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Controlling Chaotic Fluctuations through Monetary Policy

  • Takao Asano EMAIL logo , Akihisa Shibata and Masanori Yokoo

Abstract

This paper applies the chaos control method (the OGY method) proposed by Ott, E., C. Grebogi, and J. A. Yorke. (1990. “Controlling Chaos.” Physical Review Letters 64: 1196–9) to policy-making in macroeconomics. This paper demonstrates that the monetary equilibrium paths in a discrete-time, two-dimensional overlapping generations model exhibit chaotic fluctuations depending on the money supply rate and the elasticity of substitution between capital and labor under the assumption of the constant elasticity of substitution (CES) production function. We also show that the chaotic fluctuations can be stabilized by controlling the money supply rate by using the OGY method and that even when the OGY method does not work due to periodic attractors, adding moderate stochastic shocks to the model can successfully stabilize the economy.


Corresponding author: Takao Asano, Faculty of Economics, Okayama University, Tsushimanaka 3-1-1, Kita-ku, Okayama 700-8530, Japan, E-mail:

Award Identifier / Grant number: 20H01507

Award Identifier / Grant number: 20H05631

Award Identifier / Grant number: 20K01745

Award Identifier / Grant number: 21K01388

Acknowledgment

The authors would like to thank an anonymous referee for helpful comments.

  1. Research funding: This research was financially supported by joint research programs of KIER and ISER and the Grants-in-Aid for Scientific Research, JSPS (20H05631, 20K01745, 20H01507, and 21K01388).

A Appendix

A.1 Derivation of (25)

In linearizing (21) around the golden-rule steady state with μ = 0, we have used the fact that f(1) = 1/α, f′(1) = 1, w(1) = 1/α − 1, and w′(1) = −f″(1) = γ = (1 − α)(1 + β). Actually,

J = D X G ( X ̄ , μ ̄ ) = D X G ( X , μ ) | X = ( 1,1 ) T , μ = 0 = 0 1 s f ( 1 ) w ( 1 ) s ( w ( 1 ) w ( 1 ) ) + f ( 1 ) + f ( 1 ) = 0 1 s γ 1 γ ( α s ) / α

and

B = D μ G ( X ̄ , μ ̄ ) = D μ G ( X , μ ) | X = ( 1,1 ) T , μ = 0 = 0 f ( 1 ) ( 1 s w ( 1 ) ) = 0 1 s ( 1 α ) / α .

A.2 Derivation of (26)

For the two eigenvalues of J, denoted by λ s and λ u , note that the determinant and the trace of J are given by

det J = s γ = λ s λ u > 0

and

tr J = 1 + γ ( s α ) / α = λ s + λ u ,

respectively. Let s = 0 and let β be sufficiently large. Then det J = 0 and trJ = 1 − γ = −(1 − α)β + α < − 1. This implies that λ s = λ s (s) = 0 and λ u = λ u (s) < − 1 for s = 0. Because λs,u(s) are continuous with respect to s when s is small, we obtain the result.

Of course, one can directly calculate the eigenvalues of J. Indeed, the characteristic equation is given by

| J λ I | = λ 1 s γ 1 + γ ( s α ) / α λ = λ 2 ( 1 + γ ( s α ) / α ) λ + s γ = 0 .

Solving this for λ yields

λ = λ ( s ) = 1 + γ ( s α ) / α ± ( 1 + γ ( s α ) / α ) 2 4 s γ 2 .

In particular,

λ ( 0 ) = λ u ( 0 ) = 1 γ = 1 ( 1 α ) ( 1 + β ) , λ s ( 0 ) = 0 .

Thus, for the golden-rule steady state to be a saddle for sufficiently small s, it must hold that λ u (0) < − 1 or β > (1 + α)/(1 − α) > 1.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2023-0015).


Received: 2023-02-19
Accepted: 2023-12-18
Published Online: 2024-01-10

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