 Marginal Rate of Substitution (MRS)

## Concept and Meaning of MRS

The concept of the marginal rate of substitution (MRS) is an important tool of indifference curve analysis of consumer’s demand. In the analysis of consumer behavior, the marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade-off or exchange one good for another. In the case of two goods, MRS answers the question, how much of one good would a consumer be willing to give up getting one more unit of the other good. Thus, MRS is the amount of a good that a consumer is ready to give up to obtain one additional unit of another good to gain the same level of total utility. It means the MRS of good X for good Y is the maximum amount of good Y that a person is willing to sacrifice to get one extra additional unit of good X.

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

Allen and Hicks argue that it is needless to measure the utility of a commodity. They find it is essential to study the behavior of the consumer to know how he favors one good to another and upholds the same level of fulfillment.

For instance, X and Y two goods are there and they are not the perfect substitute for each other. Suppose that the buyer is arranged to trade-off or to exchange good X for Y. Here the number of units of good Y he is ready to leave for one unit of good X to keep the same satisfaction is related to the marginal rate of substitution. Thus, the good X and Y exchanging rate is known as the marginal rate of substitution (MRS).

The marginal rate of substitution (MRS) measures the trade-off between two goods along an indifference curve. The MRS measures the value that the consumer places on one extra unit of a good, where the opportunity cost is measured by the amount of another good given up. Mathematically MRS is the negative of the slope of the indifference curve. Therefore, the MRS at any point on an indifference curve is equal to the size of the slope of that indifference curve, which is the slope of the tangent to the IC at a specific point.

Symbolically,

MRSXY= –ΔY/ΔX=-MUX/MUY

Where ΔX change in the unit of good X; ΔY is the Change in the units of good Y; MRSXY is the marginal rate of substitution between goods X and Y.

So, MRS is the slope of the indifference curve, i.e. slope of the indifference curve at any point is MRSXY at that point.

Similarly,

When consumer substitute good Y for good X.  Then the operation symbolically is shown as;

MRSYX= – ΔX/ΔY=- MUY/MUX

## The Doctrine of Diminishing Marginal Rate of Substitution (MRS)

As we know the MRS refers to the amount of one good that an individual is willing to give up for an additional unit of another good while maintaining the same level of satisfaction. According to this principle, the MRS diminishes. It means, suppose that the consumer substitute X for Y such that his total utility remains the same. When he gives up some units of Y, his stock of Y decreases by ΔY. For another unit of X, he has to sacrifice further certain units of commodity Y, denotes by ΔY1. Here according to the principle of diminishing marginal rate of substitution, the ΔY1 is less than ΔY.

The given schedule helps to explain the concept of diminishing marginal rate of substitution (MRS).

In the given above, all combinations of good X and good Y yield the identical pleasure or utility to the buyers. If the consumer chooses the 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 leave 4 units of good Y to keep the same level of satisfaction. Here, he MRS is, therefore, 4:1.

Similarly, the buyer is eager to leave 3 units of good Y only to achieve one extra unit of good X in the third combination. The MRS is 3:1. Similarly, when the consumer moves from 4th to 5th combination, the MRS of good X for good Y falls to one (1:1). This demonstrates the diminishing marginal rate of substitution.

The following diagram also helps to explain the concept of diminishing marginal rate of substitution (MRS).

In the above figure we can see that, when the buyer travels down from combination A to B, he is ready to leave 4 units of good Y as stands by ∆Y to receive one more unit of good X stands by ∆X. Again, when he travels from blend C to D and from D to E, the distance of ∆Y converts minor and minor.  But at the same time, the span of ∆X is lasting the same. This displays that as units of good X increases, the units of good Y declines. Hence, the buyer is keen to leave fewer units of Y to acquire another unit of good X. It means that the MRS of good X for good Y drops as the consumer has more of good X and less of good Y. due to such fact, the IC slopes downhill from left to the right.

## Causes of Diminishing Marginal Rate of Substitution (MRS)

From our discussion, we got to know that the marginal rate of substitution (MRS) is diminishing. The following are the major reasons behind the diminishing MRS.

The want for a specific good is satiable

If we believe that for a particular commodity is satiable or satisfied, then the more a consumer consumes that commodity, his desire for that commodity diminishes. It is due to the universal fact that the more we have of a commodity, the less we want to have more of it. Thus, MRS diminishes.

Goods are imperfect substitutes of each other

Goods that we consume regularly are expected to be an imperfect substitute to each other and so that result MRS diminishes. If goods under a consumption bundle are a perfect substitute for each other, MRS would be constant. But in the real world, perfect substitutes are rare, so MRS diminishes.

The increase in the quantity of one good doesn’t increase the want satisfying power of the other commodity

The increase in the consumption of one good can’t support increasing the want satisfying power of other goods. So that MRS diminishes.

## Mathematical Derivation of Marginal Rate of Substitution (MRS)

Let us assume that the utility function of the consumer is given as;

U=f (X, Y) ———–(i)

Where U is the utility of the consumer; X and Y are two goods, and f is the utility function. By taking total differentiation of utility function (i) we will get the following equation;

dU= ∂{f (X, Y)}/ ∂X*dX +∂{f (X, Y)}/ ∂Y*dY——–(ii)

According to the definition of marginal utility,

∂{f (X, Y)}/ ∂X= ∂U/ ∂X=MUX  and ∂{f (X, Y)}/ ∂Y=∂U/ ∂Y=MUY

Now, using such concepts of marginal utility on equation (ii), we get

dU= MUX. dX + MUY. dY————-(iii)

But according to the definition of the indifference curve, it represents the same level of total utility so dU=0 and thus equation (iii) will become;

0= MUX. dX + MUY. dY

Or, MUX. dX =- MUY. dY

Or, dY/dX=-MUX/MUY=MRSX, Y (MRS X for Y)

Therefore, MRS is the ratio of marginal utilities of X and Y. The relationship between marginal utility and the marginal rate of substitution is summarized with the following equation;

MRS X, Y=-MUX/MUY

For instance, if the value of MRS X, Y=7, it means the consumer will give up 7 units of good Y to obtain 1 extra unit of good X.  As a consumer moves down a convex indifference curve, the marginal rate of substitution decreases as the size of the slope of the indifference curve decreases.

In our above figure, by following the definition of indifference curve (IC), along the indifference curve,

Level of satisfaction enlarged= Level of satisfaction reduced

Or, (+) MUX * ΔY= (-) MUY* ΔY

Therefore, the marginal rate of substitution (MRS) is the absolute value of the slope of the indifference curve. It measures the rate at which a consumer is willing to give up one good for one more unit of another good, holding utility constant.