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Delay dynamics in nonlinear monopoly with gradient adjustment

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Abstract

Two types of boundedly rational monopolists are considered, when they are unable to determine the profit maximizing output levels. In the first case, the monopolist knows the price function and in the second case it can access only past output and price values. In applying gradient dynamics, the marginal profit is either known or approximated by finite differences based on two past profit data. Stability conditions are derived first with discrete time scales, which are also applied in a special case. Two models of continuous-time dynamics are then introduced. The first is a natural modification of the discrete model, and the other includes an inertia coefficient with the derivative. In each case, a delay differential equation is obtained with two delays. Stability conditions are derived and the stability-switching curves are constructed and illustrated.

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Notes

  1. Even if the price function is known, it might be possible that a monopolist is endowed with limited computational skills to solve the profit maximization problem.

  2. This is a very extreme assumption; however, it is useful to see the effects caused by the information differences.

  3. Matsumoto and Szidarovszky (2014b) build a continuous model to adopt this method.

  4. Clower (1959) calls a monopolist "knowledgeable monopolist" under similar circumstances.

  5. Equation (6) is a simple numerical differentiation formula, which can be replaced with any more complex rule based on more previous points without changing the essence of the paper but making the mathematical derivations more lengthy and complicated.

  6. Notice that the \(\ell \)-monopolist is unable to manipulate the following calculations since it does not know the form of the profit function.

  7. It has been checked that the convex cost function \(c(x)=x^{\beta },\ 2\le \beta \le \alpha \ \)does not affect the general properties of the result to be obtained. See Matsumoto and Szidarovszky (2014b).

  8. Gori et al. (2016) specify the price function as the cubic equation in (13) and arrive at the same result. Notice that the price function is not specified but general in Theorem 3.

  9. Using the cubic price function, Gori et al. (2016) derives essentially the same characteristic equation as (28). However, they solve it in a completely different way. Further they take no notice of the constraint \( \tau _{1}<\tau _{2}\) that should hold in equation (23).

  10. Mathematically, we have the same result if \(\tau _{1}>0\) and \(\tau _{2}=0.\ \) However, this symmetric case is assumed away by assumption \(\tau _{1}<\tau _{2}\).

  11. \(\tau _{1}>\tau _{2}\) holds in the white region and it violates the constraint \(\tau _{1}<\tau _{2}.\ \)We eliminate this region from further investigation.

  12. From (37) and (38), solving \(2=\tau _{2,0}^{+}(\omega )\) gices \(\omega _{a}\simeq 1.214\ \)that is substituted into \(\tau _{1,0}^{-}(\omega _{a})\simeq 0.589.\)

  13. See Berezowski (2001) for more details.

  14. This dynamic equation is identical with the one considered by Gori et al. (2016) . As a result, our Theorem 6 is essentially the same as their result.

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Authors and Affiliations

  1. Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan

    Akio Matsumoto

  2. Department of Mathematics, Corvinus University, Fövám tér 8, Budapest, 1093, Hungary

    Ferenc Szidarovszky

Authors
  1. Akio Matsumoto
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  2. Ferenc Szidarovszky
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Correspondence to Akio Matsumoto.

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The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions. The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566)

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Matsumoto, A., Szidarovszky, F. Delay dynamics in nonlinear monopoly with gradient adjustment. Decisions Econ Finan 44, 533–557 (2021). https://doi.org/10.1007/s10203-021-00342-x

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