On the comparative analysis of linear and nonlinear business cycle model: Effect on system dynamics, economy and policy making in general

Firdos Karim; Sudipa Chauhan; Joydip Dhar; Firdos Karim; Sudipa Chauhan; Joydip Dhar

doi:10.3934/QFE.2020008

Quantitative Finance and Economics

2020, Volume 4, Issue 1: 172-203. doi: 10.3934/QFE.2020008

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Research article

On the comparative analysis of linear and nonlinear business cycle model: Effect on system dynamics, economy and policy making in general

1.
Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida Sector-125, 201301, India
2.
Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior, Gwalior-474015, M.P., India

Received: 25 February 2020 Accepted: 21 March 2020 Published: 23 March 2020
JEL Codes: C62, P51

Research on linear and nonlinear IS-LM models has been resonating under synonymous perspectives, confined to bifurcations and intangible relations to economic work systems. Trifle discussion exists on how choice of linear/nonlinear models affects policy making and almost no elaboration on framing an economic system within a linear and nonlinear structure to analyze their effect separately. Parameters surrounding IS-LM model like adjustment coefficients, depreciation of capital stock etc. have not been given due spotlight, given the audacity they possess to modulate system dynamics. In counteraction, we have investigated an augmented IS-LM model with two-time delays in capital accumulation equation. This model is subjected to linear and nonlinear arguments of investment, savings and liquidity function giving rise to $M_1$(linear) and $M_2$(nonlinear) models. They undergo hopf bifurcation for different values of delay parameters $\tau_{1}$ and $\tau_{2}$. Our study accentuates the following aspects(1) In a neophyte attempt, comparing the dynamics of a linear and nonlinear business cycle model in an environment as similar as possible, when $\tau_{1}$ and $\tau_{2}$ are the bifurcating parameters. (2) Parameter sensitivity analysis for both models. (3) Non linearity in savings function, which is a sparse event so far. Our findings reveal that (1) Non-linearity elevates system sensitivity and $M_2$ model attains stability easily in the long run for dual delays, while for single delay $M_1$ model has this feat. (2) $M_2$ model encapsulates recurring cyclic behavior while $M_1$ model is not capable of generating the same and demonstrates motifs of either stability or instability. (3) Parameter sensitivity analysis reveals that both the models are most vulnerable when (3a)Value of depreciation of capital stock is decreased. (3b) Money supply and propensities to investment are increased. (4) how aforementioned information can be utilized for crafting economic policies for linear/nonlinear economies, especially curated for their modus operandi. Numerical simulations follow.
- IS-LM model,
- capital accumulation,
- hopf bifurcation,
- parameter sensitivity
Citation: Firdos Karim, Sudipa Chauhan, Joydip Dhar. On the comparative analysis of linear and nonlinear business cycle model: Effect on system dynamics, economy and policy making in general[J]. Quantitative Finance and Economics, 2020, 4(1): 172-203. doi: 10.3934/QFE.2020008

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Abstract

Research on linear and nonlinear IS-LM models has been resonating under synonymous perspectives, confined to bifurcations and intangible relations to economic work systems. Trifle discussion exists on how choice of linear/nonlinear models affects policy making and almost no elaboration on framing an economic system within a linear and nonlinear structure to analyze their effect separately. Parameters surrounding IS-LM model like adjustment coefficients, depreciation of capital stock etc. have not been given due spotlight, given the audacity they possess to modulate system dynamics. In counteraction, we have investigated an augmented IS-LM model with two-time delays in capital accumulation equation. This model is subjected to linear and nonlinear arguments of investment, savings and liquidity function giving rise to $M_1$(linear) and $M_2$(nonlinear) models. They undergo hopf bifurcation for different values of delay parameters $\tau_{1}$ and $\tau_{2}$. Our study accentuates the following aspects(1) In a neophyte attempt, comparing the dynamics of a linear and nonlinear business cycle model in an environment as similar as possible, when $\tau_{1}$ and $\tau_{2}$ are the bifurcating parameters. (2) Parameter sensitivity analysis for both models. (3) Non linearity in savings function, which is a sparse event so far. Our findings reveal that (1) Non-linearity elevates system sensitivity and $M_2$ model attains stability easily in the long run for dual delays, while for single delay $M_1$ model has this feat. (2) $M_2$ model encapsulates recurring cyclic behavior while $M_1$ model is not capable of generating the same and demonstrates motifs of either stability or instability. (3) Parameter sensitivity analysis reveals that both the models are most vulnerable when (3a)Value of depreciation of capital stock is decreased. (3b) Money supply and propensities to investment are increased. (4) how aforementioned information can be utilized for crafting economic policies for linear/nonlinear economies, especially curated for their modus operandi. Numerical simulations follow.

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