Endogenous Growth, Spatial Dynamics and Convergence: A Refinement

Raouf Boucekkine (), Carmen Camacho and Weihua Ruan

Annals of Economics and Statistics, 2025, issue 159, 79-106

Abstract: The dynamics of capital distribution across space are an important topic in economic geography and, more recently, in growth theory. In particular, the spatial AK model has been intensively studied in the latter stream. It turns out that the positivity of optimal capital stocks over time and space for any initial capital spatial distribution has not been entirely settled even in the simple linear AK case. We use Ekeland's variational principle together with Pontryagin's maximum principle to solve an optimal spatiotemporal AK model with a state constraint (non-negative capital stock), where the capital law of motion follows a diffusion equation. We derive the necessary optimality conditions to ensure the solution satisfies the state constraints for all times and locations. The maximum principle enables the reduction of the infinite-horizon optimal control problem to a finite-horizon problem, ultimately proving the uniqueness of the optimal solution with positive capital and the non-existence of such a solution when the time discount rate is either too large or too small.

Keywords: Diffusion and Growth; Convergence; Optimal Control; State Constraint; Ekeland's Variational Principle. (search for similar items in EconPapers)
JEL-codes: C61 O44 Q15 Q56 R11 (search for similar items in EconPapers)
Date: 2025
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Working Paper: Endogenous growth, spatial dynamics and convergence: A refinement (2025)
Working Paper: Endogenous growth, spatial dynamics and convergence: A refinement (2025)
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Persistent link: https://EconPapers.repec.org/RePEc:adr:anecst:y:2025:i:159:p:79-106

DOI: 10.2307/48839155

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