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Dr. Alfred Marshal was of the view that the law of demand and the demand curve can be derived with the help of utility analysis. He explained the derivation of demand curve in following two cases:

(i) In case of a single commodity.

(ii) In case of two or more than two commodities.

Assumptions:

In the derivation of demand curve by utility analysis, the following assumptions are made:

(i) Utility is cardinally measurable.

(ii) Utilities of different commodities are independent.

(iii) The marginal utility of money to the consumer remains constant.

(iv) Utility gained from the successive units of a commodity diminishes.

Explanation with Diagram:

Marshall explained the derivation of demand curve in following two cases:

(1) Derivation of Demand Curve in Case of Single Commodity (Law of Diminishing Marginal Utility):

Dr. Alfred Marshall derived the demand curve with the aid of law of diminishing marginal utility. The law of diminishing marginal utility states that as the consumer purchases more and more units of a commodity, he gets less and less utility from the successive units of the expenditure. At the same time, as the consumer purchases more and more units of one commodity, then lesser and lesser amount of money is left with him to buy other goods and services.

A rational consumer, before, while purchasing a commodity compares the price of the commodity which he has to pay with the utility of a commodity he receives from it. So long as the marginal utility of a commodity is higher than its price (MUx > Px), the consumer would demand more and more units of it till its marginal utility is equal to its price MUx = Px or the equilibrium condition is established.

To put it differently, as the consumer, consumes more and more units of a commodity, its marginal utility goes on diminishing. So it is only at a diminishing price at which the consumer would like to demand more and more units of a commodity.

Diagram:

In fig. 2.4 (a) the MUx is negatively slopped. It shows that as the consumer acquires larger quantities of good x, its marginal utility diminishes. Consequently at diminishing price, the quantity demanded of the good x increases as is shown in fig. 2.4 (b).

At X1 quantity, the marginal utility of the good is MU1. This is equal to P1 by definition. The consumer here demands OX1 quantity of the commodity at P1 price. In the same way X2 quantity of the good is equal to P2. Here, at P2 price, the consumer will buy OX2 quantity of commodity. At X3 quantity the marginal utility is MU3, which is equal to P3. At P3, the consumer will buy OX3 quantity and so on.

We conclude from above, that as the purchase of the units of the commodity X are increased, its marginal utility diminishes. So at diminishing price, the quantity demanded of good X increases as is evident from fig. 2.4 (b). The rational supports the notion of down slopping demand curve that when price falls, other things remaining the same, the quantity demanded of a good increases and vice verse. (The negative section of the MU curve does not form part of the demand curve, since negative quantities do not make sense in economics).

(2) Derivation of the Demand Curve in Case of Two or more than Two Commodities (Law of Equilibrium Marginal Utility):

The law of diminishing marginal utility can also be applied in case of two or more than two goods. When a consumer has to spend a certain given income on a number of goods, he attains maximum satisfaction when the marginal utilities of the goods are proportional to their prices as stated below:

Diagram:

In the fig. 2.5 (a), (b) and (c) given the money income, the price of X commodity (Px) and the price of Y commodity (PY) and constant marginal utility of money (MUM), the demand curve derived is illustrated. The consumer allocates his money income between X and Y commodities to get OQ1 units of good X and OY unit of good Y commodities because the combination correspondence to constant at the OM level.

Let us assume that money income and price of Y commodity remain constant but the price of X commodity decreases. As a result of this money expenditure on commodity X rises resulting (MUx / Px) curve to shift towards right. The consumer now allocates his income to OQ2 quantity of X commodity and OY quantity of Y commodity because the combinations correspondence to constant at OM level.

Thus in response to decrease in the price from PX to PX1 (fig. 2.5 (c)) the quantity demanded of a good X increases from OQ1 to OQ2. The DD is a negatively sloped demand curve.