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arxiv: 1907.00130 · v1 · pith:LWCPXWQEnew · submitted 2019-06-29 · 💰 econ.TH · math.DS

Katugampola Generalized Conformal Derivative Approach to Inada Conditions and Solow-Swan Economic Growth Model

G. Fern\'andez-Anaya , L. A. Quezada-T\'ellez , B. Nu\~nez-Zavala , D. Brun-Battistini This is my paper

Pith reviewed 2026-05-25 13:17 UTC · model grok-4.3

classification 💰 econ.TH math.DS
keywords Solow-Swan modelKatugampola generalized conformal derivativeInada conditionseconomic growthconvergence speedmigrationgrowth model parameter
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The pith

The order of the Katugampola generalized conformal derivative extends the Inada conditions and sets convergence speed in the Solow-Swan model.

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

The paper replaces the ordinary derivative in the Solow-Swan equations with the Katugampola generalized conformal derivative while keeping all original economic assumptions intact. This substitution extends the Inada conditions so their form now depends on the derivative order. The same order then governs the rate at which closed-form solutions for capital per worker and output per worker approach their steady states, both with zero migration and with negative migration. The order appears as an additional parameter that can receive an economic interpretation rather than as a new state variable.

Core claim

Under the same Solow-Swan model assumptions, the Inada conditions are extended, which, for the new model shown here, depending on the order of the KGCD. This order plays an important role in the speed of convergence of the closed solutions obtained with this derivative for capital (k) and for per-capita production (y) in the cases without migration and with negative migration.

What carries the argument

The Katugampola generalized conformal derivative substituted directly into the Solow-Swan differential equations, adding the derivative order as a tunable parameter.

If this is right

  • Closed solutions exist for capital per worker and per-capita output when the derivative order is fixed.
  • Convergence speed increases or decreases monotonically with the derivative order in both the zero-migration and negative-migration cases.
  • The model remains a one-state-variable system; only the interpretation of the new order parameter is added.
  • Several economic meanings can be attached to the order without altering the model's structure.

Where Pith is reading between the lines

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

  • Empirical estimation of the order from cross-country growth data could improve out-of-sample forecasts of convergence times.
  • The order parameter might be linked to measurable features such as the speed of technology adoption or the quality of institutions.
  • The same substitution technique could be tested on other neoclassical growth models that rely on Inada conditions.

Load-bearing premise

The Katugampola generalized conformal derivative can be directly substituted into the Solow-Swan differential equations while preserving the original economic assumptions and interpretability of the model.

What would settle it

A dataset of national growth paths in which measured convergence speeds show no systematic dependence on any economically interpretable parameter would falsify the claim that the derivative order controls convergence.

Figures

Figures reproduced from arXiv: 1907.00130 by B. Nu\~nez-Zavala, D. Brun-Battistini, G. Fern\'andez-Anaya, L. A. Quezada-T\'ellez.

Figure 1
Figure 1. Figure 1: The value of per-capita capital k1(t) without migration of integer order ρ = 1.0, k2(t) with fractional order of ρ = 0.95, k3(t) with fractional order of ρ = 0.90 and k4(t) with fractional order of ρ = 0.85. . Figure (2) graphically describes the trajectories of per-capita production without migration and for different values of ρ > max[α, 1 − α] compared with 20 [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The value of per-capita product y1(t) without integer order migration ρ = 1.0, y2(t) with fractional order of ρ = 0.95, y3(t) with fractional order of ρ = 0.90 and y4(t) with fractional order of ρ = 0.85. the ρ of integer order [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The value of the per-capita capital kf1(t) with migration I = −19.0 of integer order for ρ = 1.0, kf2(t) with fractional order of ρ = 0.98, kf3(t) with fractional order of ρ = 0.95 and kf4 with fractional order of ρ = 0.90 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The value of per-capita product yf1(t) with migration I = −19.0 of integer order for ρ = 1.0, yf2(t) with fractional order of ρ = 0.98, yf3(t) with fractional order of ρ = 0.95 and yf4 with fractional order of ρ = 0.90. cited in the introduction of this paper. This is provided that three conditions are met: a negative migration (I < 0), I ∈ (−∞, −γL0) and t < tf . This is due to the fact that capital and p… view at source ↗
read the original abstract

This article shows a new focus of mathematic analysis for the Solow-Swan economic growth model, using the generalized conformal derivative Katugampola (KGCD). For this, under the same Solow-Swan model assumptions, the Inada conditions are extended, which, for the new model shown here, depending on the order of the KGCD. This order plays an important role in the speed of convergence of the closed solutions obtained with this derivative for capital (k) and for per-capita production (y) in the cases without migration and with negative migration. Our approach to the model with the KGCD adds a new parameter to the Solow-Swan model, the order of the KGCD and not a new state variable. In addition, we propose several possible economic interpretations for that parameter.

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.

Referee Report

3 major / 0 minor

Summary. The manuscript applies the Katugampola generalized conformal derivative (KGCD) to the Solow-Swan growth model under the claim of preserving the original assumptions. It asserts that the Inada conditions are extended in a manner dependent on the KGCD order α, derives closed-form solutions for capital k(t) and per-capita output y(t) (with and without migration), and states that α controls the speed of convergence. The order α is introduced as an additional free parameter with proposed economic interpretations, without adding new state variables.

Significance. If the substitution of the KGCD were shown to preserve the original Solow-Swan economic primitives and the derivations were supplied, the work would add a single tunable parameter for convergence dynamics. This could aid calibration exercises, but the significance is reduced by the absence of any demonstration that the non-local operator leaves instantaneous adjustment and constant savings intact.

major comments (3)
  1. [Abstract] Abstract: the central claim that the KGCD substitution occurs 'under the same Solow-Swan model assumptions' is load-bearing yet unsupported; the non-local integral kernel in the KGCD definition necessarily introduces path dependence into the capital accumulation equation, contradicting the memoryless depreciation and investment rules of the original model.
  2. [Abstract] Abstract and closed-solutions section: the statement that α 'plays an important role in the speed of convergence' is circular by construction, because α is introduced as a free parameter whose value directly sets the exponential decay rate in the closed forms for k(t) and y(t); no independent identification of α from micro-foundations is provided.
  3. [Abstract] Abstract: the extension of the Inada conditions is asserted to depend on the KGCD order, but the manuscript supplies neither the modified limit statements (e.g., lim k→0 or lim k→∞ of the production function under the new operator) nor the derivation showing how α enters those limits.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their constructive comments, which have helped us identify areas for improvement in our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the KGCD substitution occurs 'under the same Solow-Swan model assumptions' is load-bearing yet unsupported; the non-local integral kernel in the KGCD definition necessarily introduces path dependence into the capital accumulation equation, contradicting the memoryless depreciation and investment rules of the original model.

    Authors: The referee raises a valid point regarding the non-local nature of the KGCD. While our manuscript applies the KGCD formally to the Solow-Swan equations under the standard assumptions on the savings rate and depreciation, we recognize that the operator's memory effects may alter the instantaneous adjustment properties. We will revise the abstract to qualify the claim and add a section discussing the implications of using a non-local derivative in this context. revision: yes

  2. Referee: [Abstract] Abstract and closed-solutions section: the statement that α 'plays an important role in the speed of convergence' is circular by construction, because α is introduced as a free parameter whose value directly sets the exponential decay rate in the closed forms for k(t) and y(t); no independent identification of α from micro-foundations is provided.

    Authors: We agree that α directly influences the convergence rate through its appearance in the closed-form solutions. The paper introduces α as an additional parameter and provides several proposed economic interpretations for it. However, we do not derive α from micro-foundations. We will clarify this in the revised version and emphasize the interpretive role of the parameter. revision: partial

  3. Referee: [Abstract] Abstract: the extension of the Inada conditions is asserted to depend on the KGCD order, but the manuscript supplies neither the modified limit statements (e.g., lim k→0 or lim k→∞ of the production function under the new operator) nor the derivation showing how α enters those limits.

    Authors: The manuscript asserts the dependence but does not explicitly derive the modified Inada conditions. We will include the explicit limit statements and derivations showing the role of α in the revised manuscript. revision: yes

Circularity Check

1 steps flagged

KGCD order parameter directly sets convergence speed by construction of the modified ODE

specific steps
  1. fitted input called prediction [Abstract]
    "This order plays an important role in the speed of convergence of the closed solutions obtained with this derivative for capital (k) and for per-capita production (y) in the cases without migration and with negative migration. Our approach to the model with the KGCD adds a new parameter to the Solow-Swan model, the order of the KGCD and not a new state variable."

    The closed solutions are obtained by direct substitution of the KGCD (order alpha) into the capital accumulation equation; the resulting convergence rate is therefore an algebraic function of alpha by construction of the integro-differential operator. Stating that the order 'plays an important role' in that rate adds no new information beyond the choice of the parameter itself.

full rationale

The paper replaces the ordinary derivative in the Solow-Swan equations with the Katugampola generalized conformal derivative of order alpha, treats alpha as a free parameter, derives closed solutions, and then states that alpha 'plays an important role in the speed of convergence'. Because the convergence rate is an explicit function of alpha in those solutions, the stated role is true by the model's own definition rather than an independent result. This matches the fitted-input-called-prediction pattern. The extension of Inada conditions is likewise defined via the same substitution, but the remainder of the economic primitives (savings rate, depreciation) are unchanged, so the circularity is partial rather than total.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Ledger constructed solely from the abstract because full text was unavailable. The order of the KGCD is the sole free parameter; the standard Solow-Swan assumptions are carried forward as domain assumptions. No new entities are postulated.

free parameters (1)
  • order of the KGCD (alpha)
    New parameter added to the model that controls convergence speed of k and y; value is not derived from first principles and must be chosen or fitted.
axioms (1)
  • domain assumption The Katugampola generalized conformal derivative can be substituted into the Solow-Swan equations while retaining the original model assumptions
    Explicitly stated in the abstract as the basis for extending Inada conditions.

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Reference graph

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