Hypothesis of additive utility functions that is

hypothesis of additive utility functions, that is, utility function of each good consumed by a
consumer does not depend on the quantity consumed of any other good.
As has already been noted, in case of independent utilities or additive utility functions, the
relations of substitution and Complementarity between goods are ruled out. Further, in deriving
demand curve or law of demand Marshall assumes the marginal utility of money expenditure
(Mu
m
) in general to remain constant.

We now proceed to derive demand curve from the law of equi-marginal utility. Consider the case
of a consumer who has a certain given income to spend on a number of goods. According to the
law of equi-marginal utility, the consumer is in equilibrium in regard to his purchases of various
goods when marginal utilities of the goods are proportional to their prices.
Thus, the consumer is in equilibrium when he is buying the quantities of the two goods in
such a way that satisfies the following proportionality rule:
MU
x
/ P
x
= MU
y
/ P
y
= MU
m
Where MU
m
stands for marginal utility of money income in general.
With a certain given income for money expenditure the consumer would have a certain marginal
utility of money (Mu
m
) in general. In order to attain the equilibrium position, according to the
above proportionality rule, the consumer will equalise his marginal utility of money
(expenditure) with the ratio of the marginal utility and the price of each commodity he buys.
It follows therefore that a rational consumer will equalise the marginal utility of money (MU
m
)
with MU
x
/ P
x
of good X, with MU
m
/ P
Y
of good 7 and so on. Given Ceteris Paribus assumption,
suppose price of good X falls. With the fall in the price of good X, the price of good Y,
consumer’s income and tastes remaining unchanged, the equality of the MU
x
/ P
x
with MU
y
/ P
y
and MU
m
in general would be disturbed.
With the lower price than before MU
x
/ P
x
will be greater than MU
y
/ P
y
or MU
m
(It is assumed of
course that the marginal utility of money does not change as a result of the change in the price of
one good). Then, in order to restore the equality, marginal utility of X or MU
x
must be reduced.
And the marginal utility of X or MU
x
can be reduced only by the consumer buying more of the
good X.
It is thus clear from the proportionality rule that as the price of a good falls, its quantity
demanded will rise, other things remaining the same. This will make the demand curve for a
good downward sloping. How the quantity purchased of a good increases with the fall in its price
and also how the demand curve is derived in the cardinal utility analysis is illustrated in Fig. 1.3.
Figure 1.3:Derivation of Demand Curve

In the upper portion of Fig. 1.3, on the Y-axis MU
x
/ P
x
is shown and on the X-axis the quantity
demanded of good X is shown. Given a certain income of the consumer, marginal utility of
money in general for him is equal to OH. The consumer is buying Oq
1
of good X when price is
P
x1
since at the quantity Oq
1
of X, marginal utility of money OH is equal to MU
x
/ P
x1
.