Paper: arXiv 2504.02835

Authors: Alain M. Dikande, H. Ntahombagana Matabaro

Abstract

The Behrens-Feichtinger model provides a deterministic picture for the co-evolution of sales of two firms, producing the same goods and competing in a common market. The model involves an active investment strategy such that the temporary investment of each of the two firms depends on its relative position in the market. In this work we are interested in a specific regime of evolution referred to as leapfrogging regime, in which each firm has the possibility to dominate the market alternately during some finite period of time. We examine conditions favoring the leapfrogging dynamics of the model by introducing two appropriate variables, namely the sale difference and total sale of the two firms at any time. An analysis of stability of fixed points of the resulting coupled discrete nonlinear equations is carried out, and the bifurcation diagrams of the sale difference and sum with respect to the elasticity coefficient, are generated. A time-series analysis suggests that the leapfrogging regime, characterized by periodic oscillations of the sale difference from positive to negative branches, is stabilized by specific values of characteristic parameters of the model and when in addition the elasticity coefficient, related to the difference in investment strategies of the two firms, is reasonably high. A positive sign of cumulated sales at any time is required to ensure the availability of goods during the leapfrogging dynamics.

Complexity vs Empirical Score

  • Math Complexity: 5.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a decent theoretical and mathematical analysis of the Behrens-Feichtinger model by reformulating it into coupled difference equations for sale differences and sums. However, it is entirely synthetic and theoretical, lacking any real-world market data validation, empirical backtesting, or practical applicability to real-world financial/economic microstructures.

Research Flowchart

  flowchart TD
    A[Research Goal: Identify conditions for leapfrogging dynamics in Behrens-Feichtinger model] --> B[Model Formulation: Behrens-Feichtinger coupled discrete nonlinear equations]
    B --> C[Variable Transformation: Introduce Sale Difference & Total Sale variables]
    C --> D[Computational Analysis: Stability analysis & Bifurcation diagrams vs Elasticity coefficient]
    D --> E[Time-Series Analysis: Evaluate periodic oscillations of sales difference]
    E --> F[Key Outcome 1: Leapfrogging stabilizes at high elasticity & specific parameter values]
    E --> G[Key Outcome 2: Positive cumulated sales required to guarantee goods availability]