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

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.

💡 Research Summary

This paper revisits the seminal model of Jean‑Tirole (1985) that introduced a dividend‑paying long‑lived asset into the Diamond (1965) overlapping‑generations (OLG) framework. While Tirole’s original analysis showed that pure bubbles (assets with zero dividends) can resolve the well‑known over‑accumulation of capital, the present study demonstrates that the version of the model with a dividend‑paying asset contains several subtle but important issues that affect equilibrium existence, uniqueness, and long‑run dynamics.

The authors first formalize the environment: discrete time, infinite horizon, a representative firm with a neoclassical production function (F(K,L)) that is homogeneous of degree one, strictly concave, twice continuously differentiable, and satisfies the Inada conditions (f’(0)=\infty) and (f’(\infty)<G). Population grows at a constant rate (G>0); the dividend stream of the asset grows at rate (G_d); and the steady‑state interest rate that would prevail without the asset is denoted (R). The asset is in unit supply, pays an exogenous dividend (D_t), and its price (P_t) can be decomposed into a fundamental value (V_t) (the discounted present value of future dividends) and a bubble component (B_t).

Using a series of lemmas, the authors reduce the full general‑equilibrium definition to four core conditions: (i) each young agent chooses savings (s_t) to maximize utility given the wage (w_t) and the future return (R_{t+1}); (ii) the firm maximizes profit; (iii) a no‑arbitrage condition (P_t = (P_{t+1}+D_{t+1})/R_{t+1}); and (iv) market‑clearing for labor, capital, and the asset. This compact representation allows the authors to prove a general existence theorem (Theorem 1) that requires only standard quasi‑concavity of preferences and the usual properties of the production function—significantly weaker than the “gross‑substitutes” assumption used in Tirole’s original proof.

The paper’s core contribution is a systematic re‑examination of the three parameter regimes identified by Tirole:

1. High‑interest regime ((R>G)). The authors show that, contrary to Tirole’s claim of a unique bubble‑less equilibrium with interest rate converging to (R), the interest rate may actually exceed (R) and even diverge to infinity. Example 2 provides a closed‑form solution where (R_t) grows without bound while the capital stock (K_t) collapses to zero. This phenomenon is interpreted as a “resource curse”: the introduction of an asset with negligible dividends can drive the economy toward vanishing capital because the asset price inflates dramatically, raising the return on savings and discouraging investment.

2. Intermediate regime ((G_d<R<G)). Here a continuum of equilibria exists, indexed by the initial asset price (p_0). The authors confirm that a bubble‑less equilibrium exists at the lower bound of the interval, while a fully bubbly equilibrium exists at the upper bound. However, they demonstrate that additional, fairly strong exogenous conditions—such as a sufficiently large initial capital stock and sufficiently small dividends—are required to guarantee these results. Theorem 5 and Lemma 4.1 formalize these sufficient conditions, highlighting that the original proposition in Tirole lacked explicit assumptions about the initial state.

3. Low‑interest, high‑dividend regime ((R<G_d<G)). Tirole argued that a bubble is inevitable in this case. The present analysis overturns that claim. Theorem 3 and Theorem 4 prove that, depending on the initial capital and dividend levels, the economy can either (i) converge to a bubble‑less equilibrium with the capital stock collapsing to zero, or (ii) converge to a unique asymptotically bubbly equilibrium where the interest rate converges to the population growth rate (G). Example 3 illustrates the first outcome, while Example 4 shows a bubbly but asymptotically bubble‑less path. Thus, the necessity of bubbles is not a universal feature of the model.

The authors summarize their refined propositions as (a′), (b′), and (c′), which replace Tirole’s original Proposition 1. (a′) retains uniqueness in the high‑interest case but adds the possibility of divergent interest rates. (b′) clarifies that the intermediate case requires explicit bounds on initial conditions. (c′) replaces the “bubble is necessary” statement with a bifurcation result contingent on the initial capital‑dividend configuration.

Beyond the technical contributions, the paper draws two broader implications. First, the “resource curse” mechanism derived from the dividend‑poor asset mirrors macro‑econometric findings that resource‑rich economies may under‑invest and experience lower growth. Second, the nuanced conditions for bubble existence suggest that policy prescriptions aimed at suppressing or exploiting bubbles must account for the underlying dividend structure and the economy’s initial asset distribution.

In conclusion, this work provides a more complete and rigorous characterization of equilibrium in Tirole’s OLG model with dividend‑paying assets. It corrects several overstated claims in the original paper, establishes clear sufficient conditions for the existence and uniqueness of various equilibrium types, and highlights the pivotal role of initial conditions in determining long‑run outcomes. The findings enrich the rational‑bubble literature and offer fresh insights into the interaction between asset markets and long‑run economic growth.