The Theory of Water and Utility Pricing - Global Drinking Water Management and Conservation - page 79

possible if the capital cost is not provided by the central government. In such a
situation, it is appropriate to consider the “ second-best ” Ramsey Pricing.

y, Ramsey prices are prices that are Pareto optimal subject to a constraint
on the total pro

Brie

ts of a single supplier or group of suppliers. In particular, because a
utility whose activities are characterized by scale economies will lose money if it
sets the prices of its products equal to their marginal costs, Ramsey prices become
for that utility the prices that are optimal (economically ef cient) given the financial

feasibility requirement that the ts be non-negative. The same Ramsey

prices can also be shown to be those necessary for maximization of the sum of
consumers

rm ' s pro

' and producers ' surpluses.

5.5.1 Derivation of Ramsey Prices

The exposition of Ramsey pricing given here follows that of Baumoland Bradford
( 1970 ).
As before let x 1 ; ...; x n be the quantities of n goods produced by a natural
monopoly and let p 1 ; ...; p n be the corresponding prices. Let Z ð p 1 ; ...; p n Þ be the
consumer

' s indirect utility function. The natural monopolist now has a pro t

constraint:

P ð p 1 ; ...; p n Þ ¼ M ð 5 : 13 Þ

Maximize Z ðÞ subject to ( 5.13 ):

Max Zp 1 ; ...; p n ð Þ þ k ½M P p 1 ; ...; p n ð Þ

For a maximum,

Z

p i ¼ k o P

o

p i ;

i ¼ 1 ; 2 ; ...; n ð 5 : 14 Þ

o o

Equation ( 5.14 ) says that the marginal gain from a given price change must be
proportionate to the marginal profit cost. Equivalently, for all goods produced, the
ratio of marginal gain to marginal pro

t cost must be the same.
From consumer demand theory, we can also show that

o Z

p i ¼ x i

ð 5 : 15 Þ

o

Substitute into ( 5.14 ) to get

x i ¼ k o P
o p i

ð 5 : 16 Þ