Does buyer heterogeneity steepen or flatten quantity discounts?

This article argues that as the distribution of a firm's buyers becomes more heterogeneous, the firm's profit-maximizing quantity-discount schedule becomes less steep. First, we note that one measure of heterogeneity is the slope of the hazard function, expressed in terms of a simple crossing condition. We then show that marginal price schedules, for distributions of buyers which are more heterogeneous by this measure, are less negatively sloped in that they cross schedules for more homogeneous distributions from below. Intuitively, quantity discounts are a response to an individual buyer's declining marginal utilities, and buyer heterogeneity interferes with this response.