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Oligopoly models with different learning and production time scales

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Abstract

We propose a modelling approach to study Cournotian oligopolies of boundedly rational firms which continuously update production decisions on the basis of information collected periodically. The model consists of a system of differential equations with piecewise constant arguments, which can be recast into a system of difference equations. Considering different economic settings, we study the local stability of equilibrium, proving the destabilizing role of the time lag between two consecutive learning activities. We investigate some particular families of oligopolies showing the occurrence of both flip and Neimark–Sacker bifurcations, as well as the evidence of multistability with the coexistence between different attractors, occurring when oligopolies consisting of both technologically different and identical firms are studied.

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Notes

  1. For the sake of simplicity, in this section we assume that the involved functions are all defined on \([0, + \infty )\), but this can be indeed relaxed and all results still hold with minor adjustments.

  2. For more details about the economic description and interpretation of the gradient adjustment mechanism, we refer to Bischi et al. (2010) and Cavalli and Naimzada (2015).

  3. We note that if \(q_i (0)=0\), from (5) we have \(q_i(t)=0\), so the ith firm actually does not take part in the market.

  4. We do not enter into mathematical details about conditions under which (5) is well defined and can be solved for any \(t>0\). For a detailed mathematical description on the whole process of transformation of a DEPCA into a discrete time difference equation, we refer to Wiener (1993) and Cavalli and Naimzada (2016).

  5. As evident from the previous considerations and from the mathematical structure of (7), the role of \(\sigma \) and \(v_i\) is very similar, so Proposition 2 and the subsequent comments can be rephrased in terms of the agents’ reactivity, too.

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Acknowledgements

The authors wish also to thank the anonymous Reviewers and Professor Anufriev, Guest Editor of the Special Issue on “Stability and Bifurcations in Nonlinear Economic Systems”, for the useful suggestions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Mathematical Finance and Econometrics, Catholic University of Sacred Heart, Via Necchi 9, 20123, Milan, Italy

    Fausto Cavalli

  2. DEMS, University of Milano-Bicocca, Piazza dell’Ateneo Nuovo 1, 20126, Milan, Italy

    Ahmad Naimzada

  3. Department of Economics and Management, University of Pisa, Via C. Ridolfi 10, 56124, Pisa, Italy

    Mauro Sodini

Authors
  1. Fausto Cavalli
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  2. Ahmad Naimzada
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  3. Mauro Sodini
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Correspondence to Fausto Cavalli.

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Cavalli, F., Naimzada, A. & Sodini, M. Oligopoly models with different learning and production time scales. Decisions Econ Finan 41, 297–312 (2018). https://doi.org/10.1007/s10203-018-0225-0

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  • DOI: https://doi.org/10.1007/s10203-018-0225-0

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