The Dynamics of Prices and Market Shares Over the Product Life Cycle

We analyze a duopoly through a differential game, in which the players set prices as functions of time. Under reasonable assumptions, we find that prices first decline, then increase. The market share of the biggest firm grows initially but decreases later. It is demonstrated that a firm may growth maximize early, but never late, in the product life cycle. Finally we show that only the low price firm will pay for informative advertising, whereas both firms will pay for persuasive advertising, though less if their market shares are very different.