Bubble Formation Dynamics
Returns (r): Asset's past performance
Population (n): Excited speculators
r* = 1/θ: Critical threshold
n*: Steady-state population
Key Insights
1
The Model: Speculators become excited at rate θr(1-n)n and "recover their senses" at rate n.
These competing processes determine whether the excited-speculator population grows or vanishes.
2
Critical Threshold: When returns cross above r* = 1/θ, the excited-speculator population becomes
self-sustaining at n* = (r-r*)/r > 0, potentially triggering a bubble.
3
Parameter Stability: θ is stable across time for a given asset, allowing it to be
estimated during normal times when no bubble is present.
4
Predictive Power: Assets with higher θ values have lower thresholds (r*), making them more
susceptible to a bubble following an initial run-up in returns.
Central Insight: This model explains both why bubbles sometimes happen and why they usually do not.
The parameter θ creates a critical threshold that must be crossed for bubble formation, allowing ex-ante prediction
of bubble likelihood across different assets.