Knowledge that Transforms

In their recent work on public utility pricing in this Review, Gardner Brown, Jr. and M. Bruce Johnson failed to point out one surprising implication of their modelnamely, that peak-load pricing is inefficient under uncertainty. To see this we simply postulate the existence of peak and off-peak stochastic demand functions, denoted by X(P)+u and Y(P)+v, respectively, where u and v are random disturbances with zero mean and finite variance.' Since X(P)+u is the peak demand, we assume X(P) > Y(P) for all P satisfying 0< P <P*, where X(P*) =0. The period of analysis is divided into peak and off-peak periods which take up fractions of the total period equal to e. and (1-er), respectively. Finally, let P. be the peak price while Pv is the off-peak price. The public utility seeks to maximize the weighted sum of expected consumers' surplus and expected net revenue during peak and off-peak periods, with the weights being ex and (1 -ex). We retain the assumption of constant marginal operating cost (= b) and constant marginal capacity cost. Where W is the weighted sum of expectations described above, first-order conditions for a maximum of W with respect to Px and P, are: