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The concept of marginal rate of substitution (MRS) and diminishing marginal rate of substitution was introduced by Dr. J.R. Hicks and Prof. R.G.D. Allen to take the place of the law of diminishing marginal utility.

## Definition and Explanation of MRS:

Allen and Hicks are of the opinion that it is unnecessary to measure the utility of a commodity. The necessity is to study the behavior of the consumer as to how he prefers one commodity to another and maintains the same level of satisfaction. Marginal rate of substitution (MRS) can be defined as:

“The ratio of exchange between small units of two commodities, which are equally valued or preferred by a consumer”.

## Example:

For example, there are two goods X and Y which are not perfect substitute of each other. The consumer is prepared to exchange goods X for Y. How many units of Y should be given for one unit of X to the consumer so that his level of satisfaction remains the same?

The rate or ratio at which goods X and Y are to be exchanged is known as the marginal rate of substitution (MRS).

In the words of Hicks:

The marginal rate of substitution (MRS) of X for Y measures the number of units of Y that must be scarified for unit of X gained so as to maintain a constant level of satisfaction”.

## Formula (Equation) for MRS:

The marginal rate of substitution (MRS) can be represented by the following equation or formula:

It may here be noted that the marginal rate of substitution (MRS) is the personal exchange rate of the consumer in contrast to the market exchange rate.

## Calculation of MRS with the help of a Table and Diagram:

The calculation of MRS can be easily explained with the help of a table and a diagram, given below:

In the table given above, all the five combinations of good X and good Y give the same satisfaction to the consumer. If he chooses first combination, he gets 1 unit of good X and 13 units of good Y.

In the second combination, he gets one more unit of good X and is prepared to give 4 units of good Y for it to maintain the same level of satisfaction. The MRS is therefore, 4:1.

In the third combination, the consumer is willing to sacrifice only 3 units of good Y for getting another unit of good X. The MRS is 3:1.

Likewise, when the consumer moves from 4th to 5th combination, the MRS of good X for good Y falls to one (1:1). This illustrates that there will be a diminishing marginal rate of substitution (explain in the last paragraph of this article).

The concept of marginal rate of substitution (MRS) can also be illustrated with the help of the diagram.

In above fig. 3.3, as the consumer moves down from combination 1 to combination 2, the consumer is willing to give up 4 units of good Y (∆Y) to get an additional unit of good X (∆X).

When the consumer slides down from combinations 2, 3 and 4, the length of ∆Y becomes smaller and smaller, while the length of ∆X is remain the same. This shows that as the stock of the consumer for good X increases, his stock of good Y decreases.

He, therefore, is willing to give less units of Y to obtain an additional unit of good X. In other words, the MRS of good X for good Y falls as the consumer has more of good X and less of good Y. The indifference curve IC slopes downward from left to the right. This means a negative and diminishing rate of substitution of one commodity for the other.

## Importance and Uses:

Marginal rate of substitution (MRS) have much importance and following uses:

(i) Measures utility ordinally: The concept of MRS is superior to that of utility concept because it is more realistic and scientific than the theory of utility. It does not measure the utility of a commodity in isolation without reference to other commodities but takes into consideration the combination of related goods to which a consumer is interested to purchase.

(ii) A relative concept: The concept of marginal rate of substitution has the advantage that it is relative and not absolute like the utility concept given by Marshall. It is free from any assumptions concerning the possibility of a quantitative measurement of utility.

## Diminishing Marginal Rate of Substitution – Concept and Definition:

In the above schedule, we have seen that as the consumer moves from combination first to fifth, the rate of substitution of good X for good Y goes down. In other words, as the consumer has more and more units of good X, he is prepared to forego less and less of good Y.

For example, in the 2nd combination, the consumer is willing to give 4 units of good Y in exchange for one unit of good X, in the fifth combination only one unit of Y is offered for obtaining one unit of X.

This behavior showing falling MRS of good X for good Y and yet to remain at the same level of satisfaction is known as diminishing marginal rate of substitution.