-peak consumption requirements. The time-of-use price in any period equals the marginal benefit in that time period if consumption is at full capacity, as shown in Figure 3.8. Prices are greater in periods with higher benefits of consump- tion but are positive in all periods if there are positive marginal benefits at Qt=K*. Thus, it should not be assumed that capacity is free off peak. Off-peak usage at full Networks in Telecommunications B|(Q) Figure 3.8. The time-of-use price in period t equals the marginal benefit of consumption at the full-capacity level. capacity still signifies scarce capacity, even if marginal benefits are lower. Because all periods with scarce capacity contribute to capacity payments, the efficient level of capacity is not just dependent on peak demand. Accordingly, additional usage result- ing from off-peak access should require additions to the capacity of the network. B. Network Capacity Pricing under Asymmetric Information 1. Vickrey Auctions of Capacity Suppose that the demand functions of individual consumers are unobservable and suppose that the distribution of consumer types is unknown to the firm. Then, as a consequence, the firm does not know aggregate demand. Suppose, for ease of presentation, that each consumer has a demand for only one unit of service. 5 A consumer i has a benefit function defined by Suppose that the taste parameter b is a positive, real-valued random draw from a cumulative distribution function F(b), with density of f(b), defined on the unit interval. Capacity is available in K discrete units. Suppose that the number of 5 For generalization to downward sloping demand, see Spulber (1992, 1993a, 1993b). Pricing Network Services bI- 1 p1-K*=b1-K Figure 3.9. Demand for network capacity when the number of consumers I exceeds the number of units of network capacity K. consumers is greater than the capacity, SO that some type of rationing is necessary, I>) K. Given these assumptions, the demand for capacity is a step function composed of the list of realized valuat