Optimal Dynamic Advertising Policies in Digital and Traditional Channels: A Control-Theoretic Approach
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.007 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".