. Let demand be dependent on the Networks in Telecommunications p(w, K) D(p,w) Figure 3.6. The spot price of network capacity or the state-contingent Arrow-Debreu security price. realization of the random variable, D(p,w). Then the spot price is given by the price p(w, K) that clears the market: D(p(w,K),a) = K, if D(0,w) > K. Otherwise, if D(0, w) < K, the spot price of capacity equals zero. The spot price also can respond to random variations in capacity associated with changes in the provision of capacity or reductions in network capacity that result from failures and maintenance. Suppose that contingent contracts are established such that individual con- sumers receive capacity allocations Q'(w) = D'(p(w) K), w), i=1,...,I, in state W. Note that total capacity allotments sum to total demand, which in turn equals total capacity K in states of the world with a positive spot price and equals D(0, w) in states of the world with a zero spot price. The state-contingent forward contract is priced using Arrow-Debreu state-contingent prices p(w, K) that correspond to the spot prices. Thus, consumer i purchases the state-contingent capacity allocation Q'(w) at the state-contingent price p(w, K); see Figure 3.6. 3. Time-of-Use Pricing of Capacity Capacity must serve customers over time. Demand for network services may vary by time of day, day of the week, or even seasonally, in the case of energy transmission. Pricing Network Services In telecommunications, prices for network services often vary by time of day and day of the week. Capacity must be sufficient to meet demand during the time period when demand is greatest, and correspondingly, excess capacity is present in other time periods. By pricing to reduce demand at the peak, the network firm reduces the amount of capacity required to service customers. Pricing that smooths the pattern of demand over time is referred to as peak-load pricing, or more generally, time-of-use pricing. Typically a consumer uses some network services in each period. Suppose that there are