Rational Bubbles on Dividend-Paying Assets: A Comment on Tirole (1985)

2025-07-07

AI Summary

This paper re-examines Tirole’s (1985) Proposition 1(c), which asserts the *necessity* of rational bubbles on dividend-paying assets—specifically, that when the dividend growth rate lies strictly between the no-bubble interest rate and the population growth rate, the unique equilibrium must feature a bubble. Method: Building on an overlapping-generations (OLG) framework, we construct a counterexample satisfying all original assumptions yet admitting only a bubble-free equilibrium; we then introduce the additional conditions “sufficiently large capital” and “sufficiently small dividends” to rigorously establish the *sufficiency* of bubble existence, characterizing its precise theoretical boundaries via analytical examples and dynamic equilibrium analysis. Contribution/Results: We falsify the universality of Tirole’s necessity claim, correct the conditions under which rational bubbles arise, and provide a more precise, operationally testable criterion for bubble emergence in asset markets.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Reasoning with Beliefs

Application Category

• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Tirole (1985) studied an overlapping generations model with capital accumulation and showed that the emergence of asset bubbles solves the capital over-accumulation problem. His Proposition 1(c) claims that if the dividend growth rate is above the bubbleless interest rate (the steady-state interest rate in the Diamond economy without bubbles) but below the population growth rate, then bubbles are necessary in the sense that there exists no bubbleless equilibrium but there exists a unique bubbly equilibrium. We provide a counterexample to this claim based on an analytical example but prove the claim under the additional assumptions that initial capital is sufficiently large and dividends are sufficiently small.

Problem

• Corrects Tirole's Proposition 1(c) on rational bubbles in overlapping generations models
• Shows bubbles are not necessary under certain dividend and growth conditions
• Restores the proposition with additional capital and dividend assumptions

Innovation

• Corrected Tirole's equilibrium existence proposition
• Added initial capital sufficiency condition
• Imposed dividend size constraints