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Long-Run Behavior of Equilibrium in Tirole (1985)'s Model with Dividend-Paying Asset

Ngoc-Sang Pham, Alexis Akira Toda

TL;DR

This paper reexamines Tirole's 1985 model of asset bubbles in an overlapping-generations framework with a dividend-paying asset, clarifying existence, uniqueness, and long-run behavior under explicit, weaker conditions. It develops a deterministic, stationary system with detrended variables $(k_t,p_t,d_t)$ and a reduced dynamic $(k_{t+1}=g(k_t,p_t),\,p_{t+1}=(f'(k_{t+1})/G)\,p_t-d_{t+1})$, decomposing asset prices into fundamental and bubble components $P_t=V_t+B_t$. The authors prove existence of equilibria, establish monotonicity and conditions for bubbleless vs. bubbly outcomes, and show that the bubble literature's propositions require qualifications; they provide counterexamples to prior claims and offer explicit analytical examples illustrating bubbleless, bubbly, and mixed long-run paths. The results sharpen understanding of when bubbles are necessary, how initial asset prices and dividend dynamics drive outcomes, and how resource-curse-type dynamics can emerge in dividend-payings settings, enriching the rational bubble literature with stronger, more transparent conditions and constructive examples.

Abstract

We revisit the classic paper of Tirole "Asset Bubbles and Overlapping Generations" (1985, Econometrica), which shows that the emergence of asset bubbles solves the capital over-accumulation problem. While Tirole's main insight holds with pure bubbles (assets without dividends), we argue that his original analysis with a dividend-paying asset contains some issues. We provide a fairly complete analysis of Tirole's model with general dividends such as equilibrium existence, uniqueness, and long-run behavior under weaker but explicit assumptions and complement with examples based on closed-form solutions. Some of the claims in Tirole (1985) require qualifications including (i) after the introduction of an asset with negligible dividends, the economy may collapse towards zero capital stock ("resource curse") and (ii) the necessity of bubbles is less clear-cut.

Long-Run Behavior of Equilibrium in Tirole (1985)'s Model with Dividend-Paying Asset

TL;DR

This paper reexamines Tirole's 1985 model of asset bubbles in an overlapping-generations framework with a dividend-paying asset, clarifying existence, uniqueness, and long-run behavior under explicit, weaker conditions. It develops a deterministic, stationary system with detrended variables and a reduced dynamic , decomposing asset prices into fundamental and bubble components . The authors prove existence of equilibria, establish monotonicity and conditions for bubbleless vs. bubbly outcomes, and show that the bubble literature's propositions require qualifications; they provide counterexamples to prior claims and offer explicit analytical examples illustrating bubbleless, bubbly, and mixed long-run paths. The results sharpen understanding of when bubbles are necessary, how initial asset prices and dividend dynamics drive outcomes, and how resource-curse-type dynamics can emerge in dividend-payings settings, enriching the rational bubble literature with stronger, more transparent conditions and constructive examples.

Abstract

We revisit the classic paper of Tirole "Asset Bubbles and Overlapping Generations" (1985, Econometrica), which shows that the emergence of asset bubbles solves the capital over-accumulation problem. While Tirole's main insight holds with pure bubbles (assets without dividends), we argue that his original analysis with a dividend-paying asset contains some issues. We provide a fairly complete analysis of Tirole's model with general dividends such as equilibrium existence, uniqueness, and long-run behavior under weaker but explicit assumptions and complement with examples based on closed-form solutions. Some of the claims in Tirole (1985) require qualifications including (i) after the introduction of an asset with negligible dividends, the economy may collapse towards zero capital stock ("resource curse") and (ii) the necessity of bubbles is less clear-cut.

Paper Structure

This paper contains 36 sections, 31 theorems, 108 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Agents
  4. Production
  5. Capital investment and asset
  6. Individual problem
  7. Equilibrium
  8. Equilibrium system
  9. Asset prices and long-run behavior
  10. Existence of bubbleless equilibria
  11. Bubble possibility and necessity
  12. Counterexample to Proposition 1 of
  13. Analytical examples
  14. Case \ref{['item:eqset1']} of Theorem \ref{['thm:eqset']}
  15. Case \ref{['item:eqset2']} of Theorem \ref{['thm:eqset']}
  16. ...and 21 more sections

Key Result

Lemma 2.1

A rational expectations equilibrium is equivalent to a nonnegative sequence $\left\{ { (P_t,R_{t+1},w_t,s_t,K_t)}\right\}_{t=0}^\infty$ such that, for each $t$,

Figures (3)

  • Figure 1: Bubbleless equilibrium with $k_t\to 0$.
  • Figure 2: Asymptotically bubbly equilibrium.
  • Figure 3: Bubbly but asymptotically bubbleless equilibrium.

Theorems & Definitions (64)

  • Remark 1
  • Definition 1
  • Lemma 2.1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Example 1: Log utility
  • ...and 54 more