Relationship between Marginal Rate of Substitution and Marginal Utility of Ordinal Utility Approach

Diminishing marginal utility law under utility analysis has been replaced by the principle of diminishing marginal rate of substitution. Diminishing marginal rate of substitution is the main force behind the consumer’s equilibrium. According to Hicks, equilibrium will not be stable, unless at that point, the marginal rate of substitution is diminishing. Marginal rate of substi­tution can be known from the ratio of the marginal utilities of the two commodities.

As the total satisfaction remains same on every point of the indifference curve, so loss in satisfaction (utility) from a small reduction in commodity ‘Y’ will be equal to the gain in satisfaction (utility) from a small increment in commodity ‘X’. It can be expressed as

Loss of satisfaction = Gain of satisfaction

i.e., reduction in Y x MU_y = increment in X × MU_X

The relationship between MRS and MU can also he established by considering the indifference curve U = f (X, Y) for two commodities ‘X’ and ‘Y’.

Since along a particular indifference curve, the level of satisfaction (U) is constant, dU = 0, Therefore,

Marginal rate of substitution of ‘X’ for ‘Y’ (MRS_{X Y}) is the rate at which consumer gives up successive units of commodity ‘Y’ in exchange for each extra unit of commodity ‘X’. As the quantity of ‘X’ increases, its marginal significance (MU_X) to the consumer decreases.

Further, with the decrease in quantity of Y, the marginal significance of Y (MU_y) increases. Therefore, consumer is willing to sacrifice lesser and lesser units of commodity ‘Y’ for each additional unit gain of commodity ‘X’. Thus, MRS_{X Y} falls, when quantity of ‘X’ is increased.

The conclusions arrived at with the help of marginal rate of substitution (MRS) are similar to the conclusions arrived at with the help of marginal utility (MU). Both tend to diminish and are not independent of each other. Substitution of one commodity for the other is based on the marginal utility or marginal significance.

Even Hicks himself recognised that the principles of diminishing marginal utility and that of diminishing marginal rate of substitution have consid­erable resemblance. However, the latter principle is not just an alternative way of presenting the former. Following are the points of differences between the two principles:

First, the conventional diminishing marginal utility principle is based on the assumption of measurability of utility on the cardinal scale. On the other hand, the proponents of the diminishing marginal rate of substitution principle believe that utility cannot be measured.

Secondly, exponents of the diminishing marginal utility principle assume that the utility of money is constant and a consumer is willing to pay lower price for additional units of the commodity due to lower utility of the additional units and not because of rise in the utility of money.

This unrealistic assumption is not there under the principle of diminishing marginal rate of substitution. Here, the marginal rate of substitution diminishes due to decrease in the utility of commodity in question and increase in the utility of other commodity (or money).

If commod­ity ‘Y’ is treated as money income, then marginal rate of substitution of ‘Y’ for ‘X’ will decline indicating that the consumer would offer a smaller and smaller amount of money for every additional unit of X.

This implies that as the consumer spends more and more money on ‘X’, the marginal utility of money rises. Thus, the marginal utility of a commodity is related to the quantity of that commodity only, while the marginal rate of substitution is affected by the quantities of other commodities also.

Thirdly, marginal utility depends on the particular form of the utility function. Marginal rate of substitution is, however, independent of the nature of utility functions chosen, provided they are all positive monotonic transformation of each other. It is determined by the consumer’s preference ordering that uniquely determines that indifference curve. The following example will make this point more clear.

Example:

Find out marginal rate of substitution for the following utility functions

In the above two utility functions, marginal utilities are different, while marginal rate of substitution is the same. Thus, marginal rate of substitution cannot be taken as mere translation of marginal utility. The concept of marginal rate of substitution is more fundamental than marginal utilities.