Ramsey model of optimal growth with Allee effect

Zihan Wang; Vladimir Shikhman; Zihan Wang; Vladimir Shikhman

doi:10.3934/jimo.2026034

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 2: 923-943. doi: 10.3934/jimo.2026034

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Research article

Ramsey model of optimal growth with Allee effect

Zihan Wang ,
Vladimir Shikhman ^,

Chemnitz University of Technology, Reichenhainer Str. 41, 09126 Chemnitz, Germany

Received: 18 September 2025 Revised: 24 December 2025 Accepted: 05 January 2026 Published: 19 January 2026
Primary: 91B62

For the Ramsey model of economic growth, which describes the optimal allocation of consumption and saving over time, we assume the underlying population dynamics due to the Allee effect. The so-called Allee threshold separates two regimes from each other. If starting below the threshold, the population decreases to zero. Above this threshold, it gradually saturates. We show that the corresponding consumption per labor stabilizes at two different levels. As for our main result, the steady state consumption per labor is higher in cases of population's decrease rather than of saturation. This is in line with our previous results on the capital per labor for the Solow–Swan model of economic growth with the Allee effect. However, the comparison of consumption-to-capital ratios at the both steady states crucially depends on the curvature of the production function.
- economic growth,
- Ramsey model,
- Allee effect,
- Pontryagin's maximum principle,
- consumption-to-capital ration
Citation: Zihan Wang, Vladimir Shikhman. Ramsey model of optimal growth with Allee effect[J]. Journal of Industrial and Management Optimization, 2026, 22(2): 923-943. doi: 10.3934/jimo.2026034

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Abstract

For the Ramsey model of economic growth, which describes the optimal allocation of consumption and saving over time, we assume the underlying population dynamics due to the Allee effect. The so-called Allee threshold separates two regimes from each other. If starting below the threshold, the population decreases to zero. Above this threshold, it gradually saturates. We show that the corresponding consumption per labor stabilizes at two different levels. As for our main result, the steady state consumption per labor is higher in cases of population's decrease rather than of saturation. This is in line with our previous results on the capital per labor for the Solow–Swan model of economic growth with the Allee effect. However, the comparison of consumption-to-capital ratios at the both steady states crucially depends on the curvature of the production function.

References

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