# Necessary and sufficient condition for equilibrium of the Hotelling model

## TL;DR

This paper provides a precise mathematical condition that characterizes the equilibrium in Hotelling's linear city model, enhancing understanding of vendor competition and location strategies.

## Contribution

It establishes a necessary and sufficient condition for equilibrium in the Hotelling model, offering a clear representation of equilibrium configurations.

## Key findings

- Derived a mathematical condition for equilibrium
- Proved the condition is both necessary and sufficient
- Provided a new representation for equilibrium

## Abstract

We study a model of vendors competing to sell a homogeneous product to customers spread evenly along a linear city. This model is based on Hotelling's celebrated paper in 1929. Our aim in this paper is to present a necessary and sufficient condition for the equilibrium. This yields a representation for the equilibrium. To achieve this, we first formulate the model mathematically. Next, we prove that the condition holds if and only if vendors are equilibrium.