Appendix 1: Mathematical Explanation for the Regulation of Natural Monopolies
The numerical explanation for regulation of natural monopoly is as follows. First, we consider simple situations. The inverse demand function for the public utility service is p (Q ) Q , where the price of the service is p (Q ) , and the quantity of the public utility service is Q . This monopoly firm decides a service output level attaining the maximum profit. The monopoly firm’s cost function is expressed as C (Q) c Q f . Therefore, the profit of the monopoly firm is expressed as (Q) p(Q) Q C(Q) ( Q) Q (c Q f ) . The behavior of the monopoly firm is to maximize profit. Therefore, the object function of the firm is as follows. m a x (Q) ( Q) Q (c Q f ) . Q (A.1.1) The first order condition for profit maximization is d dQ ( 2 Q) c 0 . (A.1.2) It is worth noting that this profit maximizing condition shows marginal revenues ( MR 2 Q ) as equal to marginal cost ( MC c ). From the equation above, the service output provided by the monopoly firm, QM , and the price, p M , are QM ( c) 2 p M ( c) 2 . (A.1.3) (A.1.4) As for the social optimal, we assume that social welfare, W (Q) , is the sum of consumer surplus, CS (Q) p(Q)dQ p(Q)Q , and the profit 237 238 Regulatory Reform of Public Utilities of the firm, (Q) . Therefore, the welfare function for society is expressed as W (Q) CS(Q) (Q) . The regulator decides the service output...
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