A simple model for the population dynamics in OTC wholesale fresh product markets

TL;DR

The paper develops a discrete-time, behavioural population-dynamics model for OTC wholesale markets with $N$ competing sellers, capturing how loyal buyers and price-driven adjustments interact to shape daily clienteles and prices. By defining a map on the state space and imposing structured assumptions on the buyer-attractiveness function $f$ and the seller-reaction function $g$, the authors prove that price ratios and clienteles remain bounded and that the system exhibits perpetual oscillations driven by negative feedback, with long-run convergence to equilibrium in the simplest case of $N=2$ under symmetry and regularity conditions. They also show that mean volumes tend to stabilize within a soft interval and that individual seller shares stay away from degeneracy, while oscillations persist in general. The results provide a mathematically rigorous account of how simple, localized buyer–seller interactions can yield realistic market dynamics, including bounded dispersion, normalization tendencies for $N=2$, and conditions leading to damped oscillations or persistent periodic behavior. The work offers a foundation for further refinement (e.g., heterogeneity, extended time scales) and has potential implications for understanding price dispersion and liquidity in non-centralized perishable markets.

Abstract

Given the combined evidences of bounded rationality, limited information and short-term optimization, over-the-counter (OTC) fresh product markets provide a perfect instance where to develop a behavioural approach to the analysis of micro-economic systems. Aiming at characterizing via a rigorous mathematical analysis, the main features of the spontaneous organization and functioning of such markets, we introduce and we study a stylized dynamical model for the time evolution of buyers populations and prices/attractiveness at each wholesaler. The dynamics is governed by immediate reactions of the actors to changes in basic indicators. Buyers are influenced by some degree of loyalty to their regular suppliers. Yet, at times, they also prospect for potential better offers. On the other hand, sellers primarily aim at maximising their profit. Yet, they can be also prone to improving their competitiveness in case of clientele deficit. Our results reveal that, in spite of being governed by simple and immediate rules, the competition between sellers self-regulates in time, as it constrains to bounded ranges the dispersion of both prices and clientele volumes, does similarly for the mean clientele volume, and it generates oscillatory behaviours that prevent any seller to dominate permanently its competitors (and to be dominated forever). Long-term behaviours are also investigated, with focus on asymptotic convergence to an equilibrium, as can be expected for a standard functioning mode. In particular, in the simplest case of 2 competing sellers, a normal-form-like analysis proves that such convergence holds, provided that the buyer's loyalty is sufficiently high or the sellers' reactivity is sufficiently low. In other words, this result identifies and proves those characteristics of the system that are responsible for long term stability and asymptotic damping of the oscillations.

Figures (4)

• Figure 1: Graphs of $f(\rho,\cdot)$ for the piecewise affine example \ref{['LINEARMAP']} with $c(\rho)=e^{-|\ln \rho|}$ (so that $c(\rho)=\frac{1}{\rho}$ for $\rho\geq 1$ and $c(\rho)=\rho$ for $\rho<1$) and $\rho\in \{\frac{1}{4},\frac{1}{2},1,2,4\}$. These plots illustrate the assumptions (Hf1-2), and in particular, the monotonicity in $x$ and $\rho$, that the graph lies above the diagonal, excepted at $x=1$, when $\rho<1$, and it lies below the diagonal, excepted at $x=0$, when $\rho>1$, and that we have $f'_x(\rho,\cdot)<1$ for all $\rho\neq 1$. The symmetry in (Hf3) is also evident.
• Figure 2: Illustration for $N=4$ of the various results on clientele fractions and prices, their asymptotic behaviours and their perpetual crossings. Plots of the times series of clientele fractions $\{x_i^t\}_{i=1}^4$ (left), mean value $\langle \mathbf x^t\rangle$ (inset) and prices ratios $\{\rho_i^t\}_{i=1}^3$ (right) associated with an orbit of the system \ref{['DEFDYNAM']} with $\alpha=0.9$, $f$ as in Fig. \ref{['GRAPHF-AFFINE']} and $g(x)=\frac{x}{2}$. The initial condition has been chosen at random. The picture illustrates in particular the perpetual crossings claimed in Proposition \ref{['ALTERNATIONS']} and it suggests that the long-term convergence to equilibrium of Proposition \ref{['REALIST-PRICES']} holds in this case. The time series in the inset indicates that monotonous decay of the mean value $\langle \mathbf x^t\rangle$ prevails below the value $N-\frac{1}{N-1}$ given in Proposition \ref{['CONSTRAINTS-X']}.
• Figure 3: Illustration of Theorem \ref{['STABILITY']}. Plot of fragments of an orbit for $f$ as in Fig. \ref{['GRAPHF-SMOOTH']} and $g(x)=\frac{x}{2}$. The parameter $\alpha$ has been chosen so that $\lambda_\pm(\frac{1}{2})=e^{\frac{i\pi}{6}}$ (NB: For such $f$, we actually have $\lambda_\pm(x)=\lambda_\pm(\frac{1}{2})$ for each $x\in [x_0,1-x_0]$). The dots represent points of coordinates $(x_1^{12 k}-x_2^{12 k},\ln \rho_1^{12 k})$ for $k\in [1,500]$. Initial condition: $x_1^0=0.51,x_2^0=0.48$ and $\rho_1^0=1.1$. Color code: Dark blue for $k=1$ to dark red for $k=500$. The picture clearly illustrates the convergence to $(0,0)$ as a consequence of the Theorem. Notice also that, due the nonlinear effects, the rotation angle slightly deviates from the one, namely $\frac{\pi}{6}$, of the linearized dynamics in the neighbourhood of the fixed points (NB: had the dynamics be that linearized one, the dots would have been aligned on a ray issued from the origin); yet, as expected, the spiralling drift is smaller as the dots come closer to the origin.
• Figure 4: Illustration of $f(\rho,\cdot)$ for the example that satisfies all conditions at once with $\rho_0=3$, $c(\rho)=e^{-(\ln \rho)^2}$, $x_0=\frac{1}{3}$ and $b(\rho)=\frac{1-c(\rho_0)}{4}(\rho-1)$. The lower (resp. upper) grey region is where $f$ coincides with the linear map in the example \ref{['LINEARMAP']} for $\rho\geq 3$ (resp. $\rho\leq \frac{1}{3}$). The left blue triangle is the domain $(\rho,x)\in [1,\rho_0)\times [0,\frac{1}{2}-x_0]$ where the map smoothly interpolates between 0 and $f\left(\rho,\frac{1}{2}-x_0\right)$ and similarly for the upper red triangle. In the remaining domains $[1,\rho_0)\times [\frac{1}{2}-x_0,1]$ and $[\frac{1}{\rho_0},1]\times [0,\frac{1}{2}+x_0]$ the map is affine and defined in a way so that it is of class $C^4$ in $U$.

Theorems & Definitions (27)

• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Remark 3.4
• Lemma 3.5
• Proposition 3.6
• Corollary 3.7
• Proposition 3.8
• Proposition 3.9
• Theorem 3.10