national multiple family submetering and allocation billing program ...

∂L
∂g
∂L
∂x
∂L
∂λ
= 0 ⇒
∂u
∂g
= λ p
= 0 ⇒
∂u
∂x
= λ p
= 0 ⇒ m = p g + p x.
g
g
x
x
∂u
and
∂g
∂u
are the marginal utilities of water and other goods respectively. λ is the
∂x
Lagrange multiplier that can be interpreted as the additional utility that a consumer can gain as a
result of having one more dollar to spend. Partial differentiation of the Lagrangean function with
respect to λ yields the budget constraint. By rearranging the first two equations we have
∂u ∂g ∂u ∂x
= λ and = λ
p
p
or
g
∂u ∂g ∂u ∂x
= .
p p
g
x
x
This indicates that the consumer achieves maximum satisfaction when the marginal
utility per dollar from consumption is the same across all goods and services. The solution to the
consumer choice problem is a set of demand equations for all goods and services involved. The
demand for each good is a function of all prices and the amount of money available for spending.
g = g( p , p , m)
x=
x( p , p , m)
g
g
x
x
where
g
and x are demand for water and all other goods respectively.
207