Optimal advertising for a generalized Vidale–Wolfe response model

In this research, we formulate budget allocation decisions as an optimal control problem using a generalized Vidale–Wolfe model (GVW) as its advertising dynamics under a finite time horizon. One key element of our modeling work is that the proposed optimal budget allocation model (called GVW-OB) takes into account the roles of two useful indexes of the GVW model representing the advertising elasticity and the word-of-mouth (WoM) effect, respectively, in determining optimal budget. Moreover, we discuss desirable properties and provide a feasible solution to our GVW-OB model. We conduct computational experiments to assess our model’s performance and its identified properties, based on real-world datasets obtained from advertising campaigns by three e-commerce companies on Google AdWords, Facebook Ads and Baidu Ads, respectively. Experimental results show that (1) our GVW-OB strategy outperforms four baselines in terms of both payoff and ROI in either concave or S-shaped settings; (2) linear budget allocation strategies favor concave advertising responses, while nonlinear strategies support S-shaped responses; (3) a larger ad elasticity empowers higher levels of optimal budget and corresponding market share and thus achieves higher payoff and ROI, so does a larger WoM effect; and (4) as the total budget increases, the resulting payoff by the GVW-OB strategy increases monotonically, but the ROI decreases, which is consistent with the law of diminishing marginal utility. From a methodological perspective, our GVW-OB strategy provides a feasible solution for advertisers to make optimal budget allocation over time, which can be easily applied to a variety of advertising media. The identified properties and experimental findings of this research illuminate critical managerial insights for advertisers and media providers.