Diffusion-Induced Instability and Pattern Formation in Infinite Horizon Recursive Optimal Control

50 Pages Posted: 14 Apr 2006

See all articles by William A. Brock

William A. Brock

University of Wisconsin, Madison - Department of Economics; University of Missouri at Columbia - Department of Economics

Anastasios Xepapadeas

Athens University of Economics and Business; University of Bologna - School of Economics, Management, and Statistics

Date Written: April 7, 2006

Abstract

This paper develops local stability analysis for deterministic optimal control theory for recursive infinite horizon intertemporal optimization problems where there is a continuum of spatial sites and the state variable can diffuse over these sites. We identify sufficient conditions for a type of local instability which emerges from the interaction of the discount rate on the future, the curvature of the Hamiltonian, and the spatial features of the problem. We call this phenomenon diffusion-induced instability. We give a self-contained treatment of the control theory needed for our particular context because, although it is covered in the mathematical literature, our treatment is much more transparent for economists readers. We illustrate our analytical methods with three stylized applications. The first application is the optimal management of spatially connected human dominated ecosystems. The second and third applications are harvesting of spatially interconnected renewable resources.

Keywords: Spatial analysis, Pattern formation, Optimal Control, Diffusion-Induced Instability.

JEL Classification: C6, Q2

Suggested Citation

Brock, William A. and Xepapadeas, Anastasios, Diffusion-Induced Instability and Pattern Formation in Infinite Horizon Recursive Optimal Control (April 7, 2006). Available at SSRN: https://ssrn.com/abstract=895682 or http://dx.doi.org/10.2139/ssrn.895682

William A. Brock

University of Wisconsin, Madison - Department of Economics ( email )

1180 Observatory Drive
Madison, WI 53706
United States
608-263-6655 (Phone)
608-263-3876 (Fax)

University of Missouri at Columbia - Department of Economics ( email )

118 Professional Building
Columbia, MO 65211
United States

Anastasios Xepapadeas (Contact Author)

Athens University of Economics and Business ( email )

76 Patission Street
Athens, 104 34
Greece

University of Bologna - School of Economics, Management, and Statistics ( email )

Piazza Scaravilli 1
40126 Bologna, fc 47100
Italy

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