Spatiotemporal Allocation of Advertising Budgets
基于实证模型，研究如何最优分配全国和区域广告预算以最大化利润，通过求解HJB方程得到闭式解，并用德国化妆品公司数据验证，利润提升5.07%。
This article develops a method for optimal allocation of resources based on an empirically validated model of how national and regional advertising generate sales over time. The authors derive the profit-maximizing total budget, its optimal split between national and regional spends, and its optimal allocation across multiple regions. They formulate a spatiotemporal model that accounts for spatial and serial dependence, spatial heterogeneity, neighborhood effects, and sales dynamics. Because of spatial and serial dependence, correlated multivariate Brownian motion drives the sales dynamics, resulting in a second-order differential equation for the Hamilton–Jacobi–Bellman (HJB) equation with multiple states (i.e., regional sales) and multiple controls (i.e., regional and national advertising expenditures). By solving the HJB equation analytically, the authors furnish closed-form expressions for the optimal total budget and its regional allocations. In addition, they develop a method to estimate the proposed model and apply it to market data from a leading German cosmetics company. Using the estimated parameters, they evaluate the optimal budget and allocations. Comparing them with actual company policy, the proposed approach enhances profit by 5.07%, and it not only identifies which regions under-or overspend but also reveals how much budget to shift from national to regional advertising (or vice versa).