First and Second-Best Pricing with Uncertain Demand

As has been well known, since at least the work of Hotelling [6], marginal cost pricing is the ideal regulatory policy in a simple certainty setting. However, when economies of scale exist in production, such a policy will in general entail a revenue shortfall for the firm. Ideally, the regulator should meet such a short fall by raising lump-sum taxes; but that solution is liable to prove infeasible. Ramsey pricing, popularized by Baumol and Bradford [4], examines the second-best pricing policy when firms receive no tax subsidy, so prices must exceed marginal costs. The celebrated inverse elasticity rule shows that with demands for the various goods independent, goods whose elasticity of demand is proportionately less should have their price exceed marginal cost proportionately more. Further complications arise when one takes into account the uncertainty involved in economic decisions. In recognition of this, there has recently been considerable discussion of first-best pricing with stochastic demand [5]. The general conclusion of this discussion is that marginal cost pricing continues to be the first-best policy. In order to reach this conclusion however, the literature has in effect assumed that regulators are risk neutral. Compared to this large literature on stochastic first-best pricing, the only two papers known to us that treat second-best pricing with stochastic demand are by Sherman & Visscher [11] and by Wecker & Upton [15]. Both these treatments assume the marginal utility of income of the regulator is constant and so also require that there be risk neutrality. In this work, on the other hand, we will consider both first and second-best pricing policies when the regulator is explicitly allowed to be risk-averse. In the former case we will be interested in whether marginal cost pricing will continue to be the first-best policy, whereas in the latter case we will examine whether the simple inverse elasticity rule can hold despite considerations of risk and the risk behavior of regulators. Finally, we will directly relate the results of our first-best analysis to our study of the second-best problem. In much of the paper, the source of demand uncertainty will be arbitrary; we will