The equilibrium price of bubble assets

TL;DR

This paper derives a new mathematical equation to determine the stable equilibrium price of bubble assets in an economy, showing that such prices can be uniquely stable if the bubble provides better returns than fiat money.

Contribution

It introduces a novel Hamilton-Jacobi equation for bubble asset valuation and proves the existence of a unique stable solution under specific economic assumptions.

Findings

A new Hamilton-Jacobi equation characterizes bubble asset values.

Existence of a unique stable bubble price under certain conditions.

Bubble assets can stabilize prices by offering better returns than fiat money.

Abstract

Considering a simple economy, we derive a new Hamilton-Jacobi equation which is satisfied by the value of a ''bubble'' asset. We then show, by providing a rigorous mathematical analysis of this equation, that a unique non-zero stable solution exists under certain assumptions. The economic interpretation of this result is that, if the bubble asset can produce more stable returns than fiat money, agents protect themselves from hazardous situations through the bubble asset, thus forming a bubble's consensus value. Our mathematical analysis uses different ideas coming from the study of semi-linear elliptic equations.