Optimal dynamic pricing of multi-generation products

We study the optimal dynamic pricing of multi-generation products. A two-stage sales diffusion model is proposed: only the original product is sold in the market in the first stage, and the updated product enters the market in the second stage. The optimal control model is converted into a non-linear programming model and solved by using Guass-Pseudospectral method. The numerical results show that the price of the original product stays unchanged in the first stage, and drops at the beginning of the second stage; the price of updated product is higher than that of the original product; the prices of both products decrease with similar speed in the second stage. It shows that the firm can promote sales and increase profit by enhancing the coefficient of innovation, shortening the updating cycle, and reducing the marginal cost of updated product.