Suppose that a public utility supplies a service, whose demand varies with the time of day. For simplicity, assume that demand in each period is independent of the price in other periods. The inverse demand function for each period is pi(yi). Assume that marginal production costs ci are constant, independent of capacity and independent across periods. Further assume that the marginal cost of capacity c0 is constant. With these assumptions, the total cost function is

The objective is to determine outputs yi(and hence prices pi) and production capacity Y to maximize social welfare as measured by total consumer and producer surplus. In any period i, total surplus is measured by the area between the demand and cost curves, that is,

So aggregate surplus is

The optimization problem is to choose nonnegative yi and Y so as to maximize (79) subject to the constraints
yi Show that it is optimal to price at marginal cost during o¨-peak periods, and extract a premium during peak periods, where the total premium is equal to the marginal cost of capacity c0. Furthermore, under this pricing rule, the enterprise will break even. Note that

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P(T) ci) dt co Y (79)