Comment on 'Asset Bubbles and Overlapping Generations' by Tirole
What this paper finds — and why it matters
Tirole (1985) studied an overlapping generations model with capital accumulation and showed that the emergence of asset bubbles can resolve the capital over-accumulation problem when the economy is dynamically inefficient. His Proposition 1(c) claims that a bubble can emerge if and only if the dividend growth rate exceeds the bubbleless steady-state interest rate. This comment identifies an error in that proposition: the stated condition is necessary but not sufficient for bubble existence. The paper constructs an explicit counterexample in which the dividend growth rate exceeds the bubbleless interest rate but no bubble equilibrium exists, and separately constructs a case in which a bubble exists even when the condition in Proposition 1(c) fails. Corrected necessary and sufficient conditions for bubble existence are derived, and the implications of the correction for the welfare results and the relationship between dynamic inefficiency and bubbles are characterized.
Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Q1. What is the error in Tirole’s Proposition 1(c)?
The error is in the sufficiency direction: Tirole argued that whenever the dividend growth rate exceeds the bubbleless interest rate, a bubble equilibrium exists; Pham and Toda construct a parameter configuration satisfying this condition where no bubble equilibrium exists, because the continuity argument used in Tirole’s proof fails at boundary parameter values. The necessity direction — that bubble existence requires this rate comparison — is not challenged.
Q2. How do the corrected conditions change the interpretation of dynamic inefficiency?
Tirole’s original result linked bubbles tightly to dynamic inefficiency (r < g), providing a clean condition for when bubbles are both feasible and welfare-improving by absorbing excess saving. The correction weakens this link: bubble existence requires additional structural conditions beyond the rate comparison, meaning dynamic inefficiency is a necessary but not sufficient condition for bubbles in the Tirole framework. Policy prescriptions based on the r < g condition for bubble welfare analysis need qualification.
Key concepts
dynamic inefficiency : the OLG condition in which the interest rate falls below the growth rate, making intergenerational transfers from young to old welfare-improving; related to but not sufficient for bubble existence under the corrected Tirole conditions.
bubble existence condition : the necessary and sufficient conditions under which an asset bubble can emerge and persist in the Tirole OLG model; the corrected version requires more than the dividend-growth-rate-exceeds-interest-rate comparison of the original Proposition 1(c).