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Pricing of Differentiated Services: Apparent Loss and Welfare

The results in this paper address the question of how to evaluate and implement optimal price-lead time menus for congested service providers when market knowledge is limited. We formulate the economic assignment problem for the allocation of lead-times to customers with two-dimensional types, heterogeneous in both value for service and sensitivity to delay, as a deterministic optimization. We identify a novel measure, apparent loss, which is the difference between current revenue and the maximum feasible revenue under the current pricing scheme. For systems where total delay is a convex function of customer arrival rate, the apparent loss is shown to have the following characteristics: it is an upper bound on the difference between current and maximum welfare, minimizing apparent loss also minimizes welfare, and it can be calculated from operational information. We argue that the apparent loss is a useful heuristic for evaluating the quality of a particular pricing scheme when market information is costly or not available. We also demonstrate how apparent loss may be used as a measure to guide an adaptive pricing algorithm. The adaptive pricing algorithm is initialized with a known set of feasible prices and shown through computational experiments to find pricing schemes which deliver near optimal welfare.