Comment on 'Asset Bubbles and Overlapping Generations'

TL;DR

This paper critically examines Tirole's 1985 model on asset bubbles in overlapping generations economies, providing counterexamples and conditions for the original proposition to hold.

Contribution

It corrects and clarifies Tirole's Proposition 1(c) by presenting counterexamples and establishing necessary conditions for the proposition's validity.

Findings

Counterexample shows Proposition 1(c) can fail under original assumptions.

Additional conditions restore the validity of Proposition 1(c).

Sufficiently large initial capital and small dividends are crucial for bubble existence.

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 economy without the asset) 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 show that this result (as stated) is incorrect by presenting an example economy that satisfies all assumptions of Proposition 1(c) but its unique equilibrium is bubbleless. We also restore Proposition 1(c) under the additional assumptions that initial capital is sufficiently large and dividends are sufficiently small. We show through examples that these conditions are essential.