An equilibrium model for matching impatient demand and patient supply over time

TL;DR

This paper develops a dynamic equilibrium model for an online exchange with stochastic arrivals, capturing the behavior of impatient buyers and patient sellers, and provides methods to compute key market quantities and distributions.

Contribution

It introduces a novel equilibrium model for online exchanges with asymmetric patience levels and offers numerical methods for computing equilibrium and related market metrics.

Findings

Equilibrium exists and can be numerically computed.

The model's order density exhibits power-law tails, matching empirical observations.

Closed-form solutions are derived for demand-independent cases.

Abstract

We present a simple dynamic equilibrium model for an online exchange where both buyers and sellers arrive according to a exogenously defined stochastic process. The structure of this exchange is motivated by the limit order book mechanism used in stock markets. Both buyers and sellers are elastic in the price-quantity space; however, only the sellers are assumed to be patient, i.e. only the sellers have a price - time elasticity, whereas the buyers are assumed to be impatient. Sellers select their selling price as a best response to all the other sellers' strategies. We define and establish the existence of the equilibrium in this model and show how to numerically compute this equilibrium. We also show how to compute other relevant quantities such as the equilibrium expected time to sale and equilibrium expected order density, as well as the expected order density conditioned on current selling price. We derive a closed form for the equilibrium distribution when the demand is price independent. At this equilibrium the selling (limit order) price distribution is power tailed as is empirically observed in order driven financial markets.