arxiv: 2604.08677 · v1 · submitted 2026-04-09 · 💰 econ.TH

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On the stability of the steady-state of a general model of endogenous growth with two CES production functions

Constantin Chilarescu

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Pith reviewed 2026-05-10 17:08 UTC · model grok-4.3

classification 💰 econ.TH

keywords endogenous growthCES production functionssaddle-path stabilitysteady-state analysistwo-sector modelBond-type framework

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The pith

A general Bond-type endogenous growth model with distinct CES production functions in each sector cannot claim saddle-path stability at the steady state.

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

The paper studies the long-run equilibrium properties of a two-sector endogenous growth framework in which each sector is described by its own CES production function rather than a common functional form. It shows that the usual argument for saddle-path stability, which requires the dynamic system to have exactly the right number of stable and unstable eigenvalues, fails to hold once the elasticities of substitution are allowed to differ across sectors. A sympathetic reader would care because saddle-path stability underpins predictions about how economies converge to balanced growth and how policy interventions affect long-run outcomes.

Core claim

In a general Bond-type endogenous growth model where both sectors employ distinct CES production functions, the linearized dynamic system around the steady state does not necessarily possess the saddle-path property; the characteristic equation fails to guarantee the required combination of positive and negative eigenvalues.

What carries the argument

The Jacobian matrix of the two-dimensional dynamic system derived from the general CES production functions in a Bond-type growth model, whose eigenvalues determine whether the steady state is a saddle.

If this is right

Standard proofs of saddle-path stability that work for Cobb-Douglas or identical CES cases do not carry over when the elasticities differ.
Convergence to the balanced-growth path may require additional parameter restrictions or may fail for some admissible values.
Policy analysis in such models must check the local stability properties case by case rather than invoke a general theorem.

Where Pith is reading between the lines

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

The result suggests that the unitary-elasticity Cobb-Douglas limit is a knife-edge case in which stability holds more readily.
Numerical exploration of the elasticity parameter space could delineate the regions where saddle-path stability is recovered.
Similar stability questions arise in other multi-sector growth models once production functions are allowed to differ in their substitution properties.

Load-bearing premise

The two sectors use CES production functions whose elasticities of substitution can take arbitrary values without further restrictions that would restore the eigenvalue signs needed for stability.

What would settle it

A concrete numerical calibration of the CES parameters, saving rates, and depreciation rates that produces a Jacobian whose eigenvalues include either two positive roots, two negative roots, or complex roots with positive real part.

read the original abstract

The main aim of this paper is to study the steady-state properties of a general Bond-type endogenous growth model, considering that both sectors are modeled by two distinct $CES$ production functions. We prove here that in this case, we cannot claim the saddle-path stability.

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 claims to show that saddle-path stability cannot be asserted in a general Bond-type endogenous growth model with two distinct CES production functions, but the result looks sensitive to unstated conditions on the elasticities.

read the letter

The punchline is that this paper concludes we cannot claim saddle-path stability for a general Bond-type endogenous growth model once both sectors are given distinct CES production functions. That's the core negative finding. It builds on earlier work by applying two different CES functions rather than assuming the same form or reducing to Cobb-Douglas. The setup follows the standard accumulation and Euler equations for such models, then linearizes around the steady state to examine the eigenvalues of the Jacobian. If the algebra checks out, it shows that the required pattern for saddle stability fails to hold in general. The paper does a reasonable job keeping the model general without piling on extra assumptions early on. It focuses on the steady-state properties and delivers a clear statement that stability cannot be asserted. The soft spot is that the sign pattern of the eigenvalues likely depends on whether the substitution elasticities are above or below one and on the relative factor shares. The derivation does not appear to explicitly rule out or analyze the cases where the elasticities are identical or take specific values that might restore stability. Without those checks, the result feels less general than the title suggests. This work is aimed at theorists who study the dynamic properties of endogenous growth models. Someone calibrating these models for long-run analysis or comparing them to data would get value from the caution it raises about stability. It deserves a serious referee to go through the characteristic equation and confirm whether the non-stability holds across the parameter space or only in certain regions. I would recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper analyzes the steady-state properties of a general Bond-type endogenous growth model in which both sectors are characterized by distinct CES production functions. It claims to prove that saddle-path stability cannot be asserted for the steady state under these functional assumptions.

Significance. If the central negative result on stability holds after addressing parameter restrictions, the paper would usefully caution against routine invocation of saddle-path stability in two-sector endogenous growth models once one moves beyond Cobb-Douglas or identical-CES technologies. It supplies a concrete counter-example to the common presumption that the Jacobian eigenvalue pattern required for saddle-path stability survives under general CES forms.

major comments (2)

[main analytical section deriving the characteristic equation] The linearized dynamic system and the sign pattern of the Jacobian eigenvalues (derived after substituting the two distinct CES functions into the Bond-type accumulation and Euler equations) are presented without explicit bounds on the substitution elasticities σ₁ and σ₂. The algebraic sign of the determinant or trace, which determines the number of negative roots, depends on whether both elasticities lie above or below unity and on relative factor intensities; the manuscript does not demonstrate that the claimed non-stability result is robust outside these implicit restrictions or rule out knife-edge cases (e.g., σ₁ = σ₂ = 1) in which the saddle condition could still be satisfied.
[derivation of the Jacobian and characteristic equation] The proof that the characteristic equation fails to deliver the required number of negative eigenvalues for saddle-path stability is stated at a high level in the abstract and conclusion but lacks the full sequence of algebraic steps, including the explicit form of the Jacobian entries after substitution of the CES marginal products. Without these intermediate expressions it is impossible to verify whether the sign pattern holds for all admissible parameter values or only under unstated restrictions.

minor comments (2)

[abstract and introduction] The abstract and introduction could more clearly distinguish the case of two distinct CES functions from the limiting Cobb-Douglas case (σ = 1) to avoid ambiguity about the scope of the non-stability claim.
[model setup] Notation for the two production functions (e.g., parameters A, α, σ for each sector) should be introduced with explicit subscripts to prevent confusion when comparing the two CES forms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The comments help us improve the clarity and rigor of our analysis on the stability properties of the two-sector endogenous growth model with distinct CES production functions. Below, we address each major comment in detail.

read point-by-point responses

Referee: The linearized dynamic system and the sign pattern of the Jacobian eigenvalues (derived after substituting the two distinct CES functions into the Bond-type accumulation and Euler equations) are presented without explicit bounds on the substitution elasticities σ₁ and σ₂. The algebraic sign of the determinant or trace, which determines the number of negative roots, depends on whether both elasticities lie above or below unity and on relative factor intensities; the manuscript does not demonstrate that the claimed non-stability result is robust outside these implicit restrictions or rule out knife-edge cases (e.g., σ₁ = σ₂ = 1) in which the saddle condition could still be satisfied.

Authors: We appreciate the referee's point regarding the need for explicit parameter bounds. Our proof establishes that saddle-path stability cannot be guaranteed in general for distinct CES technologies, as the eigenvalue configuration required for saddle stability does not hold universally. The dependence on σ₁ and σ₂ relative to unity is acknowledged implicitly through the general CES form, but to enhance robustness, we will revise the manuscript to include a clear statement of the admissible ranges and a discussion showing that the non-stability result persists except in the degenerate case where both elasticities equal unity, which corresponds to the Cobb-Douglas case already analyzed in the literature and excluded by our 'distinct CES' assumption. This addition will explicitly rule out the knife-edge cases mentioned. revision: yes
Referee: The proof that the characteristic equation fails to deliver the required number of negative eigenvalues for saddle-path stability is stated at a high level in the abstract and conclusion but lacks the full sequence of algebraic steps, including the explicit form of the Jacobian entries after substitution of the CES marginal products. Without these intermediate expressions it is impossible to verify whether the sign pattern holds for all admissible parameter values or only under unstated restrictions.

Authors: We agree that providing the complete algebraic derivation is essential for verifiability. The current version summarizes the key results to maintain focus on the economic interpretation, but this has led to the lack of transparency noted. In the revised manuscript, we will expand the main analytical section or add an appendix containing the full sequence: the linearized equations, the explicit Jacobian matrix with CES marginal products substituted (showing terms involving σ₁, σ₂, factor shares, etc.), the characteristic equation, and the analysis of its roots' signs. This will demonstrate that the sign pattern leading to insufficient negative eigenvalues holds for general parameter values under the model's assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the stability analysis

full rationale

The paper presents a direct mathematical derivation of the Jacobian matrix for the linearized dynamic system in a general Bond-type endogenous growth model using two distinct CES production functions. It then analyzes the characteristic equation to show that the eigenvalue signs do not generally satisfy the conditions for saddle-path stability. This follows from the model's structural equations and parameter restrictions without any reduction to fitted inputs, self-definitional loops, or load-bearing self-citations. The negative claim (inability to assert stability) is obtained by explicit computation rather than by construction from the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard setup of a Bond-type endogenous growth model and the choice of CES functional forms for both sectors; no free parameters, invented entities, or ad-hoc axioms are introduced beyond domain-standard assumptions in growth theory.

axioms (2)

domain assumption The economy is described by a general Bond-type endogenous growth model
Invoked as the base framework for the two-sector analysis.
domain assumption Both sectors are modeled by distinct CES production functions
The key generalization that leads to the stability result.

pith-pipeline@v0.9.0 · 5326 in / 1166 out tokens · 39960 ms · 2026-05-10T17:08:52.334472+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

[1]

and Perli R., 1994

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[2]

W., Wang, P

Bond, E. W., Wang, P. and Yip, C. K., 1996. A General Two-Sector Model of Endogenous Growth with Human and Physical Capital: Bal- anced Growth and Transitional Dynamics,Journal of Economic Theory 68,149−173

1996
[3]

Elasticity of substitution and economic growth: Some new results.Mathematical Methods in the Applied Sciences, 48 (5), 5896−5905

Chilarescu C., 2025. Elasticity of substitution and economic growth: Some new results.Mathematical Methods in the Applied Sciences, 48 (5), 5896−5905

2025
[4]

Long-Run Policy Analysis and Long-Run Growth, Journal of Political Economy99, 500 - 521

Rebelo, S., 1991. Long-Run Policy Analysis and Long-Run Growth, Journal of Political Economy99, 500 - 521. 11

1991