Basic Pricing Theory

This chapter provides an introduction to multi-product monopoly pricing when the variable costs are linear. Profit maximization problems with linear variable costs arise from capacity constraints, where the firm maximizes the expected profit net of the opportunity costs of the capacities used. We argue that under mild assumptions, both the optimal profit function and the expected consumer surplus are convex functions of the variable costs. Consequently, when variable costs are random, both the firm and the representative consumer benefit from prices that dynamically respond to changes in variable costs. Randomness in variable cost is often driven by randomness in demand in conjunction with capacity constraints, and this accounts for some of the benefits of dynamic pricing. We explore conditions for the existence and uniqueness of maximizers of the expected profit and analyze in detail problems with capacity constraints both when prices are set for the entire sales horizon a priori, and when prices are allowed to change during the sales horizon. The firm’s problem is discussed in Sect. 8.2, while the representative consumer’s problem is presented in Sect. 8.3. The case with finite capacity is discussed in Sect. 8.4. Details about existence and uniqueness for single product problems are discussed in Sect. 8.5. This section also includes applications to priority pricing, social planning, multiple market segments, and peak-load pricing. Multi-product pricing problems are discussed in Sect. 8.6.