Variable elasticity of substitution and economic growth in the neoclassical model

Manuel A. Gómez

doi:10.1515/snde-2019-0145

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Variable elasticity of substitution and economic growth in the neoclassical model

Manuel A. Gómez

Published/Copyright: September 18, 2020

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 25 Issue 5

Abstract

We study the effect of factor substitutability in the neoclassical growth model with variable elasticity of substitution. We consider two otherwise identical economies differing uniquely in their initial factor substitutability with Variable-Elasticity-of-Substitution (VES), Sobelow or Sigmoidal technologies. If the initial capital per capita is below its steady-state value, the economy with the higher initial elasticity of substitution will feature a higher steady-state income and capital per capita irrespective of whether the production technology is VES, Sobelow or Sigmoidal. Numerical results are provided to compare the effect of a higher elasticity of substitution in the Constant-Elasticity-of-Substitution (CES) model versus the models with variable-elasticity-of-substitution technology.

Keywords: economic growth; elasticity of substitution; neoclassical growth

JEL Classification: O41; E21

Corresponding author: Manuel A. Gómez, Departamento de Economía, Universidade da Coruña, Campus de A Coruña, A Coruña, 15071, Spain, E-mail: mago@udc.es

Funding source: Spanish Ministerio de Economía, Industria y Competitividad 501100010198

Award Identifier / Grant number: ECO2017-85701-P

Funding source: European Regional Development Fund 501100008530

Award Identifier / Grant number: ECO2017-85701-P

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad, and the European Regional Development Fund under Grant No. ECO2017-85701-P.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Proof of Proposition 1

Differentiating Eq. (12) with respect to −2.5065 we get that y is a an increasing function of the initial elasticity of substitution x ^old(t),

(A.1) ∂ y ∂ σ 0 = ∂ f ∂ σ 0 ( k , σ 0 ) = − ( 1 − π 0 ) π 0 [ π 0 + σ 0 ( 1 − π 0 ) ] 2 f ( k , σ 0 ) [ 1 − ( 1 − π ) k ( 1 − π 0 ) k 0 + ln ( 1 − π ) k ( 1 − π 0 ) k 0 ] ≥ 0 ,

with equality if and only if k = k 0 . To derive the former inequality we have used that the logarithmic function is strictly concave and, therefore, ln x ≤ x − 1 , with equality if and only if x = 1 . Differentiating Eq. (12) with respect to k we have that

(A.2) ∂ f ∂ k ( k , σ 0 ) = A ( σ 0 ) k α ( σ 0 ) − 1 [ B ( σ 0 ) k + 1 ] − α ( σ 0 ) [ B ( σ 0 ) k + α ( σ 0 ) ] > 0.

Differentiating Eq. (A.2) with respect to k = k ₀, after simplification, we get that

(A.3) ∂ 2 f ∂ k ∂ σ 0 ( k , σ 0 ) = ( 1 − π 0 ) f ( k , σ 0 ) π 0 α ( σ 0 ) 2 [ 1 + B ( σ 0 ) k ] k × { [ 1 + B ( σ 0 ) k + α ( σ 0 ) ] [ ( 1 − π ) k ( 1 − π 0 ) k 0 − 1 ] − [ B ( σ 0 ) k + α ( σ 0 ) ] ln ( 1 − π ) k ( 1 − π 0 ) k 0 } > 0 ,

if x = 1. To derive the last inequality in Eq. (A.3), let us define

h ( k ) = ( 1 − π ) k ( 1 − π 0 ) k 0 = [ 1 − α ( σ 0 ) ] k ( 1 − π 0 ) k 0 [ B ( σ 0 ) k + 1 ] .

The function h is strictly increasing because its derivative is strictly positive,

h ′ ( k ) = [ 1 − α ( σ 0 ) ] ( 1 − π 0 ) k 0 [ B ( σ 0 ) k + 1 ] 2 > 0 ,

and satisfies that h ( k 0 ) = 1 . Hence, we have that h ( k ) > 1 if and only if k > k 0 . Taking into account this feature and using that the logarithmic function is strictly concave and, therefore, ln x ≤ x − 1 , we have that

[ 1 + B ( σ 0 ) k + α ( σ 0 ) ] [ ( 1 − π ) k ( 1 − π 0 ) k 0 − 1 ] − [ B ( σ 0 ) k + α ( σ 0 ) ] ln ( 1 − π ) k ( 1 − π 0 ) k 0 ≥ ( 1 − π ) k ( 1 − π 0 ) k 0 − 1 = k − k 0 [ 1 + B ( σ 0 ) k ] k 0 > 0 ,

if k > k 0 , and the inequality in Eq. (A.3) follows. Differentiating Eq. (A.2) with respect to k, after simplification, we get that

∂ 2 f ∂ k 2 ( k , σ 0 ) = − [ 1 − α ( σ 0 ) ] α ( σ 0 ) A ( σ 0 ) k α ( σ 0 ) − 2 [ B ( σ 0 ) k + 1 ] − α ( σ 0 ) − 1 < 0.

We are now ready to examine the effect of the elasticity of substitution on per capita capital and income. Taking into account the former results, differentiating Eq. (8) with respect to σ 0 we have that

(A.4) d k ¯ d σ 0 ( σ 0 ) = − ∂ 2 f ∂ k ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) ∂ 2 f ∂ k 2 ( k ¯ ( σ 0 ) , σ 0 ) > 0 ,

if k ¯ ( σ 0 ) > k 0 ^[8] The effect of the initial elasticity of substitution on per capita income is given by

(A.5) d y ¯ d σ 0 ( σ 0 ) = ∂ f ∂ k ( k ¯ ( σ 0 ) , σ 0 ) d k ¯ d σ 0 ( σ 0 ) + ∂ f ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) > 0 ,

if k ¯ ( σ 0 ) > k 0 .

Let us now consider the effect of the elasticity of substitution on the capital income share

(A.6) π = π ( k , σ 0 ) = k ∂ f ∂ k ( k , σ 0 ) f ( k , σ 0 ) = B ( σ 0 ) k + α ( σ 0 ) B ( σ 0 ) k + 1 = π 0 [ k 0 + k ( σ 0 − 1 ) ] π 0 ( σ 0 − 1 ) k + [ π 0 + σ 0 ( 1 − π 0 ) ] k 0 .

Differentiating this expression, with respect to k and σ 0 , after simplification we get

(A.7) ∂ π ∂ k ( k , σ 0 ) = [ 1 − α ( σ 0 ) ] B ( σ 0 ) [ B ( σ 0 ) k + 1 ] 2 > 0 ,

(A.8) ∂ π ∂ σ 0 ( k , σ 0 ) = [ 1 − α ( σ 0 ) ] α ( σ 0 ) 2 [ B ( σ 0 ) k + 1 ] 2 [ B ( σ 0 ) k 0 + α ( σ 0 ) ] k 0 ( k − k 0 ) { > 0 , if k > k 0 , = 0 , if k = k 0 , < 0 , if k < k 0 .

Differentiating Eq. (A.6) we get that

d π ¯ d σ 0 ( σ 0 ) = ∂ π ¯ ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) + ∂ π ¯ ∂ k ( k ¯ ( σ 0 ) , σ 0 ) d k ¯ d σ 0 ( σ 0 ) = [ 1 − α ( σ 0 ) ] α ( σ 0 ) [ B ( σ 0 ) k ¯ ( σ 0 ) + α ( σ 0 ) ] [ B ( σ 0 ) k ¯ ( σ 0 ) + 1 ] [ B ( σ 0 ) k 0 + α ( σ 0 ) ] { k ¯ ( σ 0 ) k 0 − 1 − B ( σ 0 ) k ¯ ( σ 0 ) ln [ ( B ( σ 0 ) k 0 + 1 ) k ¯ ( σ 0 ) ( B ( σ 0 ) k ¯ ( σ 0 ) + 1 ) k 0 ] } .

On the one hand, using that the logarithmic function is strictly concave and, therefore, ln x ≤ x − 1 , we have that

d π ¯ d σ 0 ( σ 0 ) ≥ [ 1 − α ( σ 0 ) ] α ( σ 0 ) [ B ( σ 0 ) k ¯ ( σ 0 ) + α ( σ 0 ) ] [ B ( σ 0 ) k ¯ ( σ 0 ) + 1 ] [ B ( σ 0 ) k 0 + α ( σ 0 ) ] × { k ¯ ( σ 0 ) k 0 − 1 − B ( σ 0 ) k ¯ ( σ 0 ) [ ( B ( σ 0 ) k 0 + 1 ) k ¯ ( σ 0 ) ( B ( σ 0 ) k ¯ ( σ 0 ) + 1 ) k 0 − 1 ] } = [ 1 − α ( σ 0 ) ] α ( σ 0 ) [ B ( σ 0 ) k ¯ ( σ 0 ) + α ( σ 0 ) ] [ B ( σ 0 ) k ¯ ( σ 0 ) + 1 ] 2 k 0 [ B ( σ 0 ) k 0 + α ( σ 0 ) ] [ k ¯ ( σ 0 ) − k 0 ] > 0 ,

if k ¯ = k ¯ ( σ 0 ) > k 0 . On the other hand, using that the function x ln x is strictly convex and, therefore, x ln x ≥ x − 1 , we have that

d π ¯ d σ 0 ( σ 0 ) ≤ [ 1 − α ( σ 0 ) ] α ( σ 0 ) [ B ( σ 0 ) k ¯ ( σ 0 ) + α ( σ 0 ) ] [ B ( σ 0 ) k ¯ ( σ 0 ) + 1 ] [ B ( σ 0 ) k 0 + α ( σ 0 ) ] × { k ¯ ( σ 0 ) k 0 − 1 − B ( σ 0 ) k 0 ( B k ¯ + 1 ) [ B ( σ 0 ) k 0 + 1 ] [ ( B ( σ 0 ) k 0 + 1 ) k ¯ ( σ 0 ) [ B ( σ 0 ) k ¯ ( σ 0 ) + 1 ] k 0 − 1 ] } = [ 1 − α ( σ 0 ) ] α ( σ 0 ) [ B ( σ 0 ) k ¯ ( σ 0 ) + α ( σ 0 ) ] [ B ( σ 0 ) k ¯ ( σ 0 ) + 1 ] [ B ( σ 0 ) k 0 + α ( σ 0 ) ] [ B ( σ 0 ) k 0 + 1 ] k 0 [ k ¯ ( σ 0 ) − k 0 ] < 0 ,

if k ¯ = k ¯ ( σ 0 ) < k 0 . In summary, we get that

(A.9) d π ¯ d σ 0 ( σ 0 ) { > 0 , if k ¯ ( σ 0 ) > k 0 , = 0 , if k ¯ ( σ 0 ) = k 0 , < 0 , if k ¯ ( σ 0 ) < k 0 .

Appendix B: Proof of Proposition 2

Differentiating Eq. (17) with respect to σ 0 we get that y is a an increasing function of the initial elasticity of substitution σ 0 ,

(B.1) ∂ y ∂ σ 0 = ∂ f ∂ σ 0 ( k , σ 0 ) = − ( 1 − π 0 ) π 0 y 0 ( σ 0 − π 0 ) 2 ( k k 0 ) α ( σ 0 ) [ 1 − ( k k 0 ) 1 − α ( σ 0 ) + ln ( k k 0 ) 1 − α ( σ 0 ) ] ≥ 0 ,

with equality if and only if k = k 0 , where we have used that the logarithmic function is strictly concave and, therefore, ln x ≤ x − 1 , with equality if and only if x = 1 . Differentiating Eq. (17) with respect to k we have that

(B.2) ∂ f ∂ k ( k , σ 0 ) = A ( σ 0 ) + α ( σ 0 ) B ( σ 0 ) k α ( σ 0 ) − 1 > 0.

Differentiating Eq. (B.1) with respect to k we get that

∂ 2 f ∂ k ∂ σ 0 ( k , σ 0 ) = ( 1 − π 0 ) π 0 y 0 k 0 ( σ 0 − π 0 ) 2 [ k k 0 ] π 0 − σ 0 σ 0 { − 1 + [ k 0 k ] π 0 − σ 0 σ 0 − π 0 σ 0 ln [ k 0 k ] π 0 − σ 0 σ 0 } > 0 ,

if k > k 0 . Here we have used that ln x ≤ x − 1 entails that

∂ 2 f ∂ k ∂ σ 0 ( k , σ 0 ) ≥ ( 1 − π 0 ) π 0 y 0 k 0 ( σ 0 − π 0 ) 2 [ k 0 k ] π 0 − σ 0 σ 0 ( σ 0 − π 0 ) σ 0 { [ k 0 k ] π 0 − σ 0 σ 0 − 1 } > 0 ,

if k > k 0 (note that σ 0 > 1 > π 0 ). Differentiating Eq. (B.2) with respect to k we get that

∂ 2 f ∂ k 2 ( k , σ 0 ) = − [ 1 − α ( σ 0 ) ] α ( σ 0 ) B ( σ 0 ) k α − 2 < 0.

The capital income share is

(B.3) π = π ( k , σ 0 ) = k ∂ f ∂ k ( k , σ 0 ) f ( k , σ 0 ) = ( k k 0 ) 1 − π 0 / σ 0 ( σ 0 − 1 ) π 0 + ( 1 − π 0 ) π 0 ( k k 0 ) 1 − π 0 / σ 0 ( σ 0 − 1 ) π 0 + ( 1 − π 0 ) σ 0 < 1.

Differentiating this expression with respect to k and σ 0 , after simplification we get

(B.4) ∂ π ∂ k ( k , σ 0 ) = [ 1 − α ( σ 0 ) ] 2 A ( σ 0 ) B ( σ 0 ) k α ( σ 0 ) [ A ( σ 0 ) k + B ( σ 0 ) k α ( σ 0 ) ] 2 > 0 ,

and

(B.5) ∂ π ∂ σ 0 ( k , σ 0 ) = α ( σ 0 ) 2 A ( σ 0 ) B ( σ 0 ) k α ( σ 0 ) + 1 π 0 f ( k , σ 0 ) 2 ( k k 0 ) α ( σ 0 ) − 1 × { σ 0 ( 1 − π 0 ) π 0 ( σ 0 − 1 ) [ ( k k 0 ) 1 − α ( σ 0 ) − 1 ] + ( k k 0 ) 1 − α ( σ 0 ) ln [ ( k k 0 ) 1 − α ( σ 0 ) ] } { > 0 , if k > k 0 , = 0 , if k = k 0 , < 0 , if k < k 0 .

Differentiating Eq. (8) with respect to σ 0 we have that

(B.6) d k ¯ d σ 0 ( σ 0 ) = − ∂ 2 f ∂ k ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) ∂ 2 f ∂ k 2 ( k ¯ ( σ 0 ) , σ 0 ) > 0 ,

if k ¯ ( σ 0 ) > k 0 . The effect of the initial elasticity of substitution on per capita income is, therefore,

(B.7) d y ¯ d σ 0 ( σ 0 ) = ∂ f ∂ k ( k ¯ ( σ 0 ) , σ 0 ) d k ¯ d σ 0 ( σ 0 ) + ∂ f ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) > 0 ,

if k ¯ ( σ 0 ) > k 0 .

Differentiating Eq. (B.3) we get that

(B.8) d π ¯ d σ 0 ( σ 0 ) = ∂ π ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) + ∂ π ∂ k ( k ¯ ( σ 0 ) , σ 0 ) d k ¯ d σ 0 ( σ 0 ) = α ( σ 0 ) B ( σ 0 ) π ¯ ( σ 0 ) k ¯ ( σ 0 ) α ( σ 0 ) π 0 f ( k ¯ ( σ 0 ) , σ 0 ) [ ( k ¯ ( σ 0 ) k 0 ) 1 − α ( σ 0 ) − 1 ] { > 0 , if k ¯ ( σ 0 ) > k 0 , = 0 , if k ¯ ( σ 0 ) = k 0 , < 0 , if k ¯ ( σ 0 ) < k 0 .

Appendix C: Proof of Proposition 3

We have that

∂ y ∂ σ 0 = ∂ f ∂ σ 0 ( k , σ 0 ) = ( 1 − π 0 ) y 0 ( k k 0 ) α ( σ 0 ) π 0 σ 0 2 [ 1 + B ( σ 0 ) k α ( σ 0 ) ] 2 { − 1 + ( k k 0 ) α ( σ 0 ) − ln [ ( k k 0 ) α ( σ 0 ) ] } ≥ 0 ,

with equality if and only if k = k 0 .

After simplification, we have that

∂ 2 f ∂ k 2 ( k , σ 0 ) = − π ( k , σ 0 ) 2 f ( k , σ 0 ) [ 1 − α ( σ 0 ) + ( 1 + α ( σ 0 ) ) B ( σ 0 ) k α ( σ 0 ) ] α ( σ 0 ) k 2 ,

and

∂ 2 f ∂ k ∂ σ 0 ( k , σ 0 ) = ( 1 − π 0 ) y 0 π ( k , σ 0 ) 3 α ( σ 0 ) 2 σ 0 2 π 0 k 0 ( k k 0 ) α ( σ 0 ) − 1 × { [ ( k k 0 ) α ( σ 0 ) ( α ( σ 0 ) − π 0 π 0 ) − 1 ] ln [ ( k k 0 ) α ( σ 0 ) ] + 2 [ ( k k 0 ) α ( σ 0 ) − 1 ] } .

Using that the logarithmic function is strictly concave and, therefore, ln x ≤ x − 1 , we have that

(C.1) ∂ 2 f ∂ k ∂ σ 0 ( k , σ 0 ) ≥ ( 1 − π 0 ) y 0 π ( k , σ 0 ) 3 α ( σ 0 ) 2 σ 0 2 π 0 k 0 ( k k 0 ) α ( σ 0 ) − 1 × { [ ( k k 0 ) α ( σ 0 ) ( α ( σ 0 ) − π 0 π 0 ) + 1 ] ln [ ( k k 0 ) α ( σ 0 ) ] } ≥ 0 ,

if k ≥ k 0 , with equality if and only if k = k 0 . Here, we have used that α ( σ 0 ) − π 0 = ( 1 − π 0 ) ( 1 − σ 0 ) / σ 0 > 0 .

Let π denote the capital income share,

(C.2) π = π ( k , σ 0 ) = k ∂ f ∂ k ( k , σ 0 ) f ( k , σ 0 ) = α ( σ 0 ) 1 + B ( σ 0 ) k α ( σ 0 ) .

We have that

(C.3) ∂ π ∂ k ( k , σ 0 ) = − α ( σ 0 ) 2 B ( σ 0 ) k α ( σ 0 ) − 1 [ 1 + B ( σ 0 ) k α ( σ 0 ) ] 2 < 0.

and, after simplification,

(C.4) ∂ π ∂ σ 0 ( k , σ 0 ) = ( 1 − π 0 ) σ 0 2 [ 1 + B ( σ 0 ) k α ( σ 0 ) ] 2 × [ α ( σ 0 ) − π 0 π 0 ( k k 0 ) α ( σ 0 ) ln ( k k 0 ) α ( σ 0 ) + ( k k 0 ) α ( σ 0 ) − 1 ] { > 0 , if k ¯ ( σ 0 ) > k 0 , = 0 , if k ¯ ( σ 0 ) = k 0 , < 0 , if k ¯ ( σ 0 ) < k 0 .

Assuming that the steady state is saddle-path stable, so that ∂ 2 f ∂ k 2 ( k ¯ ( σ 0 ) , σ 0 ) < 0 , differentiating Eq. (8) with respect to σ 0 we have that

(C.5) d k ¯ d σ 0 ( σ 0 ) = − ∂ 2 f ∂ k ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) ∂ 2 f ∂ k 2 ( k ¯ ( σ 0 ) , σ 0 ) > 0

if k ¯ ( σ 0 ) > k 0 .

The effect of the elasticity of substitution on the capital income share can be derived as

(C.6) d π ¯ d σ 0 ( σ 0 ) = ∂ π ∂ σ 0 ( k ¯ ( σ 0 ) , σ 0 ) + ∂ π ∂ k ( k ¯ ( σ 0 ) , σ 0 ) d k ¯ d σ 0 ( σ 0 ) = π 0 2 π ¯ ( σ 0 ) [ 1 − α ( σ 0 ) + ( 1 + α ( σ 0 ) ) B ( σ 0 ) k 0 α ] 2 α ( σ 0 ) 3 ( 1 − π 0 ) [ 1 − α ( σ 0 ) + ( 1 + α ( σ 0 ) ) B ( σ 0 ) k ¯ ( σ 0 ) α ( σ 0 ) ] × { [ 1 − α ( σ 0 ) ] [ ( k ¯ ( σ 0 ) k 0 ) α ( σ 0 ) − 1 ] + B ( σ 0 ) k 0 α ( σ 0 ) [ k ¯ ( σ 0 ) k 0 ] α ( σ 0 ) ln [ k ¯ ( σ 0 ) k 0 ] α ( σ 0 ) } { > 0 , if k ¯ ( σ 0 ) > k 0 , = 0 , if k ¯ ( σ 0 ) = k 0 , < 0 , if k ¯ ( σ 0 ) < k 0 .

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Received: 2019-12-21

Accepted: 2020-08-04

Published Online: 2020-09-18

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https://doi.org/10.1515/snde-2019-0145

Keywords for this article

economic growth; elasticity of substitution; neoclassical growth