# Injectivity and the Law of Demand

## TL;DR

This paper explores conditions under which demand mappings are injective, linking the law of demand to injectivity criteria, and generalizing classical Jacobian conditions with differentiability assumptions.

## Contribution

It introduces new injectivity conditions for demand functions based on the law of demand and extends classical Jacobian criteria to differentiable cases.

## Key findings

- Injectivity of demand mappings can be verified via constancy over line segments when the law of demand holds.
- Differentiability allows for necessary and sufficient conditions for injectivity that generalize classical Jacobian criteria.
- The results provide a theoretical foundation for analyzing demand functions in economic models.

## Abstract

Establishing that a demand mapping is injective is core first step for a variety of methodologies. When a version of the law of demand holds, global injectivity can be checked by seeing whether the demand mapping is constant over any line segments. When we add the assumption of differentiability, we obtain necessary and sufficient conditions for injectivity that generalize classical \cite{gale1965jacobian} conditions for quasi-definite Jacobians.