The Theory of Water and Utility Pricing - Global Drinking Water Management and Conservation - page 83

5.5.2 Ramsey Pricing Expressed as Covering Capital Costs

Ramsey pricing can be expressed in a variety of formulae all of which are equiv-
alent. For the purpose of this chapter, we can outline Ramsey pricing by consid-
ering only two goods supplied by a public utility, that might receive a capital
subsidy S, faces a capital cost K, and price Pi, i , marginal cost MCi, i , and marginal
revenue MR i , for i =1,2.
Using this notation, we can summarize Ramsey as follows. Maximizing the sum
of consumer and producer surplus,

it can be shown that, at the constrained

optimum:

P 1 MC 1
P 2 MC 2 ¼

MR 1 MC 1
MR 2 MC 2

ð 5 : 22 Þ

X

2

subject to P i Q i ¼ c ð Q 1 ; Q 2 Þþ K S ð 5 : 23 Þ

i ¼ 1

where c ð Q 1 ; Q 2 Þ is the total cost of production, so that the constraint represents total

revenue equal to total cost plus capital costs (K), minus any capital subsidy (S), if
positive. An equivalent statement of the optimum in Eq. ( 5.22 ) is:

P 1 MC 1 = P 1 E 2

E 1 P 2 ¼

ð 5 : 24 Þ

P 2 MC 2 =

X

2

subject to P i Q i ¼ c ð Q 1 ; Q 2 Þþ K S ð 5 : 25 Þ

i ¼ 1

where E i is the price elasticity of user i.
In either formulation, Pi i are the Ramsey prices. Formulation ( 5.24 ) is also known
as the inverse elasticity rule. Suppose index i = 1 stands for the industrial users of
water and i = 2 represents the residential users. If the elasticity of industrial users
equals

σ , it follows that

rð P 1 MC Þ= P 1 ¼ ð P 2 MC Þ= P 2 ð 5 : 26 Þ

ð P 1 MC Þ= P is a markup over MC required to cover average costs as well as the

capital cost, net of any capital subsidy. Suppose = 2, i.e. the elasticity of industrial

users is twice that of residential users, then the markup on industrial users will be
half of that of residential users.
The difference between Pi i and MC can also be viewed as the constrained optimal
Ramsey commodity tax. If the subsidy S is zero, then the tax must fully cover all
costs.

σ