Elasticity of substitution and general model of economic growth

Constantin Chilarescu

REVIEW 5 major objections 5 minor 16 references

Elasticity of substitution and general model of economic growth

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims higher substitution elasticity raises income, growth, and capital share for both sigma below and above one.

desk verdict A useful extension of normalized-CES comparative statics to a two-sector model with distinct elasticities, but the steady-state claims are not proven as stated and the abstract overclaims. read the letter →

arxiv 2506.02936 v1 pith:6AU3IXMW submitted 2025-06-03 econ.TH

Constantin Chilarescu This is my paper

classification econ.TH MSC 91B62

keywords elasticityofsubstitutionCESproductionfunctiontwo-sectorendogenousgrowthbalancedpathnormalizationphysicalcapitalsharehumancomparativestatics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies a two-sector endogenous growth model in which the good sector and the education sector each use their own CES production function with distinct elasticities of substitution. It claims that an economy with a higher elasticity of substitution ends up with higher per-capita income, a higher relative share of physical capital, a higher common growth rate, and a larger fraction of human capital allocated to the goods sector compared with an otherwise identical economy. The novel assertion is that this ranking holds whether the elasticities are below or above one, removing the usual restriction that sigma must exceed one for long-run growth.

What carries the argument

The argument is carried by a two-sector model of the Bond-Wang-Yip type with two distinct CES production functions, together with the normalization methodology of de La Grandville and Klump. The normalized production functions are rewritten in terms of the physical capital shares pi^k, and the concavity of the logarithm is used to sign the derivatives of output and growth with respect to the substitution parameter. The steady-state growth rate formula (33) and its derivative (34) are the quantitative core of the comparative statics.

What would settle it

Choose parameter values and baseline normalizations such that the steady state satisfies w* < w_bar while the model's other assumptions hold, then evaluate the derivative in equation (34); if dr*/dσ1 becomes negative, the claimed monotonicity fails. Alternatively, find a parameter set where the function P(w) in equation (15) does not have the asserted limits of plus and minus infinity, so the unique solution w* may not exist.

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Extended reading notes

Core claim

The central result is Theorem 1 and the steady-state comparative statics given by equations (33) and (34). Theorem 1 states that at any stage of development the economy with the higher elasticity of substitution has higher per-capita income, higher physical capital per capita, and higher human capital. The steady-state analysis adds that the common growth rate r* is an increasing function of the elasticity of substitution, provided the steady-state capital ratio w* lies above the arbitrary baseline w_bar chosen during normalization. The paper argues these results are independent of whether sigma is below or above one, unlike earlier one-sector findings.

Load-bearing premise

The balanced growth path exists with a unique interior allocation u* strictly between zero and one, and the steady-state capital ratio w* lies above the arbitrary baseline w_bar used for normalization.

Editorial extensions

If this is right

If the theorem is correct, the positive effect of substitution elasticity on income and growth holds for economies with substitution elasticities below one, not just above one.
The model predicts that the physical capital share in both sectors rises with the elasticity of substitution when the steady-state capital ratio exceeds the baseline.
The common growth rate of the economy increases with sigma, so policies or technologies that raise substitutability between capital and labor can support faster long-run growth.
The result extends the two-sector analysis beyond the common-elasticity case, allowing different substitution elasticities in the goods and education sectors.
The finding that higher sigma raises the fraction of human capital allocated to goods production could affect how human capital accumulation responds to factor substitutability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The comparative static on the growth rate depends crucially on w* being above the arbitrary baseline w_bar; for w* below w_bar the derivative in equation (34) likely flips sign, which would overturn the claim for that region.
The paper cannot prove u* lies strictly between zero and one, so the interior balanced growth path is an assumption rather than a derived result; if u* hits a boundary the theorem's conclusions may fail.
A direct testable extension would simulate the model across a grid of baseline normalizations to check whether the sign of dr*/dσ1 is robust, or whether it is an artifact of the chosen w_bar.
The result suggests a similar dominance property may hold in models with more than two sectors or with variable elasticity of substitution, though such extensions are not explored here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read Chilarescu's arXiv:2506.02936. The paper aims to extend de La Grandville–Klump normalized-CES comparative statics to a two-sector Bond–Wang–Yip model with two distinct elasticities, and it claims the usual sigma>=1 restriction can be dropped. The output-level result (Theorem 1) is a restatement of the known normalized-CES monotonicity property in a two-sector setting; that part is correct, and the algebra in (29)–(32) is sound. The genuinely new contribution is the steady-state comparative statics for the growth rate and factor shares with two distinct elasticities including values below one.

That's where the soft spots are. Proposition 1's uniqueness proof asserts limits and monotonicity for P(w) that are not generally true over the admissible parameter space, and the paper admits it cannot prove u* in (0,1), so the interior BGP is not established. The stability analysis is numerical; the eigenvalues include zero, and claiming saddle-path stability from 'at least one negative eigenvalue' isn't valid — you need the count of stable eigenvalues to match the number of state variables.

The biggest issue is the steady-state comparative statics. Eq. (34) is obtained by differentiating with respect to psi1 while effectively holding the share pi1 (equivalently w) fixed, but w* moves with psi1 through Eq. (15). The omitted indirect term need not vanish, so the sign of dr*/dsigma is not proven. And the concavity inequality used to sign (34) fails for sigma1<1; with psi1=-0.1, pi=0.7, barpi=0.5 the claimed bound is violated. So the paper's central claim for sigma<1 is unsupported. The share result (29) is also conditional on w>w_bar, an arbitrary baseline that the abstract doesn't mention.

What's good: the normalization is carried out carefully, the output monotonicity proof is correct, and the simulations are consistent with the claims. The self-citations are appropriate given the direct line of work. But the abstract overstates what is proved.

I'd send this to peer review. The question matters, the setup is a natural generalization, and the gaps are technical — parameter restrictions and a correct total-derivative computation could fix them. A referee should require a rigorous existence/interiority proof and a proper derivation of the steady-state comparative statics that accounts for the w* response. My own verdict is conditional, not reject.

Referee Report

5 major / 5 minor

Summary. The paper develops a two-sector endogenous growth model with two distinct CES production functions, one for the goods sector and one for the education sector, and analyzes how the elasticity of substitution affects key economic variables. Using the de La Grandville-Klump normalization, the paper claims to prove that a higher elasticity of substitution raises per-capita income, the relative share of physical capital, the common growth rate, and the share of human capital allocated to the production sector, for both elasticities below and above one. The main theoretical results are Proposition 1 (existence and uniqueness of the balanced growth path), Theorem 1 (monotonicity of income and capital levels), and the comparative-static formulas in Eqs. (29)-(34). The paper also provides numerical simulations for economies with different combinations of elasticities below and above unity.

Significance. If the results were fully established, the paper would be a valuable generalization of recent one-sector results to a two-sector model with two distinct CES functions, and it would extend the analysis to elasticities below one, a case often excluded in the existing literature. The concavity argument in Eq. (30) that yields the monotonicity of output in the elasticity of substitution is elegant, and the explicit formulas for the normalized production functions are a useful contribution. However, as it stands, the paper's central advertised claims are not fully proven: the uniqueness of the balanced growth path rests on an unproved and generally false monotonicity assertion, the interiority of the allocation is admitted to be unproven, and the comparative-static results for the growth rate and capital share are partial derivatives that ignore the response of the steady-state allocation to a change in the elasticity. The paper also notes conditions (such as w > \bar{w}) that are absent from the abstract, so the unconditional claims are not supported.

major comments (5)

[Section 3, Proposition 1, Eq. (15)] The proof asserts without proof that the function P(w) defined in Eq. (15) is strictly decreasing on (0,∞) with limits +∞ at 0 and -∞ at ∞, and therefore that w* is unique. This is not generally true across the admissible parameter space: the derivative of P(w) consists of terms with opposite signs, and the exponents depend on ψ1 and ψ2 in a way that does not guarantee monotonicity. For instance, when ψ2<0, the factor P2^{1/ψ2 - 1} has non-monotone behavior, and the claimed limits need not hold. Since the existence and uniqueness of the steady-state allocation are foundational for all subsequent comparative statics, this gap is load-bearing.
[Section 3, Proposition 1(iii)] The paper explicitly concedes, "Unfortunately we cannot prove that u* ∈ (0,1)" and supports the claim only by numerical simulations. The interiority of u* is necessary for the meaningfulness of the balanced growth path and for the derived formula for v* in Eq. (16). Without a proof of u* ∈ (0,1), the model's steady-state is not established for the full parameter space, and the subsequent Theorem 1 and Section 4 results rest on an unverified assumption.
[Section 4, Eq. (34)] The derivative dr*/dψ1 in Eq. (34) is computed by holding πk1* (and hence w) fixed when differentiating r*(σ1) in Eq. (33). Along the balanced growth path, however, w* is determined by Eq. (15) and is itself a function of ψ1. The total derivative of r* with respect to ψ1 must include the indirect term ∂r*/∂w* · dw*/dψ1. The paper provides no argument that this indirect term vanishes or has a definite sign, and its own numerical examples in Section 5 show that the steady-state allocation changes with ψ1 (e.g., z* = 10.73 in case 1 versus z* = 5.18 in case 2). Therefore the sign of dr*/dσ1 is not proven by the displayed calculation.
[Section 4, Eq. (29)] The sign of dπk1/dψ1 in Eq. (29) is positive only when w > \bar{w}, where \bar{w} is the arbitrarily chosen normalization baseline. No proof is given that the steady-state value w* lies above \bar{w} for the admissible parameter region, and the paper's concluding remark in Section 4 explicitly conditions the result on "the ratio kv/hu being greater than the reference one." This contradicts the unconditional claim in the abstract that higher elasticity increases the relative share of physical capital. The same issue affects the growth-rate result in Eq. (34), which is stated to hold only when the capital share is increasing. A parameter restriction or a proof that w* > \bar{w} is required.
[Section 3.1] The local stability analysis claims saddle-path stability from the observation that "all the numerical simulations confirm that at least one eigenvalue is negative." In this four-dimensional system with two state variables, saddle-path stability requires exactly two eigenvalues with negative real parts and two with positive real parts. The reported eigenvalue sets (e.g., case 1: [0.0014; 0.173; 12.963; -12.788]) show only one clearly negative eigenvalue and one near zero, which is insufficient to support the claim of a unique optimal steady-state equilibrium. This is load-bearing for the model's equilibrium selection.

minor comments (5)

[Section 3, proof of Proposition 1] The phrase "It is just a simply exercise" should be corrected to "It is just a simple exercise."
[Section 5] The numerical simulations do not report the normalization baseline values (\bar{k}, \bar{h}, \bar{u}, \bar{v}, \bar{y}, m) used in the exercise, so the results cannot be replicated or checked. Please provide these values.
[Section 5] The text refers to "The four above graphs" but no graphs appear in the manuscript. Please include the figures or remove the references to them.
[Section 3, Proposition 1(iii)] In the definition of τ0, the parameter θ is used without being restated; for readability, please remind the reader that θ = α1(1-α2)/(α2(1-α1)) as defined in Eq. (8).
[Throughout] The notation x* is used both for the steady-state value of a variable and for its value at the start of the balanced growth path (t = t*). This dual use may confuse readers; please introduce distinct notation.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the comparative-statics claims are derived from the model and the de La Grandville–Klump normalization, not from fitted inputs or self-citations.

full rationale

The paper's central results are derived in-text rather than imported from fitted data or from the author's prior theorems. Proposition 1's balanced-growth-path characterization rests on the paper's own equation (15) and a claimed monotonicity of P(w); Theorem 1's income and factor-share comparisons rest on the normalized CES identities (26)-(32), with the sign of dπ/dσ explicitly made conditional on w versus the baseline w̄ in (29). The steady-state growth comparison in (34) is likewise stated as conditional on the steady-state capital ratio exceeding the reference ratio. No parameter is fitted to a subset of data and then renamed a prediction, and no uniqueness theorem is imported from earlier work by the same authors: the self-citations to Chilarescu (2024) and to Gomez are motivational rather than load-bearing. The manuscript itself flags two limitations that are correctness and robustness issues rather than circular reductions: it 'cannot prove that u∗∈(0,1)' and the Jacobian results 'do not allow an incontestable interpretation' of eigenvalue signs. The abstract's unconditional wording is stronger than the conditional results, and eq. (34) is a partial derivative that omits the induced variation of w∗ through eq. (15); these are substantive technical gaps, but they do not make the derivation equivalent to its own inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The load-bearing inputs are the assumed existence of a unique interior balanced growth path and the normalization baseline. The interiority and uniqueness are asserted rather than proven, with the paper itself conceding the u* point. The comparative statics for the growth rate and factor shares are computed relative to an arbitrary baseline and change sign depending on where the steady state sits relative to that baseline. No new entities are introduced.

free parameters (2)

Normalization baseline values (k_bar, h_bar, u_bar, v_bar, y_bar, m) = Not reported; arbitrary by construction
The signs of d(pik_1)/d(sigma_1) and d(r*)/d(sigma_1) depend on comparing the steady-state ratio w* with the baseline ratio w_bar = k_bar v_bar/(h_bar u_bar) (eqs. 29 and 34). The baseline is arbitrarily chosen and is not stated in the Section 5 simulations.
Benchmark model parameters = A1=1.05, A2=0.20, alpha1=0.6, alpha2=0.8, delta_k=0.06, delta_h=0.05, epsilon=2, rho=0.06
Hand-chosen calibration values used for the numerical simulations in Sections 3.1 and 5. They are conventional in this literature and are not fit to data, but together with the unreported baseline they determine the sign pattern of the reported results.

assumptions (6)

ad hoc to paper The function P(w) in eq. (15) is strictly decreasing on (0, infinity) with limits +infinity at 0 and -infinity at infinity, giving a unique w*.
Invoked in the Proof of Proposition 1 as a "simply exercise". The monotonicity and the stated limits do not hold across the admissible (psi1, psi2) parameter space, e.g., the goods-sector marginal product term tends to a positive constant for psi1 in (0,1).
ad hoc to paper The steady-state allocation satisfies u* in (0,1), and therefore v* in (0,1).
The paper states "Unfortunately we cannot prove that u* in (0,1)" and relies on numerical simulations (Section 3, after eq. (16)).
ad hoc to paper Local saddle-path stability holds with a unique optimal steady-state equilibrium.
Concluded in Section 3.1 from "at least one eigenvalue is negative" in numerical spectra that contain zero eigenvalues (cases 2 to 5) and a small positive eigenvalue (case 1); necessary and sufficient conditions are not given.
domain assumption The de La Grandville-Klump normalization is a valid way to compare economies with different elasticities of substitution.
Section 4 constructs normalized production functions at an arbitrary baseline (k_bar, h_bar, u_bar, v_bar, y_bar, m). The comparative statics are computed within this normalized family, and the sign results inherit the baseline choice.
domain assumption epsilon > 1, the inverse of the elasticity of intertemporal substitution, ensures the transversality limits are negative.
Used at the end of Section 3. The claim that l1 = l2 = -[rho + (epsilon - 1) r*] is "obviously negative for all epsilon > 1" also requires r* > -rho/(epsilon - 1), which is not stated.
standard math Standard CRS and concavity of CES aggregators, plus the log-concavity inequalities used in eqs. (30) to (32).
Background properties of CES production functions and of the logarithmic function. These are standard and not the source of the paper's main risk.

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Cite this review

Pith. "Pith review of Elasticity of substitution and general model of economic growth." pith.science (2026). https://pith.science/paper/6AU3IXMW

@misc{pith2026250602936,
  author       = {Pith},
  title        = {Pith review of: Elasticity of substitution and general model of economic growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AU3IXMW}},
  note         = {Machine review of arXiv:2506.02936}
}

read the original abstract

The main purpose of this paper is to generalize some recent results obtained by Chilarescu and Manuel Gomez. Essentially, we are trying to study the effect of elasticity of substitution on the parameters of economic growth, based on its two possible values - lower and higher than one. We show that a higher elasticity of substitution increases per capita income, the relative share of physical capital, the common growth rate and the share of human capital allocated to the production sector, and this property is not affected by the position of the elasticity of substitution - below or above one.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

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

work page 1996
[2]

and Ruiz-Tamarit, R., 2008

Boucekkine R. and Ruiz-Tamarit, R., 2008. Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model. J our- nal of Mathematical Economics, 44 , 33 − 54

work page 2008
[3]

Elasticity of substitution and economic growt h: Some new results

Chilarescu C., 2024. Elasticity of substitution and economic growt h: Some new results. Mathematical Methods in the Applied Sciences (In press)

work page 2024
[4]

On the Existence and Uniqueness of Solution t o the Lucas-Uzawa Model, Economic Modelling, 28, 109 − 117

Chilarescu C., 2011. On the Existence and Uniqueness of Solution t o the Lucas-Uzawa Model, Economic Modelling, 28, 109 − 117

work page 2011
[5]

de La Grandville, O. 1989. In Quest of the Slutsky Diamond, American Economic Review 79, 468 - 481

work page 1989
[6]

Gomez M. A. 2016. Factor substitution is an engine of growth in a m odel with productive public expenditure. Journal of Economics 117, 37 − 48

work page 2016
[7]

Gomez M. A. 2017. Factor substitution and long-run growth in th e Lucas model with elastic labor supply. Economics Letters 159, 180 − 184

work page 2017
[8]

Gomez M. A. 2018. Factor substitution and convergence speed in the neoclassical model with elastic labor supply. Economics Letters 172, 89− 92

work page 2018

Show all 16 references

[9]

Gomez M. A. 2020. Factor substitution, long-run growth, and s peed of convergence in the one-sector convex endogenous-growth mod el. Metroe- conomica 71, 2 − 21

2020
[10]

Gomez M. A. 2021. Variable elasticity of substitution and econom ic growth in the neoclassical model. Studies in Nonlinear Dynamics and Econometrics 25, 345 − 364

2021
[11]

Gomez M. A. 2023. Factor substitution, long-run equilibrium, an d con- vergence speed in the Lucas model. Economics Letters 232, 111358. 19

2023
[12]

Gomez M. A. 2024. Elasticity of substitution, long-run growth a nd con- vergence speed: A general framework. International Journal of Eco- nomic Theory 20(3), 352 − 370

2024
[13]

and de La Grandville, O

Klump, R. and de La Grandville, O. 2000. Economic growth and the elasticity of substitution: Two theorems and some suggestions. Ameri- can Economic Review 90(1), 282 − 291

2000
[14]

On the mechanics of economic development

Lucas R., 1988. On the mechanics of economic development. Jou rnal of Monetary Economics, 22, 3 − 42

1988
[15]

Ozkaya, A. 2022. The Elasticity of Substitution and Economic Gr owth Rate: What linkage? Florya Chronicles of Political Economy 8(2), 139− 151

2022
[16]

Long-Run Policy Analysis and Long-Run Growth , Journal of Political Economy 99, 500 − 521

Rebelo, S., 1991. Long-Run Policy Analysis and Long-Run Growth , Journal of Political Economy 99, 500 − 521. 20

1991

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

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

work page 1996

[2] [2]

and Ruiz-Tamarit, R., 2008

Boucekkine R. and Ruiz-Tamarit, R., 2008. Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model. J our- nal of Mathematical Economics, 44 , 33 − 54

work page 2008

[3] [3]

Elasticity of substitution and economic growt h: Some new results

Chilarescu C., 2024. Elasticity of substitution and economic growt h: Some new results. Mathematical Methods in the Applied Sciences (In press)

work page 2024

[4] [4]

On the Existence and Uniqueness of Solution t o the Lucas-Uzawa Model, Economic Modelling, 28, 109 − 117

Chilarescu C., 2011. On the Existence and Uniqueness of Solution t o the Lucas-Uzawa Model, Economic Modelling, 28, 109 − 117

work page 2011

[5] [5]

de La Grandville, O. 1989. In Quest of the Slutsky Diamond, American Economic Review 79, 468 - 481

work page 1989

[6] [6]

Gomez M. A. 2016. Factor substitution is an engine of growth in a m odel with productive public expenditure. Journal of Economics 117, 37 − 48

work page 2016

[7] [7]

Gomez M. A. 2017. Factor substitution and long-run growth in th e Lucas model with elastic labor supply. Economics Letters 159, 180 − 184

work page 2017

[8] [8]

Gomez M. A. 2018. Factor substitution and convergence speed in the neoclassical model with elastic labor supply. Economics Letters 172, 89− 92

work page 2018