Optimal Dynamic Advertising Policies in Digital and Traditional Channels: A Control-Theoretic Approach

This study applies optimal control theory to investigate a monopolistic firm’s optimal allocation of advertising efforts across digital and traditional channels. By considering the competitive relationship between advertising efforts in different channels in satisfying consumers’ informational needs, this study explicitly models their substitution effect. Furthermore, we propose an alternative approach to incorporate different decay rates of incremental goodwill in the two channels, allowing the system dynamics to be directly represented by the firm’s total goodwill without separating it into multiple channel-specific components. Technically, this approach leads to the system dynamics being governed by an integro-differential equation rather than an ordinary differential equation. Our analysis reveals that the marginal value of goodwill in the digital channel is greater than that in the traditional channel due to a lower decay rate. However, this comparative advantage of the digital channel progressively diminishes over time. As a result, the firm should always invest in digital advertising, while employing traditional advertising only when the comparative advantage of the digital channel becomes weak in later stages. When additionally considering the synergistic effect between the two channels, the optimal adoption timing of traditional advertising occurs earlier as the intensity of synergistic effect increases.