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Perpetual leapfrogging in Bertrand duopoly

  • Economics


In this paper we focus on the intertemporal aspects of technological adoption. We consider a simplified model: firms produce an homogeneous good and adoption decisions concern a cost reducing technology. We focus on the issue of industrial leadership reversal. Imagine an industry facing a sequence of cost reducing innovations; the appearance of newer generations of PC processors provides a good example of the sort of improvements we have in mind. Individual firms can upgrade by adopting the most recent improvement. This improvement comes at a cost, for example, of installing the new processors. Will firms choose to make these costly adoptions? How do adoption rates depend on the product market competition? Which adoption patterns will be sustainable in equilibrium? Will these adoption strategies reduce aggressive competition? We consider a duopoly where firms set prices, i.e. there is Bertrand competition in the product market. In an intertemporal, infinite horizon setting, firms can adopt alternative and complicated dynamic adoption strategies. We study Markov Perfect equilibria (MPE’s) and in addition restrict attention to equilibria with relatively simple but economically relevant patterns. These are: 1) Alternating adoptions and 2) Increasing asymmetry. An increasing asymmetry pattern of adoptions is such that the firm with the lowest unit cost adopts, while the high cost firm does not, so that existing cost asymmetry is reinforced. An alternating adoptions pattern is such that the firm with high unit cost adopts, while the low cost firm does not, existing cost asymmetry is reversed and leapfrogging takes place. We find that if adoption is profitable at a given date, and the price elasticity of demand is greater than, or equal to, one, then no asymmetry can be absorbing and technological adoption goes on forever through an infinite sequence of leapfroggings. For this case we characterize the adoption cost region where a pattern of alternating adoptions is an MPE. In this setting increasing asymmetry is never an MPE. Perpetual leapfrogging emerges therefore as a set of simple adoption strategies allowing implicit and sustainable coordination between two firms. Such coordination helps avoiding the most aggressive aspects of duopolistic price competition. Only with high price elasticity, and a market size large enough compared to adoption costs, this goes on forever. If adoption is profitable at a given date but the elasticity of demand is below one, there is a date in which the adoption process will stop. Alternating adoptions up to this date is an MPE for a range of adoption costs. Increasing asymmetry can also be an equilibrium in this case, but under very restrictive conditions. In an oligopolistic industry demand conditions play an essential role in determining both the continuation or the end of the technological adoption and the identity of the adopters. When adoption continues, long run technological improvements are only made by high cost firms, which emerge as the engine of productivity growth. This is mainly due to the nature of the incentives for the adoption of a new technology under Bertand competition. For any downward sloping demand function the increments in period profits, due to the adoption of a new cost reducing technology, are larger for the follower than for the leader, because market demand when the follower adopts is higher than when the leader adopts. In this last case there is, indeed, a higher equilibrium price2. With isoelastic demand functions the value of the elasticity determines whether the adoption process will go on forever or not. With more general demand functions one would need to study the limit behaviour of the profits’ increments, due to adoption, when the equilibrium price tends to zero. Only if demand grows proportionally faster than the decrease of the price-cost margin, adoption of new technologies can go on forever. After a brief review of some related literature, the remainder of the paper is organized as follows: in section 2 we describe the model industry: market demand, technology evolution and costs, and firms’ decision sets and objective functions. In section 3 we analyse adoption decisions in a Bertrand duopoly, first with myopic firms and then with discounting. Finally section 4 contains the conclusions of the paper. All the proofs are contained in the appendix.

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