Monopole, stratégies multiproduit et incitation à la recherche et développement

In the framework of a vertical differentiation model where consumers are continuously distributed with respect to their intensity of preference for quality and their income, we studdy the optimal strategy of a natural monopoly: how many qualities to produce? Which qualities should be produced? What prices have to be set to maximize the monopoly profit? Considering costless production, the monopoly optimal strategies are such that the consumers with the same income buy the same quality. The monopoly never discriminates with respect to the intensities of preference for quality. If intensities of preference for quality are very low, the monopoly offers the highest quality and all the consumers buy. Else, the monopoly, offers an infinity of qualities. However, according to the model's parameters, we can have a discrimination with respect to the incomes such that the "poor" consumers don't buy. If the highest quality is not sufficiently high, the monopoly can't extract all the surplus of the "rich" consumers. This may be interpreted as an incitation to investment in research and development to improve the quality.