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Optimal control theory is employed to derive explicitly the optimal (profit‐maximizing) price of a durable new product over time. The sales rate dynamics depends on the product price and on the unsold portion of the market. Specifically, the hazard rate (i.e. the probability of a purchase by a new customer) increases as the price decreases in a linear fashion. It is shown that both the price and sales rate decline over time for finite horizon problems with or without discounting. In the discounted infinite horizon case, the price remains constant over time. When there is no discounting in the infinite horizon case, there does not exist an optimal solution. However, it is possible in this case to attain a profit level which is arbitrarily close to the theoretical, albeit unattainable, maximum level profit. Economic interpretations are provided for the various quantities that arise in the course of solving the optimal pricing problem.
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Sethi et al. (2003) studied this question.
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