Stochastic bifurcation in economic growth model driven by Lévy noise

TL;DR

The paper extends the Solow economic growth model by incorporating Lévy noise to capture non-Gaussian shocks and abrupt economic events, analyzing both deterministic and stochastic dynamics in continuous and discrete time. It develops a Lévy-driven SDE for capital per effective worker with a sigmoid saving function and establishes well-posedness, then investigates stability via a threshold $\xi=\frac{\beta B}{\rho(1+\gamma)}$ and Lyapunov exponents. A slow-fast bifurcation analysis reveals a critical value $\gamma_c\approx 2.3333$ at which a monostable equilibrium becomes bistable, accompanied by a double-well potential and rapid shifts due to jumps. Numerical simulations using Euler–Maruyama show Lévy jumps substantially increase volatility and yield abrupt regime shifts relative to Gaussian noise, underlining the importance of non-Gaussian shocks for forecasting and policy design in uncertain environments.

Abstract

This paper enhances the classical Solow model of economic growth by integrating Lévy noise, a type of non-Gaussian stochastic perturbation, to capture the inherent uncertainties in economic systems. The extended model examines the impact of these random fluctuations on capital stock and output, revealing the role of jump-diffusion processes in long-term GDP fluctuations. Both continuous and discrete-time frameworks are analyzed to assess the implications for forecasting economic growth and understanding business cycles. The study compares deterministic and stochastic scenarios, providing insight into the stability of equilibrium points and the dynamics of economies subjected to random disturbances. Numerical simulations demonstrate how stochastic noise contributes to economic volatility, leading to abrupt shifts and bifurcations in growth trajectories. This research offers a comprehensive perspective on the influence of external shocks, presenting a more realistic depiction of economic development in uncertain environments.

Figures (5)

• Figure 1: The time series of both the system with $\varepsilon=0.05$ and the reduced system.
• Figure 2: The sigmoid savings function $s(k,\gamma)$.
• Figure 3: The bifurcation diagram shows equilibrium solutions $k^{\ast}$ vs $\gamma$. The equilibrium $k_0^{\ast}=1$ exists for all $\gamma\geq0$, provided $s(1,\gamma)=1/\beta$. A critical value $\gamma_c\approx2.3333$ marks a bifurcation. For $\gamma<\gamma_c$, $k_0^{\ast}=1$ is stable. For $\gamma>\gamma_c$, two new stable equilibria $k_{+}^{\ast}$ and $k_{-}^{\ast}$emerge, and $k_0^{\ast}$ becomes unstable.
• Figure 4: (a) The phase line plots plot $\frac{dk}{dt}$ against $k$ for three different $\gamma$ values. The system \ref{['kwn']} transitions from monostable to bistable as $\gamma$ exceeds $\gamma_c$, driven by the nonlinear savings function $s(k,\gamma)$; (b) The potential function $V(k,\gamma)$ for different $\gamma$, highlighting the double-well when $\gamma>\gamma_c$, illustrates bistability with minima at $k_{+}^{\ast}(\gamma)$ and $k_{-}^{\ast}(\gamma)$, and a maximum at $k_{0}^{\ast}(\gamma)=1$.
• Figure 5: Simulations of the stochastic dynamic system for capital per effective worker $k(t)$, investment $I(t)$, and the external shock process $X(t)$. The savings function is sigmoidal, $s(k) = s_1 + (s_2 - s_1)/(1 + e^{-\gamma(k - \phi)})$, with $s_1 = 0.2$, $s_2 = 0.8$, $\gamma = 0.5$, and $\phi = 1.0$. The production function is $f(k) = B k^\alpha$ with $B = 1.5$. Stochastic components include Brownian motion and compound Poisson jumps with intensity $\lambda = 0.01$. Other parameters are: capital depreciation rate $\rho = 0.02$, investment decay rate $v = 0.02$, capital-investment interaction rate $\beta = 0.4$, Brownian noise scale $\sigma = 0.1$, and mean-reversion rate $\eta_a = 0.1$. The simulation runs over $T = 50$ time units with a step size of $\Delta t = 0.01$.