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The equilibrium price of bubble assets

Charles Bertucci, Jean-Michel Lasry, Pierre Louis Lions

TL;DR

The study addresses how bubble assets can sustain a stable, nonzero equilibrium price in a simple economy by deriving a stationary Hamilton-Jacobi-Bellman equation for the bubble value, $- u \Delta u + \frac{1}{2}|\nabla u|^2 = a(x) u$. It establishes existence and uniqueness of a positive, stable solution under the spectral condition $\lambda_1(-\nu\Delta - a) < 0$, using semi-linear elliptic analysis and a control-theoretic interpretation. Economically, the paper interprets the positive solution as a bubble consensus value (C-value) that hedges against policy risk, with explicit implications for cryptocurrencies, gold, and real estate, and connects to no-arbitrage when risk aversion is removed. The results suggest robust, parameter-sensitive mechanisms for bubble persistence and lay a foundation for extensions to mean-field games, heterogeneous agents, and multi-asset coupling, offering a quantitative lens on the stability and dynamics of bubble assets.

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.

The equilibrium price of bubble assets

TL;DR

The study addresses how bubble assets can sustain a stable, nonzero equilibrium price in a simple economy by deriving a stationary Hamilton-Jacobi-Bellman equation for the bubble value, . It establishes existence and uniqueness of a positive, stable solution under the spectral condition , using semi-linear elliptic analysis and a control-theoretic interpretation. Economically, the paper interprets the positive solution as a bubble consensus value (C-value) that hedges against policy risk, with explicit implications for cryptocurrencies, gold, and real estate, and connects to no-arbitrage when risk aversion is removed. The results suggest robust, parameter-sensitive mechanisms for bubble persistence and lay a foundation for extensions to mean-field games, heterogeneous agents, and multi-asset coupling, offering a quantitative lens on the stability and dynamics of bubble assets.

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.

Paper Structure

This paper contains 27 sections, 19 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generic model and derivation of the main equation
  3. The one dimensional case
  4. The decision problem of the agents
  5. Market clearing
  6. Generalization of our model
  7. Mathematical analysis
  8. Existence and uniqueness
  9. On the necessity of the assumptions of the main theorem
  10. The case of a source term
  11. Optimal control interpretation
  12. Stability of the solution
  13. Some properties of the solution
  14. Interpretation of Theorem \ref{['thm:1']} as a non-linear version of Krein-Rutman
  15. Convergence of the time dependent problem
  16. ...and 12 more sections

Key Result

Proposition 1

If all the functions involved are smooth, the problem of maximization of the growth rate of the portfolio of an agent with wealth $q$, when the economy is in state $x \in (0,1)$, is given by where the maximization is done over $\theta$, the proportion of "other" good the agent needs to buy.

Figures (1)

  • Figure 1: Scheme of the sets of solution of \ref{['kr']} for various values of $\epsilon$, as functions of $\lambda$. The vertical axes can be interpreted as the $\|\cdot\|_\infty$ norm of the solution. In green, $\epsilon = 0$, in orange $\epsilon= \epsilon'>0$ and in blue $\epsilon > \epsilon'$.

Theorems & Definitions (47)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Remark 5
  • Lemma 1
  • proof
  • ...and 37 more