Abstract
The purpose of the paper is to determine the behavior of Nash optimal open-loop advertising strategies of a dynamic duopoly. The logarithmic two-person nonzero-sum advertising differential game analyzed byJørgensen [1982] is extended to more general nonlinear functionsg (u 1,u 2) measuring the effectiveness of advertising expendituresu 1,u 2 of two firms with respect to the development of the market shares over time. It is shown that under the assumption of zero cross effects of the advertising efficiency function the main features of Jørgensen's model are preserved. Furthermore, two models with advertising rates interacting asymmetrically with respect to the marginal advertising effectiveness are studied. For zero salvage values of terminal market shares it is shown that the Nash optimal advertising expenditures of a small firm are always downward sloping, whereas the shape of a large competitor's advertising rate can only be determined under additional assumptions.
Zusammenfassung
Zweck der vorliegenden Arbeit ist die Ermittlung des Verhaltens von Nashoptimalen open-loop Werbepolitiken in einem dynamischen Duopol. Das logarithmische Zweipersonen Nichtnullsummen Werbe-Differentialspiel vonJørgensen [1982] wird auf allgemeinere nichtlineare Funktioneng (u 1,u 2) erweitert, welche die Effizienz der Werbeausgabenu 1,u 2 der beiden Firmen im Hinblick auf die Entwicklung ihres Marktanteils beschreiben. Es wird gezeigt, daß die wesentlichen Resultate des Modells von Jørgensen erhalten bleiben, wenn man annimmt, daß die gegnerischen Werbeausgaben die marginale Effizienz der eigenen Werbung nicht beeinflussen. Weiters werden Modelle untersucht, in denen die marginale Werbeeffizienz einer Firma durch die Werbung der anderen Firma negativ bzw. positiv beeinflußt wird. Falls der Marktanteil am Ende des Planungshorizonts neutral bewertet wird, so wird gezeigt, daß die Nash-optimale Lösung einer kleinen Firam stets abnimmt, während die Gestalt der Werbeausgaben einer großen gegnerischen Firma nur unter zusätzlichen Annahmen bestimmt werden kann.
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Feichtinger, G., Dockner, E. A note to Jørgensen's logarithmic advertising differential game. Zeitschrift für Operations Research 28, B133–B153 (1984). https://doi.org/10.1007/BF01922147
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DOI: https://doi.org/10.1007/BF01922147