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Erschienen in: Dynamic Games and Applications 2/2020

29.08.2019

Evolution of Behavior When Duopolists Choose Prices and Quantities

verfasst von: Abhimanyu Khan, Ronald Peeters

Erschienen in: Dynamic Games and Applications | Ausgabe 2/2020

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Abstract

We study the stable market outcome that evolves when firms, that operate in a spatially differentiated duopolistic market, recurrently choose prices and quantities via an imitation dynamic. In particular, given that firms choose both prices and quantities, we do not explicitly impose market clearing. We find that: (i) a market-clearing outcome always belongs to the set of stable long-run outcomes, (ii) when the level of market differentiation is low, non-market-clearing outcomes (with excess supply as well as excess demand) are also stable, (iii) when the level of market differentiation is not too high or too low, the stable long-run outcomes imply market clearing at prices above monopoly level, and (iv) when the level of market differentiation is high, the stable long-run outcome implies market clearing at the monopoly price.

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Fußnoten
1
Huck et al. [12, 13], Offerman et al. [18] and Apesteguia et al. [4] find experimental evidence of firms imitating a more successful firm when information on strategies and received profits are provided.
 
2
The location of the firms is simply a modeling device to specify the demand that each firm faces in the spatially differentiated market, i.e., location is given exogenously to determine demand, and does not arise out of choices made by the firms. See Hehenkemp and Wambach [9] for a paper that studies the long-run stable choice of location of firms.
 
3
For a slightly more detailed discussion on this, see, for example, Van den Berg and Bos [23].
 
4
Herings [10] and more recently by Alós-Ferrer and Kirchsteiger [3] show that it is possible for non-market-clearing institutions to evolve.
 
5
We note that our approach shares a commonality with the literature on Bertrand–Edgeworth competition in that we assume that firms may not be able to satisfy the demand that they face. In case of Edgeworth [6], one possible reason for this was the capacity constraints faced by firms; one may see Levitan and Shubik [16] for a game theoretic treatment of this situation. Here, the reason for firms not being able to do so is that they themselves produce in advance, and so, the quantity they produce may differ from the demand they face. Thus, while in Bertrand–Edgeworth competition, firms are constrained forever by their capacity choice, in our case, firms are only ‘capacity-constrained’ in a particular period; in the ensuing periods, there exists the possibility of firms choosing a different ‘capacity’ either by means of imitation or experimentation. Interestingly, this inter-temporal flexibility in capacity gives rise to equilibria of an entirely different character than has been discussed in the literature on Bertrand–Edgeworth competition (see, e.g., Vives [25]).
 
6
We ignore the possibility of spillover demand, i.e., the possibility that a consumer who is not able to acquire the good at the preferred location travels to the other firm (which might have excess supply). Such considerations would require additional assumptions on how goods are rationed or distributed among consumers by the firm that faces an excess demand. While it may be instructive to look at this aspect, we do not pursue it in this paper.
 
7
Results are available upon request.
 
8
A technical requirement for stochastic stability analysis is finiteness of the strategy space, implying that firms choose from finite price and quantity grids. While, in the proofs, we treat the strategy space as continuous, to meet the technical requirement we only need to assume that the finite grids for prices and quantities are fine enough so as to not exclude those of any significance (i.e., prices and quantities that are used in the proofs).
 
9
The latter equation is quadratic and it is easily seen that \(\tau >\beta -c\) is a sufficient condition for the larger solution to be a price above \(\beta \).
 
10
The first inequality follows from p being in \(({\overline{p}},\beta )\), the second by the condition in the proposition, and the third from the assumption that \(\tau <c\).
 
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Metadaten
Titel
Evolution of Behavior When Duopolists Choose Prices and Quantities
verfasst von
Abhimanyu Khan
Ronald Peeters
Publikationsdatum
29.08.2019
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 2/2020
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-019-00325-z

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