# Appendix 1: Mathematical Explanation for the Regulation of Natural Monopolies

## Fumitoshi Mizutani

## Extract

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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