The Theory of Water and Utility Pricing - Global Drinking Water Management and Conservation - page 80

or

1
k x i o P

1

¼

p i

o

For any other good j, we have

1
k

x j o P

1

¼

o p j

Equating them, we get

x i o P

1

x j o P

1

p i ¼

ð 5 : 17 Þ

p j

o o

Equation ( 5.17 ) says that the ratio of marginal pro t is the same as the ratio of

the two output quantities. This is a Ramsey result.
The same result can be expressed as a price deviating from marginal cost (MC).
Let marginal revenue for product i be MRi, i , and let Ei i be the price elasticity of
demand for good i.
Note that MR i is:

MR i ¼ p i þ x i o p i
o

ð 5 : 18 Þ

x i

and

o P
o p i ¼ MR i MC i

dx i
dp i

ð Þ

dx i
dp i

ð 5 : 19 Þ

p i

p i þ x i o

¼

x i MC i o

Substitute ( 5.19 ) into ( 5.16 ):

x i dp i

dx i ¼ kð p i þ x i dp i

dx i MC i Þ

Now adding ð p i þ x i d p i

d x i MC i Þ to both sides of the above equation, we get
p i MC i ¼ ð 1 þ kÞð p i þ x i dp i

dx i MC i Þ

ð 5 : 20 Þ

p i MC i ¼ 1 þ k ð Þ MR i MC i ð Þ

Equation ( 5.20 ) says that the difference between price and marginal cost should
be proportionate to the difference between marginal revenue and marginal cost.