A new route to the rapid growth of the service sector: rise of the standard of living

Harutaka Takahashi; Kansho Piotr Otsubo

doi:10.1515/snde-2019-0018

Article

A new route to the rapid growth of the service sector: rise of the standard of living

Harutaka Takahashi and Kansho Piotr Otsubo

Published/Copyright: April 24, 2019

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 23 Issue 4

Abstract

In the present study, we set up a continuous-time two-sector optimal growth model with services and manufacturing goods and then examine structural change: the rapid growth of the service sector. Earlier studies of structural changes can be separated into two categories: preference-driven and technology-driven. Here we introduce a new and distinct category of structural change: consumption externality identified as rise of the living standard. A key assumption is that (1) a representative consumer has a non-homothetic Stone–Geary type utility function with respect to manufacturing goods and that (2) its subsistence level will be regarded as the standard of living and will be affected by the average consumption of manufacturing goods, which also affects the consumption level of services. We also assume that the manufacturing sector is more capital-intensive than the service sector, which takes an important role in our proofs. Results show that a steady state equilibrium exists that is globally stable as well as saddle-point stable. Then, given certain production parameters in a steady state, there exists optimal steady state where the value-added and employment shares by service sector will dominate those of the manufacturing sector under the condition that external effects of the service sector dominates capital-intensity effect of the manufacturing sector. In other words, through the transition process, the service sector will dominate the manufacturing sector in the steady state.

Keywords: capital intensity; global stability; saddle-point stability; steady state; subsistence level; the Stone-Geary utility function

JEL Classification: O41; O11; E21; E22

Acknowledgement

We thank an anonymous referee for helpful comments and suggestions. Especially, Kunieda and Nishimura (2016) has inspired us to write the paper. The previous version was presented at the conference “Real and financial interdependences: new approaches with dynamic equilibrium models” held at the Paris School of Economics in July 2017. We thank the conference participants, especially X. Raurich and A. Venditti for their comments and suggestions. We also thank Ngo Van Long for detailed review of the paper.

Appendix A

Rewriting (19) and (20) with the natural logarithm, one can obtain

(24)ln⁡(pS∗cS∗)=ln⁡(pS∗)+lncS∗=ln⁡(ps∗)+lnk∗−(1−β)ln⁡[(1+M)k∗−MkS∗]

and

(25)lny∗=ln⁡(k∗−kS∗)+(1−α)(1+M)−(1−α)ln⁡[(1+M)−MkS∗].

Subtracting (25) from (24) will yield

(26)lnps∗cS∗−lny∗=lnps∗+[ln⁡kS∗−ln⁡(k∗−kS∗)]+[−(1−α)(1+M)]+{−[(1−β)−(1−α)]ln⁡[(1+M)k∗−MkS∗]}.

Now we will show that each term of the right hand side of (26) will turn to be positive.

lnpS∗>0
Due to Assumption 6, it readily follows that lnpS∗>0.
[lnkS∗−ln⁡(k∗−kS∗)]>0
All we need to show is that kS∗>k∗−kS∗. Substituting (16) into the equation and after some manipulation yields
kS∗−(k∗−kS∗)=2kS∗−k∗=2(B−Ak∗)−k∗=2B−(2A+1)k∗.
Therefore, the condition (*) implies that 2B−(2A+1)k∗>0 and kS∗−(k∗−kS∗)>0.
–(1 – α)ln(1 + M) > 0
Due to Assumption 5, it follows that −1 < M < 0. Thus 0 < 1 + M < 1 holds and it implies that ln(1 + M) < 0. This is what we wanted.
−[(1−β)−(1−α)]ln⁡[(1+M)k∗−MkS∗]>0
(M+1)k∗−MkS∗=β(1−α)α(1−β)k∗−MB+MAk∗=β(1−α)α(1−β)k∗−MB−β(1−α)α(1−β)k∗=−MB,
whereMA=−β(1−α)α(1−β).

Thus we need to show that 0 < −MB < 1 where

−MB=[α−βα(1−β)](DpS∗)1α(α−β)=D1αα(1−β)(pS∗)1α.

Note that

D1αα(1−β)(pS∗)1α<1⇔[D1αα(1−β)(pS∗)1α]α=D[α(1−β)]α(pS∗)<1⇔pS∗<[α(1−β)]αD=(αβ)α(1−α1−β)1−α.

Assumption 6 will establish the last inequality and ln⁡[(1+M)k∗−MkS∗]<0.

Since Assumption 5 implies that −[(1 − β) − (1 − α)] < 0, we establish the result that we wanted. Finally, combining all the results 1. through 4. will complete the Proof. ■

Appendix B

Due to Assumption 5, let us define β = α − ɛ where ɛ > 0. Then, we have

(27)(αβ)α(1−α1−β)1−α=(αα−ε)α(1−α1−α+ε)1−α.

Rewriting Eq. (27) with natural logarithm yields,

I(ε)≡α[lnα−ln⁡(α−ε)]+(1−α)[ln⁡(1−α)−ln⁡(1−α+ε)].

Differentiate I(ɛ) with respect to ɛ yields

dIdε=αα−ε−1−α1−α+ε=αβ−1−α1−β=α−ββ(1−β)>0.

Since I(0)=0⇒(αβ)α(1−α1−β)1−α=1, then dIdε>0 implies that ε>0⇒I(0)>0. Therefor, it readily follows that (αβ)α(1−α1−β)1−α>1. We complete the Proof. ■

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Published Online: 2019-04-24

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Articles in the same Issue

https://doi.org/10.1515/snde-2019-0018

Keywords for this article

capital intensity; global stability; saddle-point stability; steady state; subsistence level; the Stone-Geary utility function