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Dynamic Games Between Firms and Infinitely Lived Consumers: A Review of the Literature

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Abstract

We review a class of models in industrial organization theory, in which consumers’ foresight takes an important role. As Coase (J Law Econ 15:143–149, 1972) makes clear, the foresight of consumers may restrain the power of monopolists and oligopolists, especially when the goods are durable and the firms lack the ability to commit to future prices and/or outputs. This important insight clearly has significant implications for antitrust policies. However, the extent to which the powers of firms are curtailed depends on a host of factors which are explored in a number of important contributions. These factors include the length of time between two consecutive offers, the rate of depreciation of the durable goods, the product lines offered by firms producing the durable goods, the nature of network externalities in the primary market and the aftermarkets, the ability of firms to rely on reputational effects, and so on. This paper also reviews issues that are closely related to product durability, such as consumption capital in the case addictive goods, anticipation of future network congestions, and sales of assets such as shares of a firm’s subsidiaries, or shares of a majority shareholder who has an influence on managers.

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Notes

  1. See Long [61] for a survey of dynamic games in the economics of natural resources.

  2. If the monopolist can rent the durable goods, she will be able to make profits. For many products, renting is not a practical option. As Bulow [16], p. 315] puts it, “when steel was first produced and began replacing iron for railroad tracts, it would have been impractical for the steel companies to rent the steel bars.” In the case of automobiles, moral hazard problems make buying more efficient than renting, as the monopolist may not be able to monitor the renter’s treatment of the goods; see also Epple and Zellenitz [31].

  3. Bulow [16] constructs a two-period model to show that the inability of a durable-goods monopolist to commit that it will not exploit period 2 profit opportunity places a constraint on the price it can charge for period 1.

  4. The durable goods monopolist’s lack of ability to commit is formally equivalent to the problem of re-optimization in the macroeconomic literature [55]. The optimization problem facing today’s policy makers must take into account two constraints: The policy makers tomorrow will re-optimize, and all agents today will choose actions based on that realization. See also Calvo’s formulation of the seignorage problem [20].

  5. Gul et al. [41] suppose that the monopolist makes price offers, which heterogeneous buyers can accept or reject. This corresponds to the infinite-horizon bargaining problems à la Rubinstein [69], where in any period, a party makes an offer which the other party can either accept or reject. This modeling strategy is also adopted by Ausubel and Deneckere [4] and Sobel [71]. Stokey [72] on the other hand assumes the monopolist sets the production level.

  6. It is not necessary to assume the linearity of the marginal product function. For illustrative purposes, the linearity simplifies the exposition. Stokey [72], p. 124] also uses the linear model for illustration.

  7. The distinction between the gap case and the no-gap case was made precise in the well-known book on game theory by Fudenberg and Tirole [35].

  8. She also shows that Coase’s argument is approximately correct in the discrete-time framework, if the period is sufficiently short.

  9. To prove uniqueness, Stokey requires that the expectation function is continuously differentiable in \(X\).

  10. This is formally stated as their Theorem 3. In the gap case, the equilibrium is higher than the marginal cost. Note that the Coase conjecture does not extend to dynamic oligopoly: As shown in Gul [42], with two or more firms, the perfection requirement imposes no restriction on the equilibrium profits. In addition, he shows that there is no tendency toward the perfect competition outcome as the number of firms increases.

  11. The solution of this equation does not yield the socially optimal path, because, with the ability to make commitment, the monopolist is able to restrict the output in order to raise the price above the socially optimal level.

  12. This time-inconsistency problem also arises in the theory of optimal tariff on exhaustible resources. See, for example, [53].

  13. Karp [48] and Bagnoli et al. [5] present two other cases where the Coase conjecture does not hold.

  14. In Karp [48], the exhaustibility of a natural resource serves as a commitment device.

  15. Karp calls his MPEs ‘Strong MPEs’ where the word ‘strong’ mean that all agents condition their actions (or their beliefs about the future) on only the current state variable, which is in this case the stock of durable good in the hand of the public.

  16. Dockner and Long [28] also find this non-uniqueness in a model of pollution game.

  17. Sobel [71] also proves this type of folk theorem in a model with entry of new consumers.

  18. In Driskill [29], a continuum of equilibria exists, but his interest is on the equilibrium which is the limit of a sequence of equilibria of finite-horizon problems.

  19. Karp’s model is Markovian and thus does not rely on the folk-theorem type of argument, such as those used in Ausubel and Deneckere [3] and Gul [43], who show that industry profit under monopoly can be lower than under duopoly, if goods are durable and agents have history-dependent beliefs.

  20. A discrete-time version of the model was considered by Malueg and Solow [62], where extraction cost is strictly convex in the extraction rate. This strict convexity generates results similar to Kahn (1987).

  21. It is assumed that the cost of the jump is \((S(0^{+} )-S_{0})c(S_{0})\). This assumption is not entirely innocuous.

  22. It is somewhat problematic to define the extraction cost of an infinitesimally small competitive firm, given that \(qc(S)\) is the extraction cost of the social planner.

  23. This is an assumption on the endogenous function \(P^{c}(S,X)\). In general, it is difficult to find sufficient conditions on the primitives of the model which ensures this condition.

  24. Uniqueness of equilibrium price functions cannot be assured.

  25. This is in sharp contrast to the literature on disadvantageous monopsony of an exhaustible resources, where a two-period suffices.

  26. The case of DeBeers is an exception; perhaps this is because the assumption of non-Markov expectations may be plausible in this case.

  27. Clarke et al. [23] also consider dynamic pricing by a monopolist, in the presence of experience effects, but consumers are not forward looking. The authors find that the price path may be non-monotonic. They do not discuss network effects.

  28. As [47] points out, “As the product is introduced, there is uncertainty associated with the product’s experience-type attributes. The value of a new product to a risk-averse customer is lower than the product’s value if all experience information was available. As more customers adopt the innovation, more of the uncertainty is removed and the valuation of the product increases.”

  29. See their Proposition 4, which unfortunately has several typos.

  30. The authors make clear that they consider only open-loop equilibria.

  31. This differential equation is equivalent to Mason’s equation (5), where \(k\equiv \lambda /r\) and \(F_{t}\equiv \theta _{t}+\lambda X_{t}\).

  32. The highest-valuation consumer (with \(\theta =1\)) is the first one to buy the good, when the network size is zero. Because of stationary network effect, she never benefits from network externalities. Her lifetime surplus is \(\left[ 1-(rc)\right] /r\) \(=(1/r)-c\). Consumers with \(\theta \in (1-\widehat{X},1)\) will eventually buy a unit of the good and enjoy a stand-alone benefit equal to \(\theta \) and a network benefit equal to \((1-\theta )\lambda \) per unit of time, starting from their date of purchase. Their lifetime surplus is \(\left\{ \left[ \theta +\lambda (1-\theta )\right] /r\right\} \hbox {e}^{-rt(\theta )}-c\hbox {e}^{-rt(\theta )}\) where \(t(\theta )\) is their date of purchase.

  33. This exception to the standard Coasian outcome has some resemblance to the “gap case” in the literature on durable-goods monopoly discussed in section 2.

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I would like to thank a reviewer for many helpful comments and suggestions.

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    Ngo Van Long

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Van Long, N. Dynamic Games Between Firms and Infinitely Lived Consumers: A Review of the Literature. Dyn Games Appl 5, 467–492 (2015). https://doi.org/10.1007/s13235-015-0155-1

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