Comment on 'Asset Bubbles and Overlapping Generations'

Alexis Akira Toda; Ngoc-Sang Pham

arxiv: 2507.12477 · v3 · pith:2STWB7N2new · submitted 2025-07-07 · 💰 econ.TH

Comment on 'Asset Bubbles and Overlapping Generations'

Ngoc-Sang Pham , Alexis Akira Toda This is my paper

Pith reviewed 2026-05-22 12:19 UTC · model grok-4.3

classification 💰 econ.TH

keywords asset bubblesoverlapping generationsTirole modelequilibrium existencecapital overaccumulationcounterexampledynamic inefficiency

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The pith

Tirole's proposition that asset bubbles are necessary under certain growth conditions fails in a counterexample economy with a unique bubbleless equilibrium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Tirole showed that asset bubbles can resolve capital overaccumulation in overlapping generations models. His Proposition 1(c) claimed that when dividend growth exceeds the bubbleless interest rate yet stays below population growth, no bubbleless equilibrium exists and a unique bubbly one does. This paper constructs an economy meeting every listed assumption of that proposition yet possessing only a bubbleless equilibrium. The authors recover the original claim by adding requirements that initial capital be large enough and dividends small enough, and demonstrate via examples that those extra conditions cannot be dropped.

Core claim

Tirole's Proposition 1(c) is incorrect as stated because there exists an example economy satisfying all its assumptions but whose unique equilibrium is bubbleless. The proposition holds if initial capital is sufficiently large and dividends are sufficiently small, but these conditions are essential as shown by further counterexamples.

What carries the argument

A constructed counterexample overlapping-generations economy with chosen parameters for capital stock, dividend stream, and growth rates that meets Tirole's stated inequalities yet admits only the bubbleless steady state.

If this is right

Bubbles need not emerge even when dividends grow faster than the bubbleless rate but slower than population growth.
The necessity of bubbles requires additional restrictions on the size of initial capital and the level of dividends.
Equilibrium uniqueness results in these models depend on initial conditions in ways not captured by the original growth-rate inequalities alone.
Policy arguments relying on Tirole's necessity claim must first verify that initial capital and dividend levels satisfy the restored conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Models of bubble necessity may need to track the entire path of capital accumulation from arbitrary starting points rather than focusing solely on steady-state growth rates.
The counterexample raises the possibility that similar gaps exist in related propositions about dynamic inefficiency and asset pricing in finite-horizon or stochastic OLG economies.
Re-deriving necessity results with explicit bounds on initial capital could produce sharper testable predictions for when bubbles appear in calibrated macroeconomic models.

Load-bearing premise

The counterexample economy truly meets every assumption in Tirole's Proposition 1(c), including the exact ordering of dividend growth relative to the bubbleless interest rate and population growth.

What would settle it

Explicit calculation of all competitive equilibria in the counterexample economy showing that market clearing holds only on the bubbleless path and not on any path with a positive bubble.

Figures

Figures reproduced from arXiv: 2507.12477 by Alexis Akira Toda, Ngoc-Sang Pham.

**Figure 1.** Figure 1: Production function. We set σ = 1.01 so that σ 1−2α 1−α = 0.9901 ∈ (Rθ/G, 1). We set C = 1 + ρ = 5, initialize to k0 = 0.01, and construct the unique equilibrium using Proposition C [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Figure 2: Equilibrium path in Proposition C. The parameter values are G = 1, A = 1/4, α = 2/3, β = 1/2, θ = 23/6, C = 5, σ = 1.01, and k0 = 0.01. 4 Restoring Proposition 1(c) Because of the existence of a counterexample, Tirole (1985)’s original claim in Proposition 1(c) cannot be true without additional assumptions. Notice that in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tirole's 1985 claim on bubble necessity fails without extra conditions on capital and dividends, and this paper supplies the counterexample plus the fix.

read the letter

The main point is that Tirole's Proposition 1(c) does not hold as stated. The authors give an explicit economy that satisfies every listed assumption yet has a unique bubbleless equilibrium, so bubbles are not necessary under those conditions alone. They then restore the original claim by adding that initial capital must be large enough and dividends small enough, and they show with further examples that dropping either extra condition breaks the result again. This is useful because it pins down exactly what was missing in the 1985 statement without overclaiming generality. The construction is direct and the logic is easy to follow once the parameters are set out. The only soft spot worth checking is whether the counterexample really places dividend growth strictly between the bubbleless interest rate and population growth while obeying all the other maintained assumptions such as concavity and market clearing. The paper's own restoration under tighter restrictions suggests the authors have verified this, so the refutation appears to land. If the numbers line up exactly, there is no deeper flaw. This is for people who work with OLG models of capital accumulation and dynamic inefficiency. A reader who already knows Tirole and wants the precise boundary conditions will find it worth reading. It deserves peer review because a targeted correction to a standard reference keeps the record straight and the math is reproducible from the examples given.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that Tirole's (1985) Proposition 1(c) is incorrect as stated: there exists an economy satisfying all listed assumptions (including dividend growth strictly between the bubbleless interest rate and population growth) whose unique equilibrium is nevertheless bubbleless. The authors construct such a counterexample, then restore the original claim by adding the requirements that initial capital be sufficiently large and dividends sufficiently small; they further provide examples showing these extra conditions are essential.

Significance. If the counterexample is shown to satisfy every hypothesis of the original proposition, including the precise ordering of dividend growth, bubbleless interest rate, and population growth, the correction would be significant for the literature on rational bubbles in OLG models. Tirole's result has shaped discussions of dynamic inefficiency and the role of bubbles in preventing capital over-accumulation; a precise refinement of its conditions would improve the sharpness of subsequent theoretical work. The use of explicit examples to demonstrate necessity of the added assumptions is a constructive feature.

major comments (2)

[Counterexample] Counterexample construction: the manuscript must report the explicit numerical or functional values of the bubbleless steady-state interest rate r, dividend growth g_d, and population growth n in the example, together with verification that g_d > r and g_d < n hold while the dynamic system admits no bubbleless equilibrium path. This verification is load-bearing for the claim that the example meets every assumption of Tirole's Proposition 1(c).
[Restored result] Restoration of the result: the additional assumptions on initial capital and dividend size are presented as sufficient to recover the original conclusion, but the paper should state the revised proposition formally, including the precise bounds or inequalities that define 'sufficiently large' capital and 'sufficiently small' dividends.

minor comments (2)

A side-by-side comparison (perhaps in a table) of Tirole's original assumptions versus the parameter values chosen for the counterexample would help readers confirm compliance at a glance.
Notation for the key rates (r, g_d, n) should be introduced with a brief reminder of their definitions when first used in the main text, even if standard in the literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The suggestions improve the clarity of the counterexample and the formal presentation of the restored result. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Counterexample] Counterexample construction: the manuscript must report the explicit numerical or functional values of the bubbleless steady-state interest rate r, dividend growth g_d, and population growth n in the example, together with verification that g_d > r and g_d < n hold while the dynamic system admits no bubbleless equilibrium path. This verification is load-bearing for the claim that the example meets every assumption of Tirole's Proposition 1(c).

Authors: We agree that explicit reporting of the parameters strengthens the verification. In the revised manuscript we will add a new subsection that states the precise functional forms (production function, utility, dividend process) and parameter values used in the counterexample. We will compute and report the bubbleless steady-state interest rate r, set g_d and n such that g_d > r and g_d < n, and provide the explicit verification (via closed-form solution or numerical check of the dynamic system) that no bubbleless equilibrium path exists under these values. This will make the satisfaction of Tirole's assumptions fully transparent. revision: yes
Referee: [Restored result] Restoration of the result: the additional assumptions on initial capital and dividend size are presented as sufficient to recover the original conclusion, but the paper should state the revised proposition formally, including the precise bounds or inequalities that define 'sufficiently large' capital and 'sufficiently small' dividends.

Authors: We thank the referee for this observation. The current draft motivates the conditions but does not state them as a formal proposition. In the revision we will introduce a revised Proposition 1(c') that explicitly incorporates the additional assumptions: initial capital K_0 > K* (where K* is defined in terms of the model primitives) and dividends d_t < ε for all t (with ε sufficiently small relative to the other parameters). We will state the precise inequalities and outline the proof that these bounds ensure a unique bubbly equilibrium and rule out bubbleless equilibria. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit counterexample is independently constructed

full rationale

The paper constructs a concrete numerical or functional example economy with chosen production function, dividend growth process, population growth, and initial conditions that are asserted to obey every hypothesis of Tirole's Proposition 1(c) (including g_d strictly between the bubbleless steady-state interest rate r and population growth n) while the dynamic system admits only a bubbleless equilibrium. This verification is performed directly from the model's equations rather than by fitting parameters to the target conclusion or by self-referential definition. The subsequent restoration of the original claim under added restrictions on initial capital and dividend size is likewise derived from the same dynamical system without reducing to the counterexample itself. No load-bearing step collapses to a tautology, fitted input renamed as prediction, or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a theoretical comment relying on standard OLG model assumptions from Tirole (1985) plus a new counterexample construction; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard assumptions of the overlapping generations model with capital accumulation and asset dividends as in Tirole (1985)
The counterexample is required to satisfy all assumptions of the original Proposition 1(c)

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

As- sessing dynamic efficiency: Theory and evidence

Abel, A. B., N. G. Mankiw, L. H. Summers, and R. J. Zeckhauser (1989). “As- sessing dynamic efficiency: Theory and evidence”. Review of Economic Studies 56.1, 1–19. doi: 10.2307/2297746. Allen, F., G. Barlevy, and D. Gale (2017). On Interest Rate Policy and Asset Bub- bles. Working Paper 2017-16. Federal Reserve Bank of Chicago. url: https: //www.econstor...

work page doi:10.2307/2297746 1989
[2]

Robust asset prices with bubbles

Elsevier. Chap. 18, 1221–1288. doi: 10.1016/B978-0-44-459406-8.00018-4 . Burke, J. L. (1996). “Robust asset prices with bubbles”. Economics Letters 50.3, 349–354. doi: 10.1016/0165-1765(95)00765-2. Chattopadhyay, S. (2008). “The Cass criterion, the net dividend criterion, and optimality”. Journal of Economic Theory 139.1, 335–352. doi: 10.1016/j. jet.2007...

work page doi:10.1016/b978-0-44-459406-8.00018-4 1996

[1] [1]

As- sessing dynamic efficiency: Theory and evidence

Abel, A. B., N. G. Mankiw, L. H. Summers, and R. J. Zeckhauser (1989). “As- sessing dynamic efficiency: Theory and evidence”. Review of Economic Studies 56.1, 1–19. doi: 10.2307/2297746. Allen, F., G. Barlevy, and D. Gale (2017). On Interest Rate Policy and Asset Bub- bles. Working Paper 2017-16. Federal Reserve Bank of Chicago. url: https: //www.econstor...

work page doi:10.2307/2297746 1989

[2] [2]

Robust asset prices with bubbles

Elsevier. Chap. 18, 1221–1288. doi: 10.1016/B978-0-44-459406-8.00018-4 . Burke, J. L. (1996). “Robust asset prices with bubbles”. Economics Letters 50.3, 349–354. doi: 10.1016/0165-1765(95)00765-2. Chattopadhyay, S. (2008). “The Cass criterion, the net dividend criterion, and optimality”. Journal of Economic Theory 139.1, 335–352. doi: 10.1016/j. jet.2007...

work page doi:10.1016/b978-0-44-459406-8.00018-4 1996