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

In today’s complex and dynamic market environment, simultaneously deploying and optimizing multiple advertising channels is crucial for firms’ success. In the present study, we apply optimal control theory to address the multichannel advertising optimization problem for a monopolistic firm that manages a digital ad channel and a traditional ad channel. 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 account for 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 ad channel is greater than that in the traditional ad channel due to a lower decay rate. However, this comparative advantage of the digital ad channel progressively diminishes over time. As a result, the firm should always invest in digital advertising, whereas employing traditional advertising only when the comparative advantage of the digital ad channel becomes weak in later stages. When additionally considering the synergistic effect between advertising efforts in the two channels, we find that as the intensity of synergistic effect increases, it is optimal to adopt traditional advertising earlier; when the synergistic effect is sufficiently strong, the optimal traditional advertising effort will remain positive throughout the planning horizon.