Abstract
We investigate a spatial economic growth model with bounded population growth to obtain the asymptotic behavior of detrended capital in a continuous space. The formation of capital accumulation is expressed by a partial differential equation with corresponding boundary conditions. The capital accumulation interacts with the morphology to affect the optimal dynamics of economic growth. After redrafting the spatial growth model in the infinite dimensional Hilbert space, we identify the unique optimal control and value function when the bounded population growth is considered. With nonnegative initial distribution of capital, the explicit solution of the model is obtained. The time behavior of the explicit solution guarantees the convergence issue of the detrended capital level across space and time.
Highlights
Distribution of economic activities across space and time has been widely investigated in many literature [1,2,3,4,5], where economic geographers apply economic growth models to analyze the relationship between production agglomeration and location choices of people, reasons for migration flows, and formation of cities
We introduce a bounded population growth factor into the spatial economic growth model presented in [10, 11] and analyze the optimal dynamics of the model in a continuous space
Let L2(Ω) denote the Hilbert space, in which we describe the dynamics of spatial capital by the adjusted model
Summary
Distribution of economic activities across space and time has been widely investigated in many literature [1,2,3,4,5], where economic geographers apply economic growth models to analyze the relationship between production agglomeration and location choices of people, reasons for migration flows, and formation of cities. In Kamihigashi and Roy [14], a discontinuous production function is applied to show the convergence of the capital stock when an optimal growth problem with the condition of discrete time is considered. We introduce a bounded population growth factor into the spatial economic growth model presented in [10, 11] and analyze the optimal dynamics of the model in a continuous space. When the bounded population growth rate is considered, we identify the unique optimal control and value function, which are used to derive the explicit solution of the model.
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