Comment on 'Asset Bubbles and Overlapping Generations'

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.

💡 Research Summary

The paper revisits Proposition 1(c) from Jean‑Tirole’s seminal 1985 overlapping‑generations (OLG) model with capital accumulation and a dividend‑paying asset. Tirole’s proposition states that when the dividend growth rate G_d lies strictly between the “bubble‑free” steady‑state interest rate R and the population growth rate G (i.e. R < G_d < G), a bubble‑free equilibrium cannot exist; instead there is a unique equilibrium in which the asset price exceeds its fundamental value (a bubble) and the long‑run interest rate converges to G. The authors of the present note demonstrate that this claim, as originally formulated, is false.

First, the authors restate the OLG framework: agents live two periods, supply one unit of labor when young, and choose consumption, savings, and holdings of a single perpetual asset that pays an exogenous dividend D_t. Firms use a neoclassical production function F(K_t, L_t) with per‑capita version f(k) satisfying f′>0, f′′<0, f′(0)=∞ and f′(∞)<G. The equilibrium conditions comprise utility maximisation, profit maximisation, a no‑arbitrage condition linking asset price and future price plus dividend, and market‑clearing for savings versus asset holdings.

Using standard algebra the authors reduce the system to a two‑dimensional recursive map for capital per‑capita k_t and detrended asset price p_t = P_t/G^t: k_{t+1}=g(k_t,p_t), p_{t+1}=f′(k_{t+1}) G p_t − d_{t+1}, where g is derived from the savings function and the wage w_t = f(k_t)−k_t f′(k_t). Lemma 2.1 guarantees at most one solution for the implicit equation defining g; Lemma 2.2 establishes monotonicity of equilibrium paths with respect to the initial price; Lemma 2.3 shows that if the limiting interest rate exceeds G, the equilibrium must be bubble‑free and unique.

The counterexample hinges on a specially constructed production function f(k)=A ϕ(k), ϕ(k)=k log(1+1/k), which satisfies all standard assumptions (strict concavity, Inada at zero, diminishing marginal product). Importantly, the wage term ω(k)=f(k)−k f′(k)=A k/(1+k) is almost linear near k=0, implying that when initial capital is small, wages are tiny, limiting savings. Assuming dividends grow at a constant geometric rate G_d∈(0,G) (i.e. D_t≥D G_t^{d}), Lemma 3.1 (“resource curse”) proves that for sufficiently small k_0 the capital sequence converges to zero and the detrended price p_t also converges to zero. In this path the asset price equals its fundamental value V_t at every date, so no bubble forms despite R < G_d < G. This directly contradicts Tirole’s Proposition 1(c).

To rescue the original result, the authors introduce two additional conditions:

1. Sufficiently large initial capital: there exists a threshold κ>0 such that if k_0≥κ the economy converges to the positive steady state k* = βA/G − 1 (with β the savings propensity from logarithmic utility). At this steady state f′(k*)<G_d, so the interest rate is below the dividend growth rate, satisfying the “bubble‑necessary” intuition.

2. Sufficiently small dividends: the dividend level D must be bounded above by a function of A, β, G, and G_d (see inequality (3.6)). When dividends are modest, the capital dynamics are not driven to zero and the unique equilibrium is bubble‑free.

Proposition C formalises these statements and shows that under the extra assumptions the economy behaves exactly as Tirole described: a unique equilibrium exists, it is bubble‑free, and the long‑run interest rate converges to G. The paper supplements this with three illustrative examples. Example 2 demonstrates convergence to the positive steady state when the initial capital is large and dividends are small. Example 3 shows that even with a large initial capital, an excessively large dividend can trigger the resource‑curse dynamics, leading again to k_t→0 and a bubble‑free outcome. These examples underline that the original proposition’s conditions are not sufficient; the additional bounds are essential.

The authors also comment on the broader literature. Most citations of Tirole’s 1985 paper invoke the result only in the context of “pure bubbles” (assets with zero dividends), where the proposition is indeed valid. The dividend‑paying case has been largely overlooked, and this note fills that gap. By exposing the missing assumptions, the paper clarifies the precise circumstances under which asset bubbles are “necessary” in OLG models.

In summary, the contribution of the paper is twofold: (i) it provides a concrete counterexample that satisfies every original assumption of Proposition 1(c) yet yields a unique bubble‑free equilibrium, thereby disproving the proposition as originally stated; (ii) it identifies and proves the minimal extra conditions (large enough initial capital and small enough dividends) that restore the proposition’s validity. The findings have important implications for theoretical work on rational bubbles, suggesting that modelers must pay careful attention to initial endowments and dividend processes when assessing the inevitability of bubbles in overlapping‑generations economies.