# Price equations with symmetric supply/demand; implications for fat tails

## TL;DR

This paper develops price dynamics equations based on symmetric supply and demand functions, showing how different functional forms influence the tail behavior of price change distributions, with implications for understanding fat tails in financial markets.

## Contribution

It introduces a framework linking supply/demand symmetry and function form to the tail behavior of price change distributions, extending microeconomic criteria to price dynamics modeling.

## Key findings

- Linear demand/supply functions lead to power-law tails with exponent -2.
- Nonlinear functions with large exponents produce tails with exponent approaching -1.
- Logarithmic functions result in exponential decay of price change probabilities.

## Abstract

Implementing a set of microeconomic criteria, we develop price dynamics equations using a function of demand/supply with key symmetry properties. The function of demand/supply can be linear or nonlinear. The type of function determines the nature of the tail of the distribution based on the randomness in the supply and demand. For example, if supply and demand are normally distributed, and the function is assumed to be linear, then the density of relative price change has behavior $x^{-2}$ for large $x$ (i.e., large deviations). The exponent approaches $-1$ if the function of supply and demand involves a large exponent. The falloff is exponential, i.e., $e^{-x}$, if the function of supply and demand is logarithmic.