Competitive equilibrium cycles for small discounting in discrete-time two-sector optimal growth models

Alain Venditti

doi:10.1515/snde-2019-0021

Article

Competitive equilibrium cycles for small discounting in discrete-time two-sector optimal growth models

Alain Venditti

Published/Copyright: April 17, 2019

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

Abstract

We study the existence of endogenous competitive equilibrium cycles under small discounting in a two-sector discrete-time optimal growth model. We provide precise concavity conditions on the indirect utility function leading to the existence of period-two cycles with a critical value for the discount factor that can be arbitrarily close to one. Contrary to the continuous-time case where the existence of periodic-cycles is obtained if the degree of concavity is close to zero, we show that in a discrete-time setting the driving condition does not require a close to zero degree of concavity but a symmetry of the indirect utility function’s concavity properties with respect to its two arguments.

Keywords: period-two cycles; small discounting; strong and weak concavity; two-sector optimal growth model

JEL Classification: C62; E32; O41

Acknowledgement

This paper has been written in honor of Professor Kazuo Nishimura, a very close friend, who has made outstanding contributions to economic theory, and in particular to the literature on endogenous fluctuations in optimal growth models. This work was supported by French National Research Agency Grants ANR-08-BLAN-0245-01 and ANR-17-EURE-0020. I would like to thank Makoto Yano and an anonymous referee for useful comments and suggestions.

This paper is dedicated to the memory of Pierre Cartigny (1946–2019), my PhD advisor and a very close friend.

A Appendix

A.1 Proof of Lemma 1

i) This is a standard result (see Montrucchio (1987), Rockafellar (1976)).

ii) From i) we get

νt[D2f(x)+αI]ν≤0

which means that the matrix [D2f(x)ν+αI] should be negative semi-definite for all x ∈ D, i.e. there must exist a ν~≠0 such that

ν~t[D2f(x)+αI]ν~=0

This means therefore that

Det[D2f(x)+αI]≡Det[D2f(x)−(−α)I]=0

It follows that −α is an eigenvalue of the matrix D2f(x). Considering λi(x) an eigenvalue of D2f(x), we may define for all x ∈ D

λ∗(x)=mini=1,…,n{|λi(x)|}

and we conclude therefore that α=infx∈Dλ∗(x). If f is α-concave over D then α ≠ 0 since the Hessian matrix is non-singular for all x ∈ D. □

A.2 Proof of Lemma 2

The arguments are the same as in the proof of Lemma 1. □

A.3 Proof of Proposition 2

Consider first the case V12δ>0. We know from (14) that P(0)=V12δ>0 and limλ→±∞P(λ)=+∞. It follows that the steady state is a saddle-point if and only if

P(1)=δV11δ+V22δ+(1+δ)V12δ<0

If V(x, y) is (α, β)-concave, we derive from Lemma 1 that the following quadratic form holds

(11)(V11δ+αV12δV12δV22δ+β)(11)≤0

which implies

α+β≤−2V12δ−V11δ−V22δ

If V(.,kδ∗) is concave-γ, we derive from Lemma 2 that γ≥−V11δ. We then derive

α+βγ≤−2V12δ+V11δ+V22δV11δ

Assume now that 1≥δ>1−(α+β)/γ. We derive from the previous inequality

δ>2V12δ+V22δV11δ

which implies

δV11δ+V22δ+2V12δ<0

Since 2V12δ≥(1+δ)V12δ we conclude that

P(1)=δV11δ+V22δ+(1+δ)V12δ<0

and the steady state is a saddle-point.

Consider now the case V12δ<0. We know from (14) that P(0)=V12δ<0 and limλ→±∞P(λ)=−∞. It follows that the steady state is a saddle-point if and only if

P(−1)=−[δV11δ+V22δ−(1+δ)V12δ]>0

If V(x, y) is (α, β)-concave, we derive from Lemma 1 that the following quadratic form holds

(1−1)(V11δ+αV12δV12δV22δ+β)(1−1)≤0

which implies

α+β≤2V12δ−V11δ−V22δ

If V(.,kδ∗) is concave-γ, we derive from Lemma 2 that γ≥−V11δ. We then derive

α+βγ≤−2V12δ+V11δ+V22δV11δ

Assume now that 1≥δ>1−(α+β)/γ. We derive from the previous inequality

δ>2V12δ−V22δV11δ

which implies

δV11δ+V22δ−2V12δ<0

Since 2V12δ≤(1+δ)V12δ we conclude that

P(−1)=−[δV11δ+V22δ−(1+δ)V12δ]>0

and the steady state is a saddle-point. The results of the Proposition follow. □

A.4 Proof of Proposition 3

Consider in a first step the case δ = 1. Assumption 4 implies that V111≠V221. Since we assume V12δ<0, we derive from the proof of Proposition 2 that

P(−1)=V111+V221−2V121

The singularity of the Hessian matrix of V at the steady state implies then V111V221−(V121)2=0 or equivalently V121=−V111V221. Substituting this into P(−1) yields

P(−1)=−(|V111|−|V221|)2<0

It follows that when δ = 1 the steady state is a saddle-point. Let us then assume that there is a value δ¯∈(0,1) such that

δ¯V11δ¯+V22δ¯−(1+δ¯)V12δ¯>0

Then there exists δ∗∈(δ¯,1) such that

(18)P(−1)=−[δ∗V11δ∗+V22δ∗−(1+δ∗)V12δ∗]=0

and there exists an eigenvalue λ(δ∗)=−1 leading to the existence of a flip bifurcation, i.e. period-two cycles in a neighborhood of δ^*. Solving equation (18), we conclude that the bifurcation value δ^* is implicitly defined by

δ∗=V22δ∗−V12δ∗V12δ∗−V11δ∗

□

A.5 Proof of Proposition 4

Using Lemma 2 and Definition 5, we have that the following matrix is positive semi-definite:

(V11δ∗+γ1V12δ∗V12δ∗V22δ∗+γ2)

This means in particular that its determinant is equal to zero:

(V11δ∗+γ1)(V22δ∗+γ2)−(V12δ∗)2=0

Using also the fact that V11δ∗V22δ∗−(V12δ∗)2=0 we obtain that:

γ1=−2V11δ∗,γ2=−2V22δ∗

Let us now consider the bifurcation value given by equation (16). We then have:

(19)δ∗=γ1γ2−γ2γ1−γ1γ2=γ2γ1

We first note that for the bifurcation value to be less than 1, we need to assume that γ2<γ1. Second, we conclude that if (γ1−γ2)→0 then the bifurcation value δ∗→1. □

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Published Online: 2019-04-17

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https://doi.org/10.1515/snde-2019-0021

Keywords for this article

period-two cycles; small discounting; strong and weak concavity; two-sector optimal growth model