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

TL;DR

The paper develops a spatially explicit, dynamic model of land value and built capital under a Land Value Tax, incorporating diffusion, centrality-driven productivity, and a profitability threshold. It demonstrates a transcritical bifurcation at a critical tax level, where active investment regimes emerge, and shows that diffusion dampens extreme clustering while preserving productive gradients. Through analytical results and robust numerical simulations, the study reveals how LVT can both erode unproductive rents and reallocate investment across space, with outcomes contingent on local centrality and financing constraints. A stochastic extension further links policy design to risk, enabling scenario-based assessment of dynamic efficiency and spatial equity under uncertainty.

Abstract

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 $μ(x, y)$, depreciation $δ$, and fiscal pressure $τ$ . Analytical characterization of the steady states reveals a transcritical bifurcation in the parameter $τ$ , 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 $α= r + τ- μ(x, y)$ exceeds a profitability threshold $θ= κ+ δ/ I_0$, and becomes locally unstable beyond this boundary. The introduction of diffusion, $D_V ΔV$, 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 $(A, μ)$, discretization schemes, and moderate differentiation of the tax rate $τ(x, y)$. 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 $τ$ 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 $τ_c$, 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.

Figures (14)

• Figure 1: Spatial diffusion and dynamic equilibrium under varying tax rates. The figure shows the contour maps of land value $V(x,y)$ (top row) and built capital $K(x,y)$ (bottom row) at the final steady state of the simulation for four tax levels $\tau \in \{0.0,\, 0.005,\, 0.01,\, 0.02\}$. For $\tau = 0$, both $V$ and $K$ concentrate strongly around the domain center, reproducing the initial Gaussian peak. As $\tau$ increases, the peak amplitude decreases significantly, indicating a progressive attenuation of land and capital accumulation. Higher tax rates reduce profitability and thus slow the concentration of built capital, leading to a more homogeneous spatial distribution, especially evident in $K(x,y)$. This pattern illustrates the erosion of built capital intensity under stronger land taxation within the diffusion–interaction framework.
• Figure 2: Temporal evolution of spatial averages of $V$ and $K$ under different LVT rates. Each pair of curves corresponds to the same tax level $\tau$, with solid lines representing the average land value $\langle V \rangle$ and dashed lines the average built capital $\langle K \rangle$. When $\tau = 0$, both $V$ and $K$ exhibit strong and sustained growth over time, reflecting an unconstrained accumulation dynamic in the absence of taxation. As $\tau$ increases, the growth trajectories flatten markedly, revealing the damping effect of the land value tax on both capital accumulation and land appreciation. Higher tax rates progressively reduce the slopes of both curves, indicating a systematic decline in the long-run growth potential of urban land and built structures.
• Figure 3: Bifurcation diagram of final spatial averages versus LVT rate. The figure illustrates the inverse relationship between the LVT rate $\tau$ and the steady-state levels of land value and built capital. In the absence of taxation ($\tau = 0$), both $\langle V \rangle$ and $\langle K \rangle$ reach high equilibrium values. As $\tau$ increases, both averages decline monotonically, confirming that the land value tax progressively reduces profitability and slows the accumulation of built capital. The decline is steeper for $K$, suggesting that LVT affects investment dynamics more intensely than pure land valuation. This outcome aligns with theoretical expectations: higher land taxation discourages excessive building investment and moderates speculative accumulation, leading to a more stable and spatially balanced urban equilibrium.
• Figure 4: Local bifurcation of the steady states $(V^*, K^*)$ with respect to the tax rate $\tau$ for three levels of local centrality $\mu$. The vertical dashed lines indicate the corresponding critical thresholds $\tau_c(x,y) = \theta - r + \mu(x,y)$. Central locations (high $\mu$) tolerate higher tax rates before the collapse of the interior branch, while peripheral locations lose the interior equilibrium for lower $\tau$, which is consistent with the analytical condition $\alpha = r + \tau - \mu = \theta$.
• Figure 5: Criticality profiles $\tau - \tau_c(d)$ for multiple tax rates $\tau$. Each curve shows how the distance to the critical threshold $\tau_c(d)$ evolves with spatial distance $d$ from the urban center. For low $\tau$, $\tau - \tau_c(d)$ remains negative across most of the space (sub–critical regime, rent persistence). As $\tau$ rises, the peripheral areas cross the critical line first (near $\tau = 0.12$), indicating the onset of investment activation outside the center. Higher tax rates ($\tau = 0.16$ and $\tau = 0.22$) shift the entire domain into a super–critical regime, consistent with a global erosion of land rents.