A Microeconomic Finance Model with a Multi-Asset Market and a Multi-Investor Heterogeneous Groups

TL;DR

This paper develops a mathematical microeconomic model of a multi-asset market with heterogeneous investor groups, analyzing stability and cyclical behaviors driven by trend and valuation motivations.

Contribution

It introduces a dynamic systems approach to analyze stability conditions and the emergence of oscillations and cycles in a multi-asset trading environment.

Findings

Stable equilibria exist without trend emphasis.

Instability leads to price oscillations.

Periodic solutions arise via Hopf bifurcation.

Abstract

We present a mathematical model of a market with $m$ shares traded across $n$ investor groups, each one with similar motivations and trading strategies. The market of each asset consists of a fixed amount of cash and shares (no additions are allowed over time, so the system is closed), and the trading groups are influenced by trend and valuation motivations when buying or selling each asset, but follow a strategy where the purchase of one asset depends on the price of another, while the sale does not. Using these assumptions and basic microeconomic principles, the mathematical model is derived using a dynamic systems approach. We analyze the stability of the model's equilibrium points and determine the parameter conditions for such stability. First, we show that all equilibria are stable in the absence of a clear emphasis on trend-based valuation for each share. Secondly, for systems where the trading group prioritizes the valuation of each stock and the trend of the other for trading purposes, we establish stability conditions and demonstrate with numerical examples that when instability occurs, it manifests as price oscillations in the stocks. Furthermore, we argue for the existence of periodic solutions via a Hopf bifurcation, taking the momentum coefficient as the bifurcation parameter. Finally, we present examples and numerical simulations to support and expand upon the analytical results. One finding in economics and finance is the existence of cyclical behavior in the absence of exogenous factors, as determined by the momentum coefficient. In particular, a stable equilibrium price becomes unstable as trend-based trading increases.