Product quality, competition, and multi-purchasing

In a Hotelling duopoly model, we introduce quality that is more appreciated by closer consumers. Then higher common quality raises equilibrium prices, in contrast to the standard neutrality result. Furthermore, we allow consumers to buy one out of two goods (single-purchase) or both (multi-purchase). Prices are strategically independent when some consumers multi-purchase because suppliers price the incremental benefit to marginal consumers. In a multi-purchase regime, there is a hump-shaped relationship between equilibrium prices and quality when quality functions overlap. If quality is sufficiently good, it might be a dominant strategy for each supplier to price high and eliminate multi-purchase.