Scaling Laws and Nonlinear Dynamics in Productive Zakat Allocation: A Complex System Modeling Approach for the Qassim Region

The research proposes a non-linear, scenario-based mathematical model for the collection and distribution of zakat in order to assess its socio-economic impacts in the Qassim region of Saudi Arabia for the period 2020-2024. The model represents the process of zakat distribution as a welfare maximization problem subject to the constraint that the entire amount collected in the form of zakat Formula: see text must be distributed among all eight asnaf groups using decision variables Formula: see text according to the budget constraint Formula: see text The aggregate welfare is defined as a weighted sum of category-specific contributions, Formula: see text where each function follows a nonlinear power-law specification Formula: see text capturing diminishing marginal returns while exhibiting scaling-law behavior characteristic of multiscale socio-economic systems. The analysis is based on two policy schemes: one for complete consumption and another for production with a mixed scheme where zakat money is used for production purposes up to Formula: see text, where Formula: see text. The welfare benefit is increased by Formula: see text where Formula: see text is the level of cumulative productive support while Formula: see text is the intensity of feedback, which adds nonlinearity to the allocation process. Under each scenario, the model calculates the optimal level of allocation together with welfare measures using artificial yet realistic data in line with aggregate levels within the region. Sensitivity analysis on allocations Formula: see text and parameters Formula: see text shows the robustness of qualitative results as well as the impact of scaling factors on welfare measures. The analysis shows that when about 30% of zakat funds are used for productive investment, then the welfare level will be higher as a result of increased incomes and decreased dependence in such a configuration. This is viewed as an example of emergent properties of nonlinear allocation systems based on power laws and feedbacks. This research adds to the existing literature a clear quantitative model that involves optimization, scaling, and policy implications, which could be helpful for policy-making organizations involved in zakat allocations in line with Saudi Vision 2030.