The dynamic of a tax on land value : concepts, models and impact scenario

This paper develops a spatial-dynamic framework to analyze the theoretical and quantitative effects of a Land Value Tax (LVT) on urban land markets, capital accumulation, and spatial redistribution. Building upon the Georgist distinction between produced value and unearned rent, the model departs from the static equilibrium tradition by introducing an explicit diffusion process for land values and a local investment dynamic governed by profitability thresholds. Land value $V (x, y, t)$ and built capital $K(x, y, t)$evolve over a two-dimensional urban domain according to coupled nonlinear partial differential equations, incorporating local productivity $A(x, y)$, centrality effects $\mu(x, y)$, depreciation $\delta$, and fiscal pressure $\tau$ . Analytical characterization of the steady states reveals a transcritical bifurcation in the parameter $\tau$ , separating inactive (low-investment) and active (self-sustaining) spatial regimes. The equilibrium pair $(V ^*, K^*)$ is shown to exist only when the effective decay rate $\alpha = r + \tau - \mu(x, y)$ exceeds a profitability threshold $\theta = \kappa + \delta / I_0$, and becomes locally unstable beyond this boundary. The introduction of diffusion, $D_V \Delta V$, stabilizes spatial dynamics and generates continuous gradients of land value and capital intensity, mitigating speculative clustering while preserving productive incentives. Numerical simulations confirm these analytical properties and display the emergence of spatially heterogeneous steady states driven by urban centrality and local productivity. The model also quantifies key aggregate outcomes, including dynamic tax revenues, adjusted capital-to-land ratios, and net present values under spatial heterogeneity and temporal discounting. Sensitivity analyses demonstrate that the main qualitative mechanisms-critical activation, spatial recomposition, and bifurcation structure-remain robust under alternative spatial profiles $(A, \mu)$, discretization schemes, and moderate differentiation of the tax rate $\tau (x, y)$. From an economic perspective, the results clarify the dual nature of the LVT: while it erodes unproductive rents and speculative land holding, its dynamic incidence on built capital depends on local profitability and financing constraints. The taxation parameter $\tau$ thus acts as a control variable in a nonlinear spatial system, shaping transitions between rent-driven and production-driven equilibria. Within a critical range around $\tau_c$, the LVT functions as an efficient spatial reallocation operator-reducing inequality in land values and investment density without impairing aggregate productivity. Beyond this range, excessive taxation induces systemic contraction and investment stagnation. Overall, this research bridges static urban tax theory with dynamic spatial economics by formalizing how a land-based fiscal instrument can reshape the geography of value creation through endogenous diffusion and nonlinear feedback. The framework provides a foundation for future extensions involving stochastic shocks, adaptive policy feedbacks, or endogenous public investment, offering a unified quantitative perspective on the dynamic efficiency and spatial equity of land value taxation.