Suppose there exists a natural monopoly firm and its regulator.
Throughout this problem (except where explicitly indicated), assume that demand for the product is perfectly inelastic at a single unit and that everyone is risk-neutral. Rather than putting equal weights on the profits of the firm and consumer welfare,the regulator places a weight 1 on the firm and (1 + Lamda) on consumers. A regulator can observe the firms cost "F" and can regulate it to charge prices at average cost if it
chooses. In particular, the regulator makes the firm charge a price alpha(F_o) + (1-alpha)"F", where alpha is the regulator's choice. A high alpha lets the firm keep most of the benefits associated with reducing its costs, while a low alpha forces it to always charge prices near its average cost.
Suppose that the firm currently has a total cost of producing the one unit of F_o but that it can be reduced to F at a smooth cost C(F_o-F) where C', C''>0, and C(0)=0. Derive an intuitive expression for the optimal value of alpha and relate it other concepts in the course thus far.