic games have been fruitfully applied in industrial organization is where firms face sophisticated customers that can foresee their future needs. For a survey of this literature, see Long (2015).

  3. 3.

    Adjustment costs are important in quite a few industries as evidenced by several empirical studies (e.g., Hamermesh and Pfann 1996; Karp and Perloff 1989, 1993a,b).

  4. 4.

    Driskill and McCafferty (1989) derive a closed-loop equilibrium in the case in which quantity-setting firms face a linear demand for a homogeneous product and bear symmetric quadratic costs for changing their output levels.

  5. 5.

    For a discussion about the nature of capacity investments (reversible vs. irreversible) and its strategic implications in games with adjustment costs, see Dockner et al. (2000, Ch. 9).

  6. 6.

    This is also referred to as Markov control-state substitutability (see Long 2010).

  7. 7.

    Figuières (2009) compares closed-loop and open-loop equilibria of a widely used class of differential games, showing how the payoff structure of the game leads to Markov substitutability or Markov complementarity. Focusing on the steady-states equilibria, he shows that competition intensifies (softens) in games with Markov substitutability (complementarity). Markov substitutability (complementarity) can be considered as the dynamic counterparts of strategic substitutability (complementarity) in Bulow et al. (1985).

  8. 8.

    The same assumption is made in the duopoly model analyzed in Reynolds (1987).

  9. 9.

    A different capital accumulation dynamics is considered in Cellini and Lambertini (1998, 2007a), who assume that firms’ unsold output is transformed into capital, i.e., firms’ capacity is accumulated à la Ramsey (1928).

  10. 10.

    In most of the literature on capacity investments with adjustment costs, it is assumed that adjustment costs are stock independent. A notable exception is Dockner and Mosburger (2007), who assume that marginal adjustment costs are either increasing or decreasing in the stock of capital.

  11. 11.

    Applications of sticky price models of oligopolistic competition with international trade can be found in Dockner and Haug (1990, 1991), Driskill and McCafferty (1996), and Fujiwara (2009).

  12. 12.

    Equation (18.18) can be derived from the inverse demand function \( p(t)=a-s\int _{0}^{t}e^{-s\left ( t-\tau \right ) }q(\tau )d\tau \). Simaan and Takayama (1978) assume that the speed of price adjustment, s, is equal to one. Fershtman and Kamien (1987) analyze the duopoly game. For the n-firm oligopoly game, see Dockner (1988) and Cellini and Lambertini (2004). Differentiated products are considered in Cellini and Lambertini (2007b).

  13. 13.

    In a finite horizon model, Fershtman and Kamien (1990) analyze the case in which firms use nonstationary feedback strategies.

  14. 14.

    Firms’ capacity constraints are considered in Tsutsui (1996).

  15. 15.

    For an off-steady-state analysis, see Wiszniewska-Matyszkiel et al. (2015).

  16. 16.

    Note that when s →∞, the differential game becomes a continuous time repeated game.

  17. 17.

    Classical references on the quasi-competitiveness property for a Cournot oligopoly are Ruffin (1971) and Okuguchi (1973).

  18. 18.

    This result can also be found in Benchekroun (2003b)

  19. 19.

    On the shadow price system approach, see also Wirl and Dockner (1995), Rincón-Zapatero et al. (1998), and Dockner and Wagener (2014), among others. Rincón-Zapatero et al. (1998) and Dockner and Wagener (2014), in particular, develop alternative solution methods that can be applied to derive symmetric Markov perfect Nash equilibria for games with a single-state variable and functional forms that can go beyond linear quadratic structures.

  20. 20.

    A similar result is obtained in Wirl (1996) considering nonlinear feedback strategies in a differential game between agents who voluntarily contribute to the provision of a public good.

  21. 21.

    Papers belonging to (i) include Levhari and Mirman (1980), Clemhout and Wan (1985), Benhabib and Radner (1992), Dutta and Sundaram (1993a,b), Fisher and Mirman (1992, 1996), and Dockner and Sorger (1996). In these papers, agents’ instantaneous payoffs do not depend on rivals’ exploitation rates. The asset is solely used as a consumption good.

  22. 22.

    Classical papers on oligopoly exploitation of nonrenewable resources are Lewis and Schmalensee (1980), Loury (1986), Reinganum and Stokey (1985), Karp (1992a,b), and Gaudet and Long (1994). For more recent contributions, see Benchekroun and Long (2006) and Benchekroun et al. (2009, 2010).

  23. 23.

    The linearized logistic growth function has been used in several other oligopoly games, including Benchekroun et al. (2014), Benchekroun and Gaudet (2015), Benchekroun and Long (2016), and Colombo and Labrecciosa (2013a, 2015). Others have considered only the increasing part of the “tent”, e.g., Benchekroun and Long (2002), Sorger (2005), Fujiwara (2008, 2011), Colombo and Labrecciosa (2013b), and Lambertini and Mantovani (2014). A nonlinear dynamics is considered in Jørgensen and Yeung (1996).

  24. 24.

    Classical examples of \(h\left ( x\right ) \) are fishery and forest stand dynamics. As to the former, with a small population and abundant food supply, the fish population is not limited by any habitat constraint. As the fish stock increases, limits on food supply and living space slow the rate of population growth, and beyond a certain threshold the growth of the population starts declining. As to the latter, the volume of a stand of trees increases at an increasing rate for very young trees. Then it slows and increases at a decreasing rate. Finally, when the trees are very old, they begin to have negative growth as they rot, decay, and become subject to disease and pests.

  25. 25.

    The impact of an increase in the number of firms on the steady-state equilibrium price is also analyzed in Colombo and Labrecciosa (2013a), who departs from Benchekroun (2008) by assuming that, instead of being common property, the asset is parcelled out (before exploitation begins). The qualitative results of the comparative statics results in Colombo and Labrecciosa (2013a) are in line with those in Benchekroun (2008).

  26. 26.

    A notable exception is represented by Jun and Vives (2004), who show that Bertrand competition with costly price adjustments leads to a steady-state price that is higher than the equilibrium price arising in the static game.

  27. 27.

    For discrete-time games of innovation, see Petit and Tolwinski (1996, 1999) and Breton et al. (2006).

  28. 28.

    Choi (1991) and Malueg and Tsutsui (1997) assume that the hazard rate is uncertain, either zero (in which case the projet is unsolvable) or equal to λ > 0 (in which case the projet is solvable). The intensity of R&D activity is fixed in the former paper and variable in the latter. Chang and Wu (2006) consider a hazard rate that does not depend only on R&D expenditures but also on the accumulated production experiences, assumed to be proportional to cumulative output.

  29. 29.

    As shown in Mehlmann and Willing (1983) and Dockner et al. (1993), there exist also other equilibria depending on the state.

  30. 30.

    Dawid et al. (2015) analyze the incentives for an incumbent firm to invest in risky R&D projects aimed to expand its own product range. They employ the same form of the hazard rate as in Doraszelski (2003), focussing on the case in which ψ > 1.

  31. 31.

    A classical reference on numerical methods is Judd (1998).

  32. 32.

    The assumption that all firms innovate is relaxed in Ben Abdelaziz et al. (2008) and Ben Brahim et al. (2016). In the former, it is assumed that not all firms in the industry pursue R&D activities. The presence of non-innovating firms (called surfers) leads to lower individual investments in R&D, a lower aggregate level of knowledge, and a higher product price. In the latter, it is shown that the presence of non-innovating firms may lead to higher welfare.

  33. 33.

    A cost function with knowledge spillovers is also considered in the homogeneous product Cournot duopoly model analyzed in Colombo and Labrecciosa (2012) and in the differentiated Bertrand duopoly model analyzed in El Ouardighi et al. (2014), where it is assumed that the spillover parameter is independent of the degree of product differentiation. In both papers, costs are linear in the stock of knowledge. For a hyperbolic cost function, see Janssens and Zaccour (2014).

  34. 34.

    Colombo and Dawid (2014) also consider the case in which all firms have the same R&D cost parameter η, but there exists one firm which, at t = 0 , has a larger stock of knowledge than all the other firms.

  35. 35.

    The case in which knowledge is a public good (β = 1) is considered in Vencatachellum (1998). In this paper, the cost function depends both on current R&D efforts and accumulated knowledge, and firms are assumed to be price-taking.

  36. 36.

    The literature on R&D cooperation is vast. Influential theoretical (static) papers include D’Aspremont and Jacquemin (1988), Choi (1993), and Goyal and Joshi (2003). Dynamic games of R&D competition vs cooperation in continuous time include Cellini and Lambertini (2009) and Dawid et al. (2013). For a discrete-time analysis, see Petit and Tolwinski (1996, 1999).

  37. 37.

    Setting n = 2, Cellini and Lambertini (2009) compare private and social incentives toward cooperation in R&D, showing that R&D cooperation is preferable to noncooperative behavior from both a private and a social point of view. On R&D cooperation in differential games see also Navas and Kort (2007), Cellini and Lambertini (2002, 2009), and Dawid et al. (2013). On R&D cooperation in multistage games, see D’Aspremont and Jacquemin (1988), Kamien et al. (1992), Salant and Shaffer (1998), Kamien and Zang (2000), Ben Youssef et al. (2013).

  38. 38.

    Studies of investment timing and capacity determination in monopoly include Dangl (1999) and Decamps et al. (2006). For surveys on strategic real option models where competition between firms is taken into account, see Chevalier-Roignant et al. (2011), Azevedo and Paxson (2014), and Huberts et al. (2015).

  39. 39.

    The idea that an incumbent has an incentive to hold excess capacity to deter entry dates back to Spence (1977, 1979).

  40. 40.

    Note that the real options approach represents a fundamental departure from the rest of this survey. Indeed, the dynamic programming problems considered in this section are of the optimal-stopping time. This implies that investments go in one lump, causing a discontinuity in the corresponding stock, instead of the more incremental control behavior considered in the previous sections.

  41. 41.

    Genc et al. (2007), in particular, use the concept of S-adapted equilibrium of Haurie and Zaccour (2005) to study different types of investment games.

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Authors and Affiliations

  1. Department of Economics, Deakin University, 3125, Burwood Campus, 221 Burwood Hwy, Burwood, VIC, Australia

    Luca Colombo

  2. Department of Economics, Monash University, 3800, Clayton Campus, Wellington Road, Clayton, VIC, Australia

    Paola Labrecciosa

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  1. Luca Colombo
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  2. Paola Labrecciosa
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Correspondence to Luca Colombo .

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  1. Coordinated Science Laboratory and Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Tamer Başar

  2. Department of Decision Sciences GERAD, HEC Montréal, Montreal, Québec, Canada

    Georges Zaccour

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Colombo, L., Labrecciosa, P. (2018). Differential Games in Industrial Organization. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44374-4_17

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