Existence and Calculation of Optimal Monetary Equilibria on Overlapping Generations Economies

Leandro Lyra Braga Dognini

arxiv: 2509.19019 · v2 · pith:CETZPHPQnew · submitted 2025-09-23 · 💰 econ.TH

Existence and Calculation of Optimal Monetary Equilibria on Overlapping Generations Economies

Leandro Lyra Braga Dognini This is my paper

Pith reviewed 2026-05-18 14:19 UTC · model grok-4.3

classification 💰 econ.TH

keywords overlapping generationsmonetary equilibriaoptimal equilibriaexistencealgorithmnon-stationary economiesprone to savings

0 comments

The pith

Sufficient conditions guarantee optimal monetary equilibria exist in non-stationary overlapping generations economies when they are prone to savings.

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

Overlapping generations economies can produce inefficient outcomes because the First Welfare Theorem does not apply. The paper supplies sufficient conditions under which optimal monetary equilibria exist in consumption-loan models that lack durable assets, cash-in-advance rules, or wealth transfers. The central requirement is that the economy must be prone to savings. It also supplies an explicit algorithm that locates these equilibria by taking the limit of nested compact sets built through backward solutions of the equilibrium conditions from the optimal equilibria of well-behaved tail economies.

Core claim

If a non-stationary overlapping generations economy satisfies the prone-to-savings condition, then optimal monetary equilibria exist and can be recovered as the limit of a nested sequence of compact sets obtained by solving the equilibrium equations backward, starting from the optimal monetary equilibria of suitable tail economies.

What carries the argument

The prone-to-savings condition, which keeps the nested compact sets nonempty and convergent, together with the backward-calculation procedure that constructs those sets from tail-economy equilibria.

If this is right

Optimal monetary equilibria exist once the prone-to-savings condition holds.
These equilibria are recoverable by an explicit algorithm of nested compact sets.
The result applies directly to non-stationary economies without assets or cash constraints.
It supplies a constructive answer to existence questions of the Hahn type in this class of models.

Where Pith is reading between the lines

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

Numerical implementations of the backward algorithm could permit quantitative policy experiments in OLG settings.
Analogous backward constructions might locate efficient equilibria in other infinite-horizon models with similar efficiency failures.
The method opens the possibility of checking optimality computationally rather than only theoretically.

Load-bearing premise

The economy must satisfy the prone-to-savings condition that keeps the nested compact sets nonempty throughout the backward construction.

What would settle it

A concrete non-stationary overlapping generations economy that meets the prone-to-savings condition yet possesses no optimal monetary equilibrium or whose backward-constructed sets fail to converge.

Figures

Figures reproduced from arXiv: 2509.19019 by Leandro Lyra Braga Dognini.

read the original abstract

A well-known feature of overlapping generations economies is that the First Welfare Theorem fails and equilibrium may be inefficient. The Cass (1972) criterion furnishes a necessary and sufficient condition for efficiency, but it does not address the existence of efficient equilibria, and Cass, Okuno, and Zilcha (1979) provide nonexistence examples. A closely related question (known as the Hahn (1965) problem) deals with the existence of monetary equilibria. In this paper, I provide sufficient conditions for the existence of optimal monetary equilibria on consumption-loan, non-stationary overlapping generations economies without durable, dividend-paying assets, cash-in-advance constraints, wealth-transfer mechanisms, or transaction costs. Essentially, the economy must be prone to savings. Furthermore, I develop an algorithm to find these optimal monetary equilibria as the limit of nested compact sets. These compact sets are the result of a backward calculation through equilibrium equations departing from the set of optimal monetary equilibria of well-behaved tail economies.

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

The paper offers sufficient conditions via a 'prone to savings' assumption plus a backward nested-compact-sets algorithm for optimal monetary equilibria in non-stationary OLG economies, but the construction's uniformity across varying tails looks shaky from the abstract.

read the letter

The main takeaway is that this paper supplies sufficient conditions for the existence of optimal monetary equilibria in non-stationary consumption-loan OLG economies and gives a concrete backward algorithm to compute them as limits of nested compact sets from finite-horizon tails. It frames the result around the Hahn problem and Cass efficiency criterion without adding durable assets, cash-in-advance rules, or transaction costs, which keeps the setup clean. The key new piece is tying existence to the economy being 'prone to savings' and showing how to build the equilibria from well-behaved tails outward. That extends the stationary-case analyses and nonexistence examples in the cited literature in a direct way. The algorithm itself is a practical addition if the compactness arguments hold, since it turns an abstract existence question into something that could be calculated step by step. The paper does a solid job stating the problem clearly and avoiding extra mechanisms that often complicate these models. On the soft spots, the abstract gives no proof details or explicit verification of why the nested sets remain nonempty and converge under the prone-to-savings condition. The stress-test concern lands: in a genuinely non-stationary environment the tail economies have time-varying endowments and preferences, so a single global condition may not automatically guarantee nonempty sets for every finite tail without extra uniformity or continuity requirements on the sequence of utilities. If those tails can deviate arbitrarily, the backward construction could fail at some horizon even if the overall economy satisfies the premise. This is not a minor gap if the algorithm is meant to be the main contribution. The work is aimed at specialists in monetary theory and dynamic macro who already know the OLG literature and want to see existence results pushed into non-stationary settings. A reader focused on the Hahn problem or computational approaches to equilibria would get value from the framing and the proposed method. It deserves a serious referee to check whether the compactness and nesting actually go through once the full proofs are examined. I would recommend sending it to peer review rather than desk rejection, with the expectation that the uniformity issue will need direct attention in revisions.

Referee Report

2 major / 2 minor

Summary. The paper claims to provide sufficient conditions for the existence of optimal monetary equilibria in non-stationary overlapping generations economies without durable assets, cash-in-advance constraints or transaction costs, with the key sufficient condition being that the economy is 'prone to savings'. It further develops an algorithm to compute these equilibria as the limit of nested compact sets constructed via backward induction starting from optimal monetary equilibria in well-behaved finite-horizon tail economies.

Significance. If the existence result and compactness arguments hold, the paper would make a useful contribution to the literature on the Hahn problem and efficiency in OLG models by extending results to genuinely non-stationary settings and offering a constructive computational procedure. The algorithmic approach via nested compact sets is a positive feature that could support numerical applications.

major comments (2)

[Prone to savings condition and backward-induction construction (as described in abstract)] The 'prone to savings' condition is invoked to guarantee that the sequence of compact sets for successively longer tail economies remains nonempty and nested. In a non-stationary environment the tail economies have time-varying endowments and preferences, so it is necessary to show that the global condition propagates uniformly to every finite tail; the abstract and construction outline do not supply this verification or additional continuity assumptions on the sequence of utility functions.
[Algorithm and compactness arguments (abstract and likely §4)] The existence argument rests on the finite-intersection property for the nested compact sets and their convergence to an optimal monetary equilibrium. The manuscript must explicitly state the topological assumptions (e.g., continuity of preferences, compactness of the feasible set in an appropriate topology) that ensure the sets remain compact in the non-stationary case; without these details the central claim cannot be assessed.

minor comments (2)

[Abstract] The abstract should include a brief, self-contained statement or reference to the precise definition of the 'prone to savings' condition.
[Notation and definitions] Notation for generations, time indices and equilibrium variables should be introduced consistently and used uniformly across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our paper. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [Prone to savings condition and backward-induction construction (as described in abstract)] The 'prone to savings' condition is invoked to guarantee that the sequence of compact sets for successively longer tail economies remains nonempty and nested. In a non-stationary environment the tail economies have time-varying endowments and preferences, so it is necessary to show that the global condition propagates uniformly to every finite tail; the abstract and construction outline do not supply this verification or additional continuity assumptions on the sequence of utility functions.

Authors: We agree that explicit verification of propagation is required in the non-stationary setting. The prone to savings condition is defined globally for the infinite economy in a manner that relies on limiting savings behavior; this definition ensures the property carries over to each finite tail under the maintained assumptions on endowments. To address the concern directly, we will add a new proposition in Section 3 establishing uniform propagation to all tails and will state the required continuity assumptions on the sequence of utility functions explicitly. revision: yes
Referee: [Algorithm and compactness arguments (abstract and likely §4)] The existence argument rests on the finite-intersection property for the nested compact sets and their convergence to an optimal monetary equilibrium. The manuscript must explicitly state the topological assumptions (e.g., continuity of preferences, compactness of the feasible set in an appropriate topology) that ensure the sets remain compact in the non-stationary case; without these details the central claim cannot be assessed.

Authors: We acknowledge that the topological assumptions should be stated more explicitly for clarity. The arguments rely on the product topology, under which the feasible consumption sets are compact by Tychonoff's theorem given the boundedness of endowments, together with continuity of preferences. We will revise the opening of Section 4 to include a dedicated statement of these assumptions and confirm that the nested sets are closed, hence compact, so that the finite-intersection property applies and yields a limit that is an optimal monetary equilibrium. revision: yes

Circularity Check

0 steps flagged

No circularity: standard existence proof via compactness and external criteria

full rationale

The paper advances a mathematical existence result for optimal monetary equilibria in non-stationary OLG economies by assuming the economy is 'prone to savings' and constructing the equilibrium as the limit of nested compact sets obtained by backward induction from well-behaved finite-horizon tails. This is a conventional topological argument resting on the external Cass (1972) efficiency criterion and standard OLG assumptions; the prone-to-savings condition is introduced explicitly as a sufficient premise that ensures the relevant sets remain nonempty, rather than being defined in terms of the target equilibrium or fitted to data. No equation or step reduces by construction to its own inputs, no self-citation chain bears the central load, and the algorithm is derived from the proof rather than presupposing the result. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions for OLG models and mathematical compactness; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard assumptions on preferences, endowments, and market clearing in consumption-loan OLG economies
Invoked implicitly as background for the equilibrium concept and the 'prone to savings' condition.
standard math Compactness of the relevant equilibrium sets in the tail economies
Used to guarantee the nested sets are nonempty and the limit exists.

pith-pipeline@v0.9.0 · 5697 in / 1268 out tokens · 42486 ms · 2026-05-18T14:19:32.679773+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 5. The economy E is prone to savings if there are ε>0, δ>0 such that ... sh(pt,pt+1)/Ht ≤δ ⇒ ||pt||/||pt+1|| ≤ αt/(1+ε)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 13 (Backward calculation algorithm). lim ∪ HP_O_k ⊆ HP_O with nonempty limit

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Nonmonetary steady states in stationary overlapping gener- ations models with long lived agents and discounting: Multiplicity, optimality, and consumption smoothing

Aiyagari, S. Rao (1988). “Nonmonetary steady states in stationary overlapping gener- ations models with long lived agents and discounting: Multiplicity, optimality, and consumption smoothing”. In:Journal of Economic Theory45.1, pp. 102–127. — (1992). “Walras’ Law and nonoptimal equilibria in overlapping generations models”. In:Journal of Mathematical Econ...

work page 1988
[2]

On the existence of optimal stationary equilibria with a fixed supply of fiat money: I. The case of a single consumer

Benveniste, Lawrence M. and David Cass (1986). “On the existence of optimal stationary equilibria with a fixed supply of fiat money: I. The case of a single consumer”. In: Journal of Political Economy94.2, pp. 402–417. Burke,JonathanL.(1987).“Inactivetransferpoliciesandefficiencyingeneraloverlapping- generations economies”. In:Journal of Mathematical Econ...

work page 1986
[3]

Social Se- curity in an Overlapping Generations Economy with Land

Elsevier. Chap. 35, pp. 1899–1960. Guillemin, Victor and Alan Pollack (1974).Differential Topology. New Jersey: Prentice- Hall. İmrohoroğlu, Ayşe, Selahattin İmrohoroğlu, and Douglas H. Joines (1999). “Social Se- curity in an Overlapping Generations Economy with Land”. In:Review of Economic Dynamics2.3, pp. 638–665. Kehoe, Timothy J. and David K. Levine (...

work page 1960

[1] [1]

Nonmonetary steady states in stationary overlapping gener- ations models with long lived agents and discounting: Multiplicity, optimality, and consumption smoothing

Aiyagari, S. Rao (1988). “Nonmonetary steady states in stationary overlapping gener- ations models with long lived agents and discounting: Multiplicity, optimality, and consumption smoothing”. In:Journal of Economic Theory45.1, pp. 102–127. — (1992). “Walras’ Law and nonoptimal equilibria in overlapping generations models”. In:Journal of Mathematical Econ...

work page 1988

[2] [2]

On the existence of optimal stationary equilibria with a fixed supply of fiat money: I. The case of a single consumer

Benveniste, Lawrence M. and David Cass (1986). “On the existence of optimal stationary equilibria with a fixed supply of fiat money: I. The case of a single consumer”. In: Journal of Political Economy94.2, pp. 402–417. Burke,JonathanL.(1987).“Inactivetransferpoliciesandefficiencyingeneraloverlapping- generations economies”. In:Journal of Mathematical Econ...

work page 1986

[3] [3]

Social Se- curity in an Overlapping Generations Economy with Land

Elsevier. Chap. 35, pp. 1899–1960. Guillemin, Victor and Alan Pollack (1974).Differential Topology. New Jersey: Prentice- Hall. İmrohoroğlu, Ayşe, Selahattin İmrohoroğlu, and Douglas H. Joines (1999). “Social Se- curity in an Overlapping Generations Economy with Land”. In:Review of Economic Dynamics2.3, pp. 638–665. Kehoe, Timothy J. and David K. Levine (...

work page 1960