hypothesis of additive utility functions, that is, utility function of each good consumed by aconsumer 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, therelations of substitution and Complementarity between goods are ruled out. Further, in derivingdemand curve or law of demand Marshall assumes the marginal utility of money expenditure(Mum) in general to remain constant.
We now proceed to derive demand curve from the law of equi-marginal utility. Consider the caseof a consumer who has a certain given income to spend on a number of goods. According to thelaw of equi-marginal utility, the consumer is in equilibrium in regard to his purchases of variousgoods 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 insuch a way that satisfies the following proportionality rule: MUx/ Px= MUy/ Py = MUmWhere MUm stands for marginal utility of money income in general.With a certain given income for money expenditure the consumer would have a certain marginalutility of money (Mum) in general. In order to attain the equilibrium position, according to theabove 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 (MUm)with MUx/ Pxof good X, with MUm/ PYof 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 MUx/ Pxwith MUy/ Pyand MUmin general would be disturbed.With the lower price than before MUx/ Pxwill be greater than MUy/ Py or MUm (It is assumed ofcourse that the marginal utility of money does not change as a result of the change in the price ofone good). Then, in order to restore the equality, marginal utility of X or MUxmust be reduced.And the marginal utility of X or MUxcan be reduced only by the consumer buying more of thegood X.It is thus clear from the proportionality rule that as the price of a good falls, its quantitydemanded will rise, other things remaining the same. This will make the demand curve for agood downward sloping. How the quantity purchased of a good increases with the fall in its priceand 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 MUx/ Pxis shown and on the X-axis the quantitydemanded of good X is shown. Given a certain income of the consumer, marginal utility ofmoney in general for him is equal to OH. The consumer is buying Oq1of good X when price isPx1 since at the quantity Oq1of X, marginal utility of money OH is equal to MUx/ Px1.